Serre fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
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**Serre fibration** $\Leftarrow$ [[Hurewicz fibration]] $\Rightarrow$ [[Dold fibration]] $\Leftarrow$ [[shrinkable map]]

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #TopologicalGeneratingAcyclicCofibrations}
###### Definition

Write

$$
  J_{Top}
    \coloneqq
  \left\{
      D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I
  \right\}_{n \in \mathbb{N}}
  \;\subset Mor(Top)
$$

for the [[set]] of inclusions of the topological [[n-disks]], into their [[cylinder objects]] (the [[product topological space]] with the [[topological interval]]), along (for definiteness) the left endpoint inclusion.

=--

+-- {: .num_defn #SerreFibration}
###### Definition

A **Serre fibration** is a $J_{Top}$-[[injective morphism]], def. \ref{TopologicalGeneratingAcyclicCofibrations}, hence a [[continuous function]] $f \colon X \longrightarrow Y$ that  has the [[right lifting property]] with respect to all inclusions of the form $(id,0) \colon D^n \hookrightarrow D^n \times I$ that include the standard topological [[n-disk]] into its standard [[cylinder object]].

=--

I.e. $f$ is a Serre fibration if for every [[commuting square]] of [[continuous functions]] of the form

$$
  \array{
    D^n  &\longrightarrow& X
    \\
    {}^{\mathllap{(id,0)}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n \times I &\longrightarrow& Y
  }
$$

then there exists a continuous function $h \colon D^n \times I \to X$ such as to make a [[commuting diagram]] of the form

$$
  \array{
    D^n  &\longrightarrow& X
    \\
    {}^{\mathllap{(id,0)}}\downarrow
    &{}^h\nearrow&
    \downarrow^{\mathrlap{f}}
    \\
    D^n \times I &\longrightarrow& Y
  }
$$

+-- {: .num_remark}
###### Remark

The class of Serre fibrations serves as the class of abstract [[fibrations]] in the [[classical model structure on topological spaces]] (whence "Serre-Quillen model structure").

=--

## Properties
 {#Properties}

### Closure properties

+-- {: .num_prop #SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes}
###### Proposition

A Serre fibration has the [[right lifting property]] against all [[retracts]] of $J_{Top}$-[[relative cell complexes]] (def. \ref{TopologicalGeneratingAcyclicCofibrations}).

=--

+-- {: .proof}
###### Proof

By general closure properties of [[projective and injective morphisms]], see there [this proposition](injective+or+projective+morphism#ClosurePropertiesOfInjectiveAndProjectiveMorphisms) for details.

=--

### Relation to Hurewicz fibrations
 {#RelationToHurewiczFibrations}

+-- {: .num_remark}
###### Remark

The condition in def. \ref{SerreFibration} is part of the condition on a [[Hurewicz fibration]], hence every [[Hurewicz fibration]] is in particular a Serre fibration.

=--

The converse is false:

+-- {: .num_example #SerreFibrationsWhichAreNotHurewiczFibrations}
###### Example
**(Serre fibrations which are not Hurewicz fibrations)**

An example of a generalized [[covering space]] which is a Serre fibration but not a Hurewicz fibration is given by Jeremy Brazas [here](https://mathoverflow.net/a/241597/381).

=--

But under some regularity condition it does becomes true:

+-- {: .num_prop #SerreFibrationsBetweenCWComplexesAreHurewiczFibrations}
###### Proposition
**(Serre fibrations of [[CW-complexes]] are [[Hurewicz fibrations]])**

In the [[convenient category of topological spaces|convenient category]] of [[compactly generated topological spaces|compactly generated]] [[weakly Hausdorff topological spaces]]) a Serre fibration in which the total space and base space are both [[CW complexes]] is a [[Hurewicz fibration]].

(No relationship between the covering map and the CW structures is required.)

=--

This is due to ([Steinberger-West 84](#SteinbergerWest84)) with the corrected proof due to ([Cauty 92](#Cauty92)) (pointers via [[Peter May]] [here](https://mathoverflow.net/a/241611/381)).  Theorem-page at [[a Serre fibration between CW-complexes is a Hurewicz fibration]].

### Long exact sequences of homotopy groups

Since Serre fibrations are the abstract [[fibrations]] in the Serre-[[classical model structure on topological spaces]], the following statement follows from general [[model category]] theory. But it may also be seen by direct inspection, as follows.

+-- {: .num_lemma #CylinderOverCWComplexIsJTopRelativeCellComplex}
###### Lemma

For $X$ a finite [[CW-complex]], then its inclusion $X \overset{(id, \delta_0)}{\longrightarrow} X \times I$ into its standard [[cylinder object|cylinder]] is a $J_{Top}$-[[relative cell complex]] (def. \ref{TopologicalGeneratingAcyclicCofibrations}).

=--

+-- {: .proof}
###### Proof

First erect a cylinder over all 0-cells

$$
  \array{
     \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X
     \\
     \downarrow &(po)& \downarrow
     \\
     \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1
  }
  \,.
$$

Assume then that the cylinder over all $n$-cells of $X$ has been erected using attachment from $J_{Top}$. Then the union of any $(n+1)$-cell $\sigma$ of $X$ with the cylinder over its boundary is homeomorphic to $D^{n+1}$ and is like the cylinder over the cell "with end and interior removed". Hence via attaching along $D^{n+1} \to D^{n+1}\times I$ the cylinder over $\sigma$ is erected.

=--

+-- {: .num_prop #SerreFibrationGivesExactSequenceOfHomotopyGroups}
###### Proposition

Let $f\colon X \longrightarrow Y$ be a [[Serre fibration]], def. \ref{SerreFibration}, let $y \colon \ast \to Y$ be any point and write

$$
  F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y
$$

for the [[fiber]] inclusion over that point. Then for every choice $x \colon \ast \to X$ of lift of the point $y$ through $f$, the induced sequence of [[homotopy groups]]

$$

  \pi_{\bullet}(F_y, x)
    \overset{\iota_\ast}{\longrightarrow}
  \pi_\bullet(X, x)
    \overset{f_\ast}{\longrightarrow}
  \pi_\bullet(Y)
$$

is [[exact sequence|exact]], in that the [[kernel]] of $f_\ast$ is canonically identified with the [[image]] of $\iota_\ast$:

$$
  ker(f_\ast) \simeq im(\iota_\ast)
  \,.
$$

=--

+-- {: .proof}
###### Proof

It is clear that the image of $\iota_\ast$ is in the kernel of $f_\ast$ (every sphere in $F_y\hookrightarrow X$ becomes constant on $y$, hence contractible, when sent forward to $Y$).

For the converse, let $[\alpha]\in \pi_{\bullet}(X,x)$ be represented by some $\alpha \colon S^{n-1} \to X$. Assume that $[\alpha]$ is in the kernel of $f_\ast$. This means equivalently that $\alpha$ fits into a [[commuting diagram]] of the form

$$
  \array{
    S^{n-1} &\overset{\alpha}{\longrightarrow}& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n &\overset{\kappa}{\longrightarrow}& Y
  }
  \,,
$$

where $\kappa$ is the contracting homotopy witnessing that $f_\ast[\alpha] = 0$.

Now since $x$ is a lift of $y$, there exists a [[left homotopy]]

$$
  \eta  \;\colon\; \kappa \Rightarrow const_y
$$

as follows:

$$
  \array{
    && S^{n-1} &\overset{\alpha}{\longrightarrow}& X
    \\
    && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    && D^n &\overset{\kappa}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{(id,1)}} && \downarrow^{\mathrlap{id}}
    \\
    D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y
    \\
    \downarrow && &&  \downarrow
    \\
    \ast && \overset{y}{\longrightarrow} && Y
  }
$$

(for instance: regard $D^n$ as embedded in $\mathbb{R}^n$ such that $0 \in \mathbb{R}^n$ is identified with the basepoint on the boundary of $D^n$ and set $\eta(\vec v,t) \coloneqq \kappa(t \vec v)$).

The [[pasting]] of the top two squares that have appeared this way is equivalent to the following commuting square

$$
  \array{
    S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X
    \\
    {}^{\mathllap{(id,1)}}\downarrow
    &&
    && \downarrow^{\mathrlap{f}}
    \\
    S^{n-1} \times I
      &\overset{(\iota_n, id)}{\longrightarrow}&
    D^n \times I
      &\overset{\eta}{\longrightarrow}&
    Y
  }
  \,.
$$

Because $f$ is a [[Serre fibration]] and by lemma \ref{CylinderOverCWComplexIsJTopRelativeCellComplex} and prop. \ref{SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes}, this has a [[lift]]

$$
  \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X
  \,.
$$

Notice that $\tilde \eta$ is a basepoint preserving [[left homotopy]] from $\alpha = \tilde \eta|_1$ to some $\alpha' \coloneqq \tilde \eta|_0$. Being homotopic, they represent the same element of $\pi_{n-1}(X,x)$:

$$
  [\alpha'] = [\alpha]
  \,.
$$

But the new representative $\alpha'$ has the special property that its image in $Y$ is not just trivializable, but trivialized:
combining $\tilde \eta$ with the previous diagram shows that it sits in the following commuting diagram

$$
  \array{
    \alpha' \colon & S^{n-1} &\overset{(id,0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X
    \\
    & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}}
    \\
    & D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y
    \\
    & \downarrow && &&  \downarrow
    \\
    & \ast && \overset{y}{\longrightarrow} && Y
  }
  \,.
$$

The commutativity of the outer square says that $f_\ast \alpha'$ is constant, hence that $\alpha'$ is entirely contained in the fiber $F_y$. Said more abstractly, the [[universal property]] of [[fibers]] gives that $\alpha'$ factors through $F_y\overset{\iota}{\hookrightarrow} X$, hence that $[\alpha'] = [\alpha]$ is in the image of $\iota_\ast$.

=--

(...)

[[long exact sequence of homotopy groups]]

(...)

### Local recognition
 {#LocalRecognition}

+-- {: .num_lemma #MapIsSerreFibrationIfLocallySo}
###### Lemma
**(map is Serre fibration if it is locally so)**

If $p\colon X \to B$ is a [[continuous function]] and $\mathcal{U} = \big\{ U_i \overset{}{\hookrightarrow} B \big\}_{i \in I}$ is an [[open cover]] of its [[codomain]], such that the restriction of $p$ to each $U_i$ (i.e. its [[pullback]] to the cover) is a [[Serre fibration]], then $p$ itself is a Serre fibration:

$$
  \array{
    X_{\vert \mathcal{U}}
    &\longrightarrow&
    X
    \\
    {}^{{}_{\mathllap{
      {Serre}
      \atop
      {fibration}
    }}}
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    {}^{{}_{\mathrlap{
      p
    }}}
    \\
    \underset{i \in I}{\sqcup}
    U_i
    &\underset{  }{\longrightarrow}&
    B
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    X
    \\
    {}^{{}_{\mathllap{
      {Serre}
      \atop
      {fibration}
    }}}
    \big\downarrow
    {}^{{}_{\mathrlap{
      p
    }}}
    \\
    B
  }
$$

=--

See [tom Dieck 08, p. 130, Thm. 6.3.3](#tomDieck2008).

\begin{remark}
  An analogous local recognition holds for [[Hurewicz fibrations]] but with [[numerable open covers]]. See [there](Hurewicz+fibration#LocalRecognition).
\end{remark}

## Examples
 {#Examples}

### General

\begin{example}
  The [[topological realization]] of a [[Kan fibration]] of [[simplicial sets]] is a Serre fibration.
\end{example}
(due to [Quillen 1968](Quillen68), see also [Goerss & Jardine 1999, Thm. 10.10](#GoerssJardine99))

### Empty bundles

\begin{example}\label{EmptyBundlesAreSerreFibrations}
**([[empty bundles]] are [[Serre fibrations]])** \linebreak
 All [[empty bundles]] $\varnothing \longrightarrow B$ are Serre fibrations,  because none of the [[commuting squares]] that one would have to [[lifting property|lift in]] actually exist:

$$
  \array{
    D^{n} &\overset{ \not \exists }{\longrightarrow}& \varnothing
    \\
    \big\downarrow && \big\downarrow
    \\
    D^n \times I &\longrightarrow& B
    \mathrlap{\,,}
  }
$$

since the [[empty topological space]] is a [[strict initial object]]: There is *no* morphism to it from any [[inhabited space]].

\end{example}

### Fiber bundles
 {#FiberBundles}

\begin{example}\label{FiberBundlesAreSerreFibrations}
**([[fiber bundles]] are Serre fibrations)**\linebreak
Every topological [[fiber bundle]] (i.e. [[locally trivial]], meaning that on some [[open cover]] it becomes a [[product space|product]] [[projection]]) is a Serre fibration.
\end{example}

This follows immediately from Lemma \ref{MapIsSerreFibrationIfLocallySo}.

Notice that the [[empty bundles]] of Example \ref{EmptyBundlesAreSerreFibrations} are examples of [[locally trivial]] [[fiber bundles]] (with [[typical fiber]] the [[empty topological space]]).

### Covering spaces

+-- {: .num_example #CoveringSpaceIsSerreFibration}
###### Example
**([[covering space]] projection is Serre fibration)**

Every [[covering space]] projection is a Serre fibration, in fact a [[Hurewicz fibration]] (by [this prop.](covering+space#HomotopyLiftingPropertyOfCoveringSpaces)).

=--

## Related concepts

* [[transport]]

* [[Serre spectral sequence]]

* [[classical model structure on topological spaces]]

## References

* {#Quillen68} [[Daniel Quillen]], *The geometric realization of a Kan fibration is a Serre fibration*, Proc. AMS 19 (1968) 1499-1500 [[doi:1968-019-06/S0002-9939-1968-0238322-1](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1), [pdf](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf)]

* {#SteinbergerWest84} M. Steinberger and J. West, _Covering homotopy properties of maps between CW complexes or ANRs_, Proc. Amer. Math. Soc. 92 (1984), 573-577.

* {#Cauty92} R. Cauty, _Sur les ouverts des CW-complexes et les fibrés de Serre_,  Colloquy Math. 63 (1992), 1--7

* [[Stephen Mitchell]], *Notes on Serre fibrations*, 2001 ([pdf](https://sites.math.washington.edu/~mitchell/Atopc/serre.pdf), [[MitchellSerreFibrations.pdf:file]])

* {#tomDieck2008} [[Tammo tom Dieck]],  _Algebraic topology_. European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86))

* {#GoerssJardine99} [[Paul Goerss]], [[J. F. Jardine]]: *[[Simplicial homotopy theory]]*, Progress in Mathematics, Birkhäuser (1999), Modern Birkhäuser  Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

For more see at _[[classical model structure on topological spaces]]_.

[[!redirects Serre fibrations]]
[[!redirects Serre fibration of topological spaces]]
[[!redirects Serre fibrations of topological spaces]]


Serre finiteness theorem

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### Context
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

There are several theorems by [[Serre]] which deserve to be called "finiteness theorems".

## In homotopy theory
 {#InHomotopyTheory}

On the [[homotopy groups of spheres]]:

+-- {: .num_theorem }
###### Theorem
**(Serre finiteness theorem)**

The [[homotopy groups of spheres]] $\pi_{n+k}(S^k)$, for $k \geq 1$, are [[finite groups]] (in fact [[finite abelian groups]]), hence pure [[torsion subgroup|torsion]], *except*

1. for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$ is the [[integers]];

1. $k = 2m$ and $n = 2m -1$ in which case

   $$
     \pi_{4m - 1}(S^{2m}) \simeq \mathbb{Z} \oplus F_m
   $$

   for $F_m$ some [[finite group]].

=--

([Serre 53](#Serre53), see [Ravenel 86, Chapter I, Lemma 1.1.8](#Ravenel86))

+-- {: .num_remark }
###### Remark

The non-torsion element in $\pi_{4k-1}(S^{2k})$ may be seen explicitly in the [[minimal Sullivan model]] for the [[rational n-sphere|rational 2k-sphere]], see there for more.

=--

## Related concepts

* [[Nishida nilpotence theorem]]

## References

The original proof is due to

* {#Serre53} [[Jean-Pierre Serre]], _Groupes d'homotopie et classes de groupes abelien_, Ann. of Math. 58 (1953), 258–294 ([jstor:1969789](https://www.jstor.org/stable/1969789))

The statement is reviewed in

* [[Douglas Ravenel]], Chapter I, Theorem 1.1.8 in: _[[Complex cobordism and stable homotopy groups of spheres]]_. Academic Press Orland (1986) reprinted as: AMS Chelsea Publishing, Volume 347 (2004) ([ISBN:978-0-8218-2967-7](https://bookstore.ams.org/chel-347-h), [webpage](http://www.math.rochester.edu/people/faculty/doug/mu.html), [pdf](https://web.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf))

An entirely different proof, using only elementary concepts, is given in

* S. S. Podkorytov, _An Alternative Proof of a Weak Form of Serre's Theorem_, Journal of Mathematical Sciences July 2002, Volume 110, Issue 4, pp 2875–2881 ([doi:10.1023/A:1015370800473](https://doi.org/10.1023/A:1015370800473))

[[!redirects Serre finiteness theorems]]


Serre functor

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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# Serre functor
* table of contents
{: toc}

## Idea

A *Serre functor* on a [[triangulated category]] $\mathcal{A}$ is an [[exact functor]] such that for any [[pair]] of [[objects]] there is a [[natural isomorphism]] $\Hom(A,B) \simeq \Hom\big(B, S(A)\big)^*$ from their [[hom-object]] to the [[linear dual]] of their reverse hom-object, "twisted" by the Serre functor.

A Serre functor does not always exist, but when it does then it is unique up to graded natural isomorphism.

The notion of Serre functors was introduced by [Bondal & Kapranov 1989](#BondalKapranov89) to study [[admissible subcategories]] of [[triangulated categories]].

Serre functors have become a powerful tool for working with the [[derived category]] of [[coherent sheaves]] on a [[variety]].

## Definition

In the original paper, the following definition was given.

+-- {: .num_defn}
###### Definition
Let $\mathcal{A}$ be a $k$-linear triangulated category with finite-dimensional [[Hom]]'s and $k$ algebraically closed.  A **Serre functor** $S : \mathcal{A} \to \mathcal{A}$ is an additive equivalence that commutes with the translation functor, with [[natural isomorphisms|natural]] (in both variables) [[isomorphisms]] $\phi_{A,B} \colon \Hom_\mathcal{A}(A,B) \stackrel{\sim}{\to} \Hom_{\mathcal{A}}(B,S(A))^*$ for any objects $A$ and $B$, such that the composite
  \[ (\phi^{-1}_{S(A),S(B)})^* \circ \phi_{A,B} : \Hom(A,B) \to \Hom(B,S(A))^* \to \Hom(S(A), S(B)) \]
coincides with the isomorphism induced by $S$.
=--

In fact, the last commutativity condition can be deduced from just the [[natural transformation|naturality]] of $\phi_{A,B}$, and commutativity with the translation functor also follows from a proposition below.  Hence, the following definition is seen in later papers.

+-- {: .num_defn}
###### Definition
Let $\mathcal{A}$ be a $k$-linear category with finite-dimensional [[Hom]]'s and $k$ an arbitrary field.  A **Serre functor** $S : \mathcal{A} \to \mathcal{A}$ is an additive equivalence with bi-functorial isomorphisms $\phi_{A,B} : \Hom_\mathcal{A}(A,B) \stackrel{\sim}{\to} \Hom_{\mathcal{A}}(B,S(A))^*$ for any objects $A$ and $B$.
=--

Of course, formally the definition could be used in categories enriched over a symmetric monoidal category with a sufficiently nice involution.

## Examples

+-- {: .num_example}
###### Example
In the [[derived category]] of [[finite-dimensional vector spaces]] over $k$, the identity functor is a Serre functor.
=--

+-- {: .num_example}
###### Example
In the derived category of [[coherent sheaves]] on a smooth projective [[variety]] $X$, the functor $(\cdot \otimes \omega_X)[n]$ is a Serre functor, in view of [[Serre-Grothendieck duality]], where $\omega_X$ is the [[canonical sheaf]] and $n$ is the dimension of $X$.
=--

## Properties

+-- {: .num_prop}
###### Proposition
Any autoequivalence $F : \mathcal{A} \to \mathcal{A}$ commutes with a Serre functor: there is a natural graded isomorphism of functors $F \circ S \stackrel{\sim}{\to} S \circ F$.
=--

+-- {: .num_prop}
###### Proposition
Any Serre functor in a [[graded category]] is [[graded]].
=--

+-- {: .num_prop}
###### Proposition
Any Serre functor in a [[triangulated category]] is [[exact]] (i.e. [[distinguished triangles]] are mapped to distinguished triangles).
=--

+-- {: .num_prop}
###### Proposition
Any two Serre functors are connected by a canonical graded functorial isomorphism that commutes with the isomorphisms $\phi_{A,B}$ in the definition of the Serre functor.
=--

## References

The original paper and English translation:

* {#BondalKapranov89}  А. И. Бондал, М. М. Капранов, _Представимые функторы, функторы Серра и перестройки_, Изв. АН СССР. Сер. матем., 53:6 (1989), 1183–1205 [pdf](http://www.mathnet.ru/links/99950da8b286808c64286a2820235651/im1153.pdf)

* {#BondalKapranov90} [[Alexei I. Bondal]], [[Mikhail M. Kapranov]], _Representable functors, Serre functors, and mutations_, Mathematics of the USSR-Izvestiya **35** 3 (1990) 519-541 [[doi:10.1070/IM1990v035n03ABEH000716](https://doi.org/10.1070/IM1990v035n03ABEH000716) [pdf](https://www.mathnet.ru/links/6ad8b932b7c8801d1b821034af7d08b2/im1153_eng.pdf)]

The following paper gives the corrected definition and also demonstrates the utility of the Serre functor as a tool for working with the derived category of coherent sheaves on a variety (c.f. [[Bondal-Orlov reconstruction theorem]]):

* [[Aleksei Bondal]], [[Dmitri Orlov]], _[Reconstruction of a variety from the derived category and groups of autoequivalences](http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf)_, Compositio Mathematica 125 (03), 327-344.

See also:

* Oleksandr Manzyk, _A-infinity-bimodules and Serre A-infinity-functors_, dissertation [pdf](https://kluedo.ub.uni-kl.de/files/1910/dissertation.pdf), [djvu](https://kluedo.ub.uni-kl.de/files/1910/dissertation.djvu);  _Serre $A_\infty$ functors_, talk at Categories in geometry and math. physics, Split 2007, slides, [pdf](http://www.irb.hr/korisnici/zskoda/manzyukslides.pdf), work with [[Volodymyr Lyubashenko]]
* Volodymyr Mazorchuk, Vanessa Miemietz, _Serre functors for Lie algebras and superalgebras_
* [[Sefi Ladkani]], _Categorification of a linear algebra identity and factorization of Serre functors_,  Mathematische Zeitschrift __285__ (2017) 879--896 [doi](https://doi.org/10.1007/s00209-016-1731-9)

[[!redirects Serre functors]]


Serre intersection formula

#Contents#
* table of contents
{:toc}

## Idea

Serre's intersection formula is a refinement of the [[intersection theory]] statement of [[Bézout's theorem]]. By replacing the plain [[tensor product]] of [[structure sheaves]] of intersecting subvarieties with the [[derived functor|derived]] tensor product -- the [[Tor]] -- it holds also when the subvarieties do not intersect transversally.

This may be understood as a first step towards formulating [[intersection theory]] properly in [[derived algebraic geometry]].

## Statement

Given a [[regular scheme]] $X$ and subschemes $Y,Z$ with defining ideal sheaves $\mathcal{I},\mathcal{J}$, respectively, the intersection multiplicity $m(x,Y,Z)$ at a generic point $x$ of (an irreducible component of) the intersection $Y\cap Z$ is given by the Serre intersection (or multiplicity) formula in terms of torsion groups:

$$
m(x,Y,Z) = \sum_{i\geq 0} (-1)^i length_{\mathcal{O}_{X,x}} Tor_i^{\mathcal{O}_{X,x}}
(\mathcal{O}_{X,x}/\mathcal{I}_x,\mathcal{O}_{X,x}/\mathcal{J}_x)
$$

The formula can be interpreted by the [[Hochschild homology]], or within the [[derived category of coherent sheaves]] on $X$. It has been one of the motivating results for the development of [[derived algebraic geometry]].

## References

* MathOverflow: [serre-intersection-formula-and-derived-algebraic-geometry](http://mathoverflow.net/questions/12236/serre-intersection-formula-and-derived-algebraic-geometry)

[[!redirects Serre multiplicity formula]]

[[!redirects Serre's intersection formula]]


Serre long exact sequence

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Serre long exact sequence_ of a [[Serre fibration]] is a [[long exact sequences]] of [[ordinary cohomology]]/[[ordinary homology]] groups associated with a [[Serre fibration]] with sufficiently highly connected base and fibers.
It arises as a special case of the information contained in the corresponding [[Serre spectral sequence]].

## Statement

+-- {: .num_prop #SerreLongExactSequenceForHighlyConnectedBaseAndFiber}
###### Proposition

Let

$$
  \array{
    F &\overset{i}{\longrightarrow}& X
    \\
    && \downarrow^{\mathrlap{p}}
    \\
    && B
  }
$$

be a [[Serre fibration]] such that

1. the base space $B$ is $(n_1-1)$-[[n-connected topological space|connected]] for $n_1 \geq 2$;

1. the [[fiber]] $F$ is $(n_2-1)$-connected, for $n_1 \geq 1$;

then for every [[abelian group]] $A$ there is a [[long exact sequence]] of [[ordinary homology]] groups of the form

$$
  H_{n_1 + n_2 - 1}(F,A)
    \overset{i_\ast}{\longrightarrow}
  H_{n_1 + n_2 - 1}(X,A)
    \overset{p_\ast}{\longrightarrow}
  H_{n_1 + n_2 - 1}(B,A)
    \overset{\tau}{\longrightarrow}
  H_{n_1 + n_2 - 2}(F,A)
    \overset{i_\ast}{\longrightarrow}
  \cdots
    \overset{i_\ast}{\longrightarrow}
  H_1(X,A)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the homology [[Serre spectral sequence]] of the given fibration

$$
  E_2 = H_q(B, H_p(F,A)) \;\Rightarrow\; H_\bullet(X,A)
  \,.
$$

By the connectedness assumptions and the [[Hurewicz theorem]], then

$$
  H_{0 \lt \bullet \lt n_1}(B,-) = 0
$$

and

$$
  H_{0 \lt \bullet \lt n_2}(F,-) = 0
$$

and hence the only possibly non-vanishing groups on the second page of the spectral sequence (hence on every higher page) in total degree $k \leq n_1 + n_2 - 1$ are $E^r_{k,0}$ and $E^r_{0,k}$.

On the second page these groups are

$$
 E^2_{k,0} \simeq H_k(B,H_0(F,A)) \simeq H_k(B,A)
$$

$$
  E^2_{0,k} \simeq H_0(B,H_k(F,A)) \simeq H_k(F,A)
$$

by the fact that both $B$ and $F$ are connected by assumption.

Now, since the differentials on the $r$th page have bidegree $(-r,r+1)$, it follows that the only possibly non-vanishing differential on the $k$th page for $k \leq n_1 + n_2$ is

$$
  \partial_k
  \;\colon\;
  E^k_{k,0}
   \longrightarrow
  E^k_{0,k-1}
  \,.
$$

This being so, it follows that the above non-vanishing groups on the $E^2$-page in total degree $k$ remain intact up to these pages, hence that

$$
  E^k_{k,0} \simeq H_k(B,A)
  \;\;\;\;\;
    \,,
  \;\;\;\;\;
  E^k_{0,k-1} \simeq H_{k-1}(F,A)
  \,.
$$

Moreover, by convergence of the spectral sequence there are [[exact sequences]] of the form

$$
  0
    \to
  E^\infty_{k,0}
    \longrightarrow
  E^k_{k,0}
   \overset{\partial_k}{\longrightarrow}
  E^k_{0,k-1}
   \longrightarrow
  E^\infty_{0,k-1}
    \to
  0
  \,,
$$

hence, by the previous statement, of the form

$$
  0
    \to
  E^\infty_{k,0}
    \longrightarrow
  H_k(B,A)
   \overset{\partial_k}{\longrightarrow}
  H_{k-1}(F,A)
   \longrightarrow
  E^\infty_{0,k-1}
    \to
  0
  \,,
$$

Finally, by convergence and the [[filtering]] condition, the only non-vanishing filter contributions to $H_k(X,A)$ in degree $k \leq n_1 + n_2 - 1$ are in filtering degree 0 and $k$, and so projection to filtering degree $k$ gives a [[short exact sequence]] of the form

$$
  0
    \to
  E^\infty_{0,k}
    \longrightarrow
  H_k(X,A)
    \longrightarrow
  E^\infty_{k,0}
    \to
  0
  \,.
$$

Splicing together the exact sequences thus obtained yields the long exact sequence in question:

$$
  \array{
    \cdots
     &\to&
    H_k(F,A)
      && \longrightarrow &&
    H_k(X,A)
      && \longrightarrow &&
    H_k(B,A)
      &\overset{}{\longrightarrow}&
    H_{k-1}(F,A)
     &\to&
    \cdots
    \\
    &&
    &
    \searrow
     && \nearrow
    &
    & \searrow && \nearrow
    \\
    &&
    &&
    E^\infty_{0,k}
    &&
    && E^\infty_{k,0}
  }
$$

=--

## Examples and Applications

+-- {: .num_prop }
###### Proposition

For $X$ ab [[n-connected topological space]], then for $k \leq 2n$ there are [[isomorphisms]]

$$
  H_k(X) \simeq H_{k-1}(\Omega X)
$$

between the [[ordinary homology]] of $X$ in degree $k$ and the ordinary homology of the [[loop space]] of $X$ in degree $k-1$.

=--

+-- {: .proof}
###### Proof

The Serre long exact sequence from prop. \ref{SerreLongExactSequenceForHighlyConnectedBaseAndFiber} applied to the based [[path space]] [[Serre fibration]] of $X$

$$
  \Omega X \longrightarrow Path_\ast(X) \longrightarrow X
$$

is of the form

$$
  H_{2n}(\Omega X)
    \overset{i_\ast}{\longrightarrow}
  H_{2n}(Path_\ast(X))
    \overset{p_\ast}{\longrightarrow}
  H_{2n}(X)
    \overset{\tau}{\longrightarrow}
  H_{2n-1}(\Omega X)
    \overset{i_\ast}{\longrightarrow}
  \cdots
    \overset{i_\ast}{\longrightarrow}
  H_1(X)
  \,.
$$

Since $Path_\ast(X)$ is [[contractible topological space|contractible]], every third group in this sequence vanishes, and hence exactness gives the isomorphisms

$$
  \tau \colon H_k(X) \simeq H_{k-1}(\Omega X)
$$

for $k \leq 2n$.

=--

## Related concepts

* [[Serre spectral sequence]]

* [[Thom-Gysin sequence]]

* [[long exact sequence in cohomology]]

* [[long exact sequence of homotopy groups]]

## References

* {#Kochmann96} [[Stanley Kochmann]], prop. 3.2.1 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#McCleary01} [[John McCleary]], example 5.D of_A User's Guide to Spectral Sequences_, Cambridge University Press (2001)

[[!redirects Serre long exact sequences]]

[[!redirects Serre exact sequence]]
[[!redirects Serre exact sequences]]


Serre microfibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[continuous function|map]] $p$ of [[topological spaces]] is called a __Serre microfibration__
if for any [[lifting problem|lifting square]] for $\{0\}\times K\to [0,1]\times K$ and $p$,
we can find $\epsilon\gt0$ such that the [[lifting property]] is satisfied
after [[restriction|restricting]] to $[0,\epsilon]\times K\subset [0,1]\times K$.

## Properties

Any [[Serre fibration]] is a Serre microfibration.

Any inclusion of [[open subspaces]] is a Serre microfibration.
It is a Serre fibration if and only it is a [[homeomorphism]].

## Related concepts

* [[microflexible sheaf]]

* [[h-principle]]

* [[Serre fibration]]

[[!redirects Serre microfibrations]]


Serre spectral sequence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### In ordinary cohomology
  {#InOrdinaryCohomology}

The _Serre spectral sequence_ or _Leray-Serre spectral sequence_
is a [[spectral sequence]] for computation of [[ordinary cohomology]] ([[ordinary homology]]) of [[topological spaces]] in a [[Serre fibration|Serre]]-[[fiber sequence]] of [[topological spaces]].

\begin{proposition}\label{OrdinaryCohomologicalSerreSpectralSequence}
**(ordinary cohomology Serre spectral sequence)**\linebreak

Given a [[homotopy fiber sequence]]

$$
  \array{
     F &\longrightarrow& E
     \\
     && \downarrow^{\mathrlap{p}}
     \\
     && X
  }
$$

over a [[connected topological space]] $X$,
such that the canonical [[group action]] of the [[fundamental group]] $\pi_1(X)$ on the [[ordinary cohomology]] of the [[fiber]] $F$ is [[trivial action|trivial]] (for instance if $X$ is a [[simply connected topological space]]), then there exists a *cohomology Serre spectral sequence* of the form:

\[
  \label{SecondPageAndConvergenceOfOrdinaryCohomologicalSerreSpectralSequence}
  E_2^{p,q}
  \;=\;
  H^p
  \big(
    X,
    \,
    H^q(F)
  \big)
  \;\Rightarrow\;
  H^{p+q}(E)
  \,.
\]

\end{proposition}

(e.g. [Hatcher, Thm. 1.14](#Hatcher))

{#ConvergenceOfTheOrdinaryCohomologySSS} Hence for $n \in \mathbb{N}$ we have a [[filtration]] of [[abelian groups]]

\[
  \label{FilteringForOrdinaryCohomologySerreSpectralSequence}
  0
  \xhookrightarrow{
    \;\;
    E^{n,0}_\infty
    \;\;
  }
  F^n_n
  \xhookrightarrow{
    E^{n-1,1}_\infty
  }
  F^n_{n-1}
  \xhookrightarrow{\;\;\;\;\;}
  \cdots
  \xhookrightarrow{\;\;\;\;\;}
  F^n_{1}
  \xhookrightarrow{
    \;\;
    E^{0,n}_\infty
    \;\;
  }
  F^n_0
  \;=\;
  H^n(E)
  \,,
\]

where

$$
  F^n_{p+1}
  \xhookrightarrow{
    E^{p,n-p}_\infty
  }
  F^n_{p}
  {\phantom{AAAA}}
  \text{means that}
  {\phantom{AAAA}}
  0
  \to
  F^n_{p+1}
  \xhookrightarrow{\;}
  F^n_{p}
  \twoheadrightarrow
  E^{p,n-p}_\infty
  \to
  0
  \,,
$$

hence that -- iteratively as $p$ *decreases* -- $F^n_p$ is an [[algebra extension|extension]] of $E^{p,n-p}_\infty$ by $F^n_{p+1}$.

### In generalized cohomology

The generalization of this from [[ordinary cohomology]] to [[generalized (Eilenberg-Steenrod) cohomology]] is the _[[Atiyah-Hirzebruch spectral sequence]]_, see there for details.

### In relative cohomology
 {#InRelativeCohomology}

There are two kinds of **relative Serre spectral sequences**.

For $F \to E \to X$ as above and $A \hookrightarrow X$ a subspace, the induced restriction of the fibration

$$
  \array{
    F & \simeq & F
    \\
    \downarrow && \downarrow
    \\
    p^{-1}(A) &\longrightarrow& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    A &\hookrightarrow& X
  }
$$

induces a spectral sequence in [[relative cohomology]] of the base space of the form

$$
  E_2^{p,q}
  =
  H^p(X,A; H^q(F))
    \;\Rightarrow\;
  H^\bullet(E, p^{-1}(A))
  \,.
$$

(e.g. [Davis 91, theorem 9.33](#Davis91))

Conversely, for

$$
  \array{
    F' & \hookrightarrow & F
    \\
    \downarrow && \downarrow
    \\
    E' &\hookrightarrow& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    X &\hookrightarrow& X
  }
$$

a sub-fibration over the same base, then this induces a spectral sequence for [[relative cohomology]] of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:

$$
  E^{p,q}_2 = H^p(X; H^q(F,F'))
    \;\Rightarrow\;
  H^\bullet(E,E')
  \,.
$$

(e.g. [Kochman 96, theorem 2.6.3](#Kochman96), [Davis 91, theorem 9.34](#Davis91))

### In equivariant cohomology

There is also a generalization to [[equivariant cohomology]]: for [cohomology with coefficients in a Mackey functor](Mackey+functor#Cohomology) with[[RO(G)-grading]] for [[representation spheres]] $S^V$, then for $E \to X$ an $F$-fibration of [[topological G-spaces]] and for $A$ any $G$-[[Mackey functor]], the equivariant Serre spectral sequence looks like ([Kronholm 10, theorem 3.1](#Kronholm10)):

$$
  E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(E,A)
  \,,
$$

where on the left in the $E_2$-page we have [[ordinary cohomology]] with [[coefficients]] in the genuine equivariant cohomology groups of the fiber.

## Details

For details on the plain Serre spectral sequence see at _[[Atiyah-Hirzebruch spectral sequence]]_ and take $E = H R$ to be ordinary cohomology.

## Examples

\begin{example}\label{IntegralCohomologyOfHomotopyQuotientOf4SphereByADEAction}
**([[integral cohomology]] of [[homotopy quotient]] of [[4-sphere]] by [[finite subgroup of SU(2)]])** \linebreak

  Let

  * $G \xhookrightarrow{\;i\;} Sp(1) \simeq SU(2) \simeq Spin(3)$ be a [[finite subgroup of SU(2)]];

  * $\mathbb{H} \,\in\, G Actions(VectorSpaces_{\mathbb{R}})$ the [[linear representation]] of $G$ on the [[quaternions]], regarded as a 4d [[real vector space]], which is the [[restricted representation]] of the defining representation of [[Sp(1)]];

  * $S^{\mathbb{H}} \in G Actions(TopologicalSpaces)$ the corresponding [[representation sphere]].

Then the [[integral cohomology]] in degree 4 of the [[homotopy quotient]] is the [[direct sum]]

\[
  \label{Integral4CohomologyOfHomotopyQuotientOf4SphereByFiniteSubgroupOfSU2}
  H^4
  \big(
    S^{\mathbb{H}} \!\sslash\! G,
    \,
    \mathbb{Z}
  \big)
  \;\;
  \simeq
  \;\;
  \mathbb{Z} \oplus (\mathbb{Z}/\left\vert G \right\vert)
\]

of the [[integers]] with the [[cyclic group]] of [[order of a group|order]] that of $G$.

\end{example}
\begin{proof}
By the [[Borel construction]] we have a [[homotopy fiber sequence]] of the form
\[
  \label{BorelConstructionFor4SphereActedOnByADESubgroup}
  \array{
    S^4 &\longrightarrow& S^{\mathbb{H}} \!\sslash\! G
    \\
    && \big\downarrow
    \\
    && B G
  }
\]

over the [[classifying space]] of $G$.

Here the [[integral cohomology]] of the [[4-sphere]] [[fiber]] is (e.g. by the nature of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},4)$)

\[
  \label{IntegralCohomologyOfThe4Sphere}
  H^n
  \big(
    S^4,
    \,
    \mathbb{Z}
  \big)
  \;\simeq\;
  \left\{
  \array{
    \mathbb{Z} & for & n \in \{0,4\}
    \\
    0 & \text{otherwise} \,.
  }
  \right.
\]

We claim that the [[group action]] of $\pi_1(B G) \simeq G$ (by [this Prop.](simplicial+classifying+space#HomotopyGroupsOfBarWG)) on the [[integral cohomology]] of the [[fiber]] is [[trivial action|trivial]]. This follows by observing that:

1. we have an [[isomorphism]] of [[topological G-spaces]] between the [[representation sphere]] of $\mathbb{H}$ and the [[unit sphere]] in $\mathbb{R} \oplus \mathbb{H}$ (by [this Prop.](representation+sphere#RepresentationSpheresAsUnitSpheres)):

   $$
     S^{\mathbb{H}}
     \;\simeq_{G}\;
     S(\mathbb{R} \oplus \mathbb{H})
     \,;
   $$

1. the [[group action]] of [[Sp(1)]] on $\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4$ is through the defining action of [[SO(4)]], hence the action on $\mathbb{R} \oplus \mathbb{H}$ is through [[SO(5)]],

   because [[quaternions]] are a [[normed division algebra]], so that left-multiplication by unit-[[norm]] quaternions $q \in$ [[Sp(1)]] $= S(\mathbb{H})$ preserves the [[norm]] (e.g [[schreiber:Equivariant homotopy and super M-branes|HSS 18, Rem. A.8]]);

1. the generator of $H^4(S^4,\mathbb{Z})$ may be identified with the [[volume form]] (under the [[Hopf degree theorem]] and the [[de Rham theorem]]) which is manifestly preserved by the action of the [[special orthogonal group]] $SO(5)$.

Therefore, the [[integral cohomology|integral]]-cohomological Serre spectral sequence (Prop. \ref{OrdinaryCohomologicalSerreSpectralSequence}) applies to the Borel fiber sequence (eq:BorelConstructionFor4SphereActedOnByADESubgroup).

Now, noticing that the [[integral cohomology]] of a [[classifying space]] of a [[discrete group]] is its [[group cohomology]]

$$
  H^\bullet(B G, \mathbb{Z})
  \;\simeq\;
  H^\bullet_{grp}(G, \mathbb{Z})
$$

we have for the given [[finite subgroup of SU(2)]] (by [this Prop](finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2)) that:

\[
  \label{IntegralCohomologyOfClassifyingSpaceOfFiniteSubgroupOfSU2}
  H^n(B G, \mathbb{Z})
  \;\simeq\;
  \left\{
  \array{
    \mathbb{Z} &\vert& n = 0
    \\
    G^{ab} &\vert&  n = 2 \, mod \, 4
    \\
    \mathbb{Z}/{\vert G \vert}
    &\vert&
    n \, \text{positive multiple of} \, 4
    \\
    0 &\vert& \text{otherwise} \,,
  }
  \right.
\]

where $G^{ab} \coloneqq G / [G,G]$ denotes the [[abelianization]] of $G$.

Using the cohomology groups
(eq:IntegralCohomologyOfThe4Sphere)
and
(eq:IntegralCohomologyOfClassifyingSpaceOfFiniteSubgroupOfSU2)
in the fomula
(eq:SecondPageAndConvergenceOfOrdinaryCohomologicalSerreSpectralSequence)
for the second page $E_2^{\bullet, \bullet}$
of the cohomology Serre spectral sequence (Prop. \ref{OrdinaryCohomologicalSerreSpectralSequence}) shows that this is of the following form:

\begin{tikzcd}[row sep={between origins, 21pt}, column sep={between origins, 21pt}]
      {}
      &
      {}
      \\[-12pt]
      {}
          &
      {}
      \\
      &
      \vdots
      &
      \vdots
      &
      \vdots
      &
      \vdots
      &
      \vdots
      &
      \vdots
      &
      \vdots
      &
      \vdots
      \\
      {}
      \ar[
        ddddddrrrrrr,
        -,
        line width=19pt,
        shift left=1pt,
        opacity=.15,
        color=blue
      ]
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      \cdots
      \\
      &
      \mathbb{Z}
      \ar[
        drr
      ]
      &
      0
      %\ar[
      %  drr
      %]
      &
      G^{\mathrm{ab}}
      \ar[
        drr
      ]
      &
      0
      &
      \mathbb{Z}_{\left\vert G\right\vert}
      \ar[
        drr
      ]
      &
      0
      &
      G^{\mathrm{ab}}
      \ar[
        drr
      ]
      &
      0
      &
      \cdots
      \\
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      \cdots
      \\
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      \cdots
      \\
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      0
      &
      \cdots
      \\
      \ar[
        rrrrrrrrrr,
        color=lightgray,
        shift right=5pt,
        "B G"{below, color=black, pos=.95}
      ]
      &
      \mathbb{Z}
      &
      0
      &
      G^{\mathrm{ab}}
      &
      0
      &
      \mathbb{Z}_{\left\vert G\right\vert}
      &
      0
      &
      G^{\mathrm{ab}}
      &
      0
      &
      \cdots
      &
      {}
      \\
      &
      \ar[
        uuuuuuuuu,
        color=lightgray,
        shift left=5pt,
        "S^4"{left, color=black, pos=.93}
      ]
      &{}&{}&{}&{}&{}
\end{tikzcd}

Since the [[codomains]] of the [[differentials]] on all the following pages are translated diagonally (downwards and rightwards, by the [general formula](spectral+sequence#eq:FormOfDifferentialInCohomologySpectralSequence)) from the codomains seen above, one sees that for every differential on every page, the [[domain]] or the [[codomain]] is the [[zero group]].

This means that all differentials are the [[zero morphism]], hence that the spectral sequence collapses already on this second page:

$$
  E^{\bullet, \bullet}_\infty
  \;\;
  \simeq
  \;\;
  E^{\bullet, \bullet}_2
  \,.
$$

Therefore the convergence statement (eq:FilteringForOrdinaryCohomologySerreSpectralSequence) says that the degree-4 cohomology group in question is a [[group extension]] of

$$
  E^{0,4}_\infty
  \;\simeq\;
  H^0\big(B G, \, H^4(S^4; \mathbb{Z}) \big)
  \;\simeq\;
  H^0\big( B G, \, \mathbb{Z}\big)
  \;\simeq\;
  \mathbb{Z}
$$

by

$$
  E^{4,0}_\infty
  \;\simeq\;
  H^4\big(B G, \, H^0(S^4; \mathbb{Z}) \big)
  \;\simeq\;
  H^4\big( B G, \, \mathbb{Z}\big)
  \;\simeq\;
  \mathbb{Z}/\left\vert G \right\vert
$$

in that we have a [[short exact sequence]] of the form

$$
  0
  \to
  \mathbb{Z}/\left\vert G\right\vert
  \hookrightarrow
  H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big)
  \twoheadrightarrow
  \mathbb{Z}
  \to
  0
  \,.
$$

But since the [[Ext-group]] of the [[integers]] is [[trivial group|trivial]] ([this Expl.](Ext#ExtensionsOfTheIntegersAreTrivial)) this extension must be the [[direct sum]]

$$
  H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big)
  \;\simeq\;
  \mathbb{Z}_{\left\vert G\right\vert}
  \oplus
  \mathbb{Z}
  \,.
$$

This is the claim (eq:Integral4CohomologyOfHomotopyQuotientOf4SphereByFiniteSubgroupOfSU2) to be proven.
\end{proof}

## Consequences

* [[Serre long exact sequence]]

* [[Thom-Gysin sequence]]

## Related concepts

* [[long exact sequence in cohomology]]

  * [[long exact sequence in generalized cohomology]]

## References

### General

The original article is

* [[Jean-Pierre Serre]], _Homologie singuliére des espaces fibrés_ Applications, Ann. of Math. 54 (1951),

Textbook accounts:

* {#Hatcher} [[Alan Hatcher]], _[Spectral sequences in algebraic topology](https://pi.math.cornell.edu/~hatcher/SSAT/SSATpage.html)_ _I: The Serre spectral sequence_ ([pdf](https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf), [[Hatcher_SerreSpectralSequence.pdf:file]])

* {#Kochman96} [[Stanley Kochman]], section 2.2. of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Davis91} Davis, _Lecture notes in algebraic topology_, 1991

Lecture notes etc. includes

* [[Greg Friedman]], _Some extremely brief notes on the Leray spectral sequence_ ([pdf](http://faculty.tcu.edu/richardson/Seminars/Gregspecseq.pdf))

Discussion in [[homotopy type theory]] includes

* [[Mike Shulman]], _[Spectral Sequences](http://homotopytypetheory.org/2013/08/08/spectral-sequences/)_, 2013

and implementation in [[Lean]] is in

* [[Floris van Doorn]], [[Egbert Rijke]], [[Ulrik Buchholtz]], [[Favonia]], [[Steve Awodey]], [[Jeremy Avigad]], [[Mike Shulman]], [[Jonas Frey]],  _Spectral_ ([github.com/cmu-phil/Spectral](https://github.com/cmu-phil/Spectral))

* [[Floris van Doorn]], §5 in: *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

### In equivariant cohomology

In [[equivariant cohomology]], for [[Bredon cohomology]]:

* [[Ieke Moerdijk]], J.-A. Svensson, _The Equivariant Serre Spectral Sequence_, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 ([JSTOR](http://www.jstor.org/stable/2160037))

and for genuine equivariant cohomology, i.e. for [[RO(G)-grading|RO(G)]]-graded [cohomology with coefficients in a Mackey functor](Mackey+functor#Cohomology):

* {#Kronholm10} [[William C. Kronholm]], *The $RO(G)$-graded Serre spectral sequence*, Homology Homotopy Appl. **12** 1 (2010) 75-92. [[arXiv:0908.3827](https://arxiv.org/abs/0908.3827), [doi:10.4310/HHA.2010.v12.n1.a7](https://doi.org/10.4310/HHA.2010.v12.n1.a7),  [euclid:hha/1296223823](https://projecteuclid.org/euclid.hha/1296223823)]

See also

* [[Megan Shulman]], _Equivariant Spectral Sequences for Local Coefficients_ ([arXiv:1005.0379](http://arxiv.org/abs/1005.0379))

[[!redirects Serre spectral sequences]]

[[!redirects Leray-Serre spectral sequence]]
[[!redirects Leray-Serre spectral sequences]]


Serre subcategory

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### Context
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[full subcategory]] $T$ of an [[abelian category]] $A$ is a __Serre subcategory__ if it is nonempty and for any [[exact sequence]]
$$M\to M'\to M''$$
$M'$ is in $T$ if $M$ and $M''$ are in $T$.

Alternative definition: a full subcategory $T$ of an abelian category such that for every short exact sequence
$$ 0\to A\to B\to C\to 0$$ $B$ is in $T$ _if and only if_ $A$ and $C$ are in $T$.

Every Serre subcategory is automatically abelian.

##Terminology

This notion in general is called a [[thick subcategory]] in Gabriel's thesis, [[Des Catégories Abéliennes]]. The terminology is a minefield as there are variants that occur in the literature, see [[thick subcategory]], and also the [Stack's Project section on this](http://stacks.math.columbia.edu/tag/02MN). Some of these variants are mentioned below.

##Serre quotient

Following Serre, one then defines the category $A/T$ whose objects are the objects of $A$ and where the morphisms in $A/T$ are defined by
$$\mathrm{Hom}_{A/T}(X,Y) := \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),$$
where the colimit is over all $X',Y'$ in $A$ such that $Y'$ and $X/X'$ are in $T$. The quotient category $A/T$ is abelian.

Very often the same definition of Serre subcategory is used in an arbitrary abelian category $A$ (we will say in that case __weakly Serre subcategory__); but in fact, at least when the abelian category is not a [[Grothendieck category]], it is more appropriate to ask for an additional condition in the definition of Serre subcategory, so that the standard theorems on correspondences with other canonical data in localization theory remain valid.

To this aim, for any subcategory of an arbitrary abelian category $A$ one denotes by $\bar{T}$ the [[full subcategory]] of $A$ generated by all objects $N$ for which any (nonzero) [[subquotient]] of $N$ in $T$
has a (nonzero) [[subobject]] from $T$.  This becomes an idempotent operation on the class of subcategories of $A$ with $T\subset \bar{T}$ iff $T$ is [[topologizing subcategory|topologizing]].
Moreover $\bar{T}$ is always [[thick subcategory|thick]] in the stronger sense (that is, thick and topologizing).

__Serre subcategories in the strong sense__ are those nonempty full subcategories which are stable under the operation $T\mapsto\bar{T}$.

## References

* [[Carl Faith]], _Algebra: Rings, Modules, Categories_, vol. 1, Chapter 15, Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]

* [[The Stacks Project]], *[Serre subcategories](https://stacks.math.columbia.edu/tag/02MN)*

[[!redirects Serre subcategories]]


Serre's criterion of affineness

Serre's criterion of affineness characterizes [[affine morphism]]s of schemes in terms of exactness properties of the corresponding functors among the categories of [[quasicoherent sheaves]].

If $f\colon X\to Y$ is a [[quasicompact morphism]] of [[algebraic scheme]]s and $X$ is [[separated scheme|separated]], then $f$ is affine iff it is cohomologically affine, that is, the direct image functor $f_*$ is [[exact functor|exact]].

* [[EGA]] II 5.2.2, EGA IV 1.7.17

* Akhil Mathew, [Serre's criterion for affineness as Morita theory](http://amathew.wordpress.com/2012/08/01/serres-criterion-for-affineness-as-morita-theory), a blog post

[[!redirects Serre's criterion of affiness]]


Serre's theorem on Proj

## Idea

Projective spectrum Proj assigns a locally ringed space $Proj(A)$ to a graded ring $A$. This construction has been introduced by Serre.

Serre proved two elementary theorems concerning the structure of sheaves of $\mathcal{O}_X$-modules on $X = Proj(A)$ relating graded modules over $A$ and sheaves of $mathcal{O}_X$-modules: one on [[coherent sheaves]] and another on [[quasicoherent sheaves]]. In both cases, the sheaves correspond to objects in the localization of a category of graded modules by a torsion (Serre's) subcategory.

There is an analogue for affine schemes, (affine) [[affine Serre's theorem|Serre's theorem on quasicoherent sheaves over affine schemes]].

## Serre's theorem for quasicoherent sheaves

There is a functor $M\mapsto \tilde{M}$ (to do: should be described here) from the category of graded modules over a $\mathbf{Z}_{\geq 0}$-graded ring $A$ to the category of quasicoherent sheaves of $\mathcal{O}_X$-modules on $X = Proj(A)$ which has a fully faithful right adjoint, hence inducing an equivalence of the category of quasicoherent modules $qcoh(X)$ with a localization of the category of graded $A$-modules. The kernel of that localization is the subcategory of modules of finite length.

## Serre's theorem for coherent sheaves

If $A$ is [[Noetherian ring|Noetherian]], the functor $M\mapsto \tilde{M}$ restricted to the category of finitely generated projective modules takes values in the category of coherent sheaves and the induced functor also has a fully faithful right adjoint, which moreover induces an equivalence between the localization of the category of finitely generated projective $A$-modules by the Serre's subcategory of modules of finite length and the category of coherent sheaves over $Proj(A)$.

## Noncommutative and derived analogues

The subject of noncommutative projective algebraic geometry by Artin and Zhang, as well as independent (and different mainly in the level of generality) attempts to
noncommutative geometry by other authors take the Serre's
theorem to the status of definition of the category of quasicoherent sheaves when the graded ring is generalized to a noncommutative one.

## References

### Commutative

This is Section 59, Proposition. 7.8, p. 252 in

* J. P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (2) (1955) 197--278

See also 3.3.5 in

* A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique II, Inst. Hautes Etudes Sci. Publ. Math. 8 (1961)

Hartshorne proves as Proposition 5.15 that the counit of the corresponding adjunction is an isomorphism (hence we have the reflective subcategory that is localization functor)

* Robin Hartshorne, _Algebraic geometry_, Graduate Texts in Mathematics 52, Springer 1977

* _Serre's theorem on Proj_, [mathOverflow](https://math.stackexchange.com/questions/333325/serres-theorem-for-operatornameproj)

### Noncommutative

* A. B. Verevkin, _On a noncommutative analogue of the category of coherent sheaves on a projective scheme_, Amer. Math. Soc. Transl. (2) 151 (1992)
* [[Michael Artin]], J. Zhang, _Noncommutative projective schemes_, Adv. Math. 109, 228--287 (1994)
* [[Fred van Oystaeyen]], L. Willaert, _Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras_, J. Pure Appl. Alg. 104 (1995) 109--122
* [[Alexander Rosenberg]], _Non-commutative algebraic geometry and representations of quantized algebras_, Mathematics and Its Applications __330__, Kluwer Academic Publishers, 1995
* [[Alexander Polishchuk]], _Noncommutative Proj and coherent algebras_, arXiv:[math.RA/0212182](https://arxiv.org/abs/math/0212182)

> We prove that an abelian category equipped with an ample sequence of objects is equivalent to the quotient of the category of coherent modules over the corresponding algebra by the subcategory of finite-dimensional modules. In the Noetherian case a similar result was proved by Artin and Zhang.

* Andrés Chacón, Armando Reyes, _Noncommutative scheme theory and the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings_, [arXiv:2301.07815](https://arxiv.org/abs/2301.07815)

category: algebraic geometry

[[!redirects Serre's theorem for Proj]]
[[!redirects projective Serre's theorem]]
[[!redirects Serre's theorem on quasicoherent sheaves]]


Serre-Swan theorem

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###Context###
#### Algebra
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#### Topology
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[[!include topology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

The _Serre-Swan theorem_ identifies suitable [[modules]] over an [[algebra of functions]] on some [[space]] with the modules of [[sections]] of [[vector bundles]] over that space and thereby identifies these modules with vector bundles themselves.

Together with theorems like _[[Gelfand duality]]_, the Serre-Swan theorem is a central part of the [[Isbell duality|general duality]] between [[geometry]] and [[algebra]].
In particular it may serve to generalize the notion of vector bundle from standard [[geometry]] to more exotic forms of geometry, such as [[noncommutative geometry]].

## Statement

There are two different original theorems of the same intuitive spirit which are usually jointly called the **Serre-Swan theorem**, the first one is in [[algebraic geometry]], the second in [[topology]]:

1) **Serre's theorem** ([Serre 55](#Serre55)): let $R$ be a commmutative unital [[Noetherian ring]] (in particular, the [[structure sheaf|coordinate ring]] of an [[affine variety]] over a [[field]]), then the [[category]] of finitely-generated _[[projective module|projective]] $R$-[[modules]] is [[equivalence of categories|equivalent]] to the category of [[algebraic vector bundles]] (= [[covering|locally]] [[free module|free]] [[sheaves]] of [[structure sheaf]]-[[modules]] of constant finite [[rank]]) on $Spec R$.

2) **Swan's theorem** ([Swan 62](#Swan)): Given a [[Hausdorff space|Hausdorff]] [[compact space]] $X$, the [[category]] of [[finitely generated module|finitely generated]] [[projective modules]] over the [[continuous function|continuous]]-[[function algebra]] $C(X)$ is [[equivalence of categories|equivalent]] to the category of finite-[[rank]] [[vector bundles]] on $X$, where the equivalence is established by sending a vector bundle to the its module of continuous [[sections]].

But there are also various variations of these theorems, for instance to [[differential geometry]]:

3) **[[smooth Serre-Swan theorem]]** ([Nestruev 03, 11.33](#Nestruev03)) For $X$ a [[smooth manifold]] with $\mathbb{R}$-algebra of [[smooth functions]] $C^\infty(X)$ there is an [[equivalence of categories]] between that of finite [[rank]] smooth [[vector bundles]] over $X$ and [[finitely generated objects|finitely generated]] [[projective modules]] over $C^\infty(X)$.

A general statement of the Serre-Swan theorems over [[ringed spaces]] is in ([Morye](#Morye)).

If one drops the condition that the [[sheaf of modules]] over the [[structure sheaf]] of a [[ringed space]] is [[covering|locally]] [[free module|free]], and allows it instad to be just _locally [[presentable module|presentable]]_, then one arrives at the notion of [[quasicoherent sheaf of modules]]. Here the Serre-Swan theorem serves to clarify in which sense precisely these are generalizations of [[vector bundles]].

The condition that the modules be [[projective module|projective]] can also naturally be relaxed.
In [[higher geometry]] the Serre-Swan theorem becomes not only more general but also conceptually simpler: if instead of [[modules]] one considers [[chain complexes]] of modules ([[(∞,1)-modules]]) then under mild assumptions (see at _[[projective resolution]]_) every chain complex of modules is [[equivalence|equivalent]] ([[quasi-isomorphism|quasi-isomorphic]]) to a chain complex of [[projective modules]], and hence this condition in the statement of the traditional Serre-Swan theorem becomes automatic. Or in other words, the non-projective modules also do correspond to [[vector bundles]], but to [[chain complexes]] of vector bundles (only that the [[chain homology]] of the complex is not itself a vector bundle again in this case). See at _[[(∞,1)-vector bundle]]_ for more on this.

## Applications

### To K-theory

The Serre-Swan theorem serves to relate [[topological K-theory]] with [[algebraic K-theory]]. (...)

## Related concepts

* [[Gelfand duality]]

* [[operator K-theory]]

* [[Quillen-Suslin theorem]]

* [[smooth Serre–Swan theorem]]

* [[modules over a ring are equivalent to quasicoherent sheaves over its spectrum]]

* [[duality between geometry and algebra]]

[[!include Isbell duality - table]]

## References

The two original articles:

* {#Serre55} [[Jean-Pierre Serre]], §50 in: *Faisceaux algebriques coherents*, Annals of Mathematics **61** 2 (1955) 197-278  [[jstor:1969915](https://www.jstor.org/stable/1969915)]

* {#Swan} [[Richard Swan]], *Vector bundles and projective modules*, Trans. AMS **105** 2 (1962) 264-277 [[doi:10.2307/1993627](https://doi.org/10.2307/1993627)]

A textbook account in the context of [[differential geometry]]:

* {#Nestruev03} [[Jet Nestruev]], _Smooth manifolds and observables_, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)

A general account of Serre-Swan-type theorems over [[ringed spaces]] is in

* {#Morye} [[Archana Morye]], _Note on the Serre-Swan Theorem_ ([arXiv:0905.0319](http://arxiv.org/abs/0905.0319))

A textbook account on the use of the theorem in [[K-theory]] is for instance

* [[Max Karoubi]], _$K$-theory. An introduction_, Grundlehren der Mathematischen Wissenschaften, Band __226__, Springer 1978. xviii+308 pp.

[[!redirects Serre–Swan theorem]]
[[!redirects Serre--Swan theorem]]


sesquialgebra

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### Context
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _sesquiunital sesquiagebra_ is an [[algebra object]] [[internalization|internal]] to the monoidal [[2-category]] [[2Mod]] of [[algebras]], [[bimodules]] and [[bimodule intertwiners]]. This means that it is an algebra equipped with an additional associative product and unit which are both exhibited by [[bimodules]]. If these bimodules come from algebra [[homomorphisms]] then the sesquialgebra is a [[bialgebra]].

The structure of a sesquialgebra is just so that the [[category of modules]] of the underlying algebra is itself a [[monoidal category]]. In this sense sesquialgebras are a placeholder for [[2-algebra|2-algebras]]. Moreover, in as far as these 2-algebras themselves are regarded as placeholders for _their_ [[2-category]] of [[2-modules]] a sesquialgebra presents a [[3-module]]/[[3-vector space]].

Sesquialgebras with an extra grouplike-property have been called _[[hopfish algebras]]_.

## Definition

+-- {: .num_defn}
###### Definition

A [[sesquiunital sesquialgebra]] over $R$ is an [[associative algebra]] $A$ over $R$ equipped with the structure of an [[algebra object]] internal to the [[2-category]] [[2Mod]] of [[associative algebras]], [[bimodules]] and [[bimodule intertwiners]].

This means that it is an $R$-algebra $A$ equipped with

* a product $A \otimes_R A$-$A$-[[bimodule]] $\Delta$;

* a unit $R$-$A$-[[bimodule]] $\epsilon$

satisfying the evident [[associative law]] and [[unit law]].

=--

## Properties

### Tannaka duality and 2-rings

Precomposition with the product and unit bimodule makes the [[category of modules]] over the underlying [[associative algebra]] of a sesquialgebra itself into a [[monoidal category]].

See for instance ([Vercruysse, 5.3.3](#Vercruysse)).

In fact, regarding the [[category of modules]] $Mod_A$ as a [[2-abelian group]], the structure of a sesquialgebra on $A$ is equivalently the structure of a [[2-ring]] (see there) on $Mod_A$.

[[!include structure on algebras and their module categories - table]]

## References

* Joost Vercruysse, _Hopf algebras---Variant notions and reconstruction theorems_ ([arXiv:1202.3613](http://arxiv.org/abs/1202.3613))
 {#Vercruysse}

See also the references at _[[hopfish algebra]]_.

[[!redirects sesquialgebra]]
[[!redirects sesquialgebras]]

[[!redirects sesquiunital sesquialgebra]]
[[!redirects sesquiunital sesquialgebras]]


sesquicategory

# Contents
* contents
{: toc}

## Idea

A **sesquicategory** is a (strict) [[2-category]] except for the fact that the [[interchange law]] need not hold.

## Definition

Sesquicategories can be straightforwardly defined just as for strict [[2-categories]] except with the interchange law left out.  (In order for this to make sense, one has to spell out the definition explicitly enough that the interchange law is a separate axiom.)  This means that composition in a sesquicategory cannot be [[functor]] $\hom(y,z)\times \hom(x,y)\to\hom(x,z)$.  So sesquicategories are more usually defined as categories [[enriched category|enriched in]] [[Cat]], where the monoidal structure for the enrichment is not the usual cartesian product but the [[funny tensor product]], i.e. the tensor adjoint to the 'unnatural' hom, in which the hom-category $[C,D]$ has morphisms given by $C$-indexed families of arrows of $D$ _without_ any naturality requirement.

Alternatively, a sesquicategory may be given as a category $C$ together with a functor $H \colon C^{op} \times C \to Cat$ whose composite $ob \circ H \colon C^{op} \times C \to Cat \to Set$ with the underlying-set functor is equal to the hom functor of $C$.  Because of the equivalence $[C, Cat(D)] \simeq Cat [C,D]$ (for [[finitely complete category|finitely complete]] $D$), this is the same as saying that a sesquicategory is given by a category $C$ together with an internal category $H$ in $[C^{op} \times C, Set]$ whose object $H_0$ of objects is the [[hom functor]] $hom_C$ of $C$.

## Remarks

A strict [[premonoidal category]] is the same as a sesquicategory with exactly one object.

A [[Gray-category]] does not have an underlying [[strict 2-category]], but it does have an underlying strict sesquicategory.  Thus, if one wants to define Gray-[[computads]], it is natural to work with "sesqui-computads" as a partway stage; see for instance [[toddtrimble:Surface diagrams]]

The name 'sesquicategory' literally means $1\frac{1}{2}$-category, although strictly speaking they are actually *more* general than $2$-categories (which are of course more general than $1$-categories).  However, one can also view a $2$-category or sesquicategory as a $1$-category with [[extra structure]] or stuff (the $2$-cells and their composition), and in this way sesquicategories are partway between $1$-categories and $2$-categories, with only one axiom left out.  (A [[strict 2-category]] can be considered directly as a 1-category with additional 2-cells added; for a weak 2-category one has instead to consider its "underlying" 1-category to be its [[homotopy category]] obtained by identifying isomorphic 1-morphisms.)

The paper [Stell (1994)](#Stell) shows the relation with rewriting.

The paper [Brown (2010)](#Brown) shows how a sesquicategory arises from a whiskered category.

## References

* [this discussion](http://www.mta.ca/~cat-dist/catlist/1999/2-cats-without) on the categories list.
{#catlist}

* {#Stell} [[John Stell]], _Modelling Term Rewriting Systems by Sesqui-Categories_, Proc. Catégories, Algèbres, Esquisses et Néo-Esquisses (1994). [pdf]

* {#Street} [[Ross Street]], _Categorical Structures_, Handbook of Algebra, Vol. 1, 1996, pp. 531-577. [doi:10.1016/S1570-7954(96)80019-2, [[Street-CategoricalStructures.pdf:file]]]

* {#Brown} [[Ronnie Brown]], Possible connections between whiskered categories and groupoids, Leibniz algebras, automorphism structures and local-to-global questions. J. Homotopy Relat. Struct., 5(1) (2010) 305--318.
[ [arXiv:0708.1677](https://arxiv.org/abs/0708.1677) [PDF](https://tcms.org.ge/Journals/JHRS/xvolumes/2010/n1a15/v5n1a15hl.pdf) ]

* [[Nelson Martins-Ferreira]], _Low-dimensional internal categorial structures in weakly Mal'cev sesquicategories_, PhD thesis, University of Cape Town (2008). [ [url](http://hdl.handle.net/11427/4907) ]

[[!redirects sesquicategory]]
[[!redirects sesquicategories]]


sesquilinear form

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### Context
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On a [[complex]] [[vector space]] $V$ a _sesquilinear map_ is a [[function]] of two arguments

$$
  \langle -,-\rangle
  \;\colon\;
  V \times V
  \longrightarrow
  \mathbb{C}
$$

which is a [[linear function]] in one argument (say the second) and complex anti-linear in the other.

\begin{defn}\label{Definity}

A sesquilinear form $\langle -,-\rangle$ is called

$$
  \left.
  \array{
    \text{positive definite} &if& \langle v^\ast,v \rangle \gt 0
    \\
    \text{negative definite} &if& \langle v^\ast,v \rangle \lt 0
    \\
    \text{positive semi-definite} &if& \langle v^\ast,v \rangle \geq 0
    \\
    \text{negative semi-definite} &if& \langle v^\ast,v \rangle \leq 0
  }
  \right\rbrace
  \;\;
  \text{for all}\; v \neq 0
  \,.
$$

Finally, it is called *indefinite* if it is neither positive nor negative semi-definite.
\end{defn}

More generally:

Let $A$ be a [[star algebra]]. Then every left $A$-[[module]] $\rho_l \colon A \otimes V \longrightarrow V$ canonically becomes a right module $\rho_r \colon V \otimes A \longrightarrow A$ by setting

$$
  \rho_r(v,a) \coloneqq a^\ast v
$$

and vice versa.

With this operation understood to turn a left module $V$ into a right module, then a sesquilinear form on $V$ is simply an element in the [[tensor product of modules]]

$$
  V^\ast \otimes_A V^\ast
$$

of the $A$-linear dual $V^\ast$ of $V$ with itself.

## Related concepts

* [[bilinear form]], [[quadratic form]]

* [[symplectic form]], [[Kähler form]], [[Hermitian form]]

## References

See also:

* [[Brian Osserman]], _A primer on sesquilinear forms_ ([[OssermanSesquilinearForms.pdf:file]])

* Wikipedia, _[Sesquilinear form](https://en.wikipedia.org/wiki/Sesquilinear_form)_

[[!redirects sesquilinear maps]]
[[!redirects sesquilinear function]]
[[!redirects sesquilinear functions]]

[[!redirects sesquilinear forms]]

[[!redirects sesquilinear map]]


Set

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#### Category theory
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#### Foundations
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#### Categories of categories
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# The category of sets
* table of contents
{: toc}

## Definition

__$Set$__ is the (or a) [[category]] with [[sets]] as [[objects]] and [[functions]] between sets as [[morphisms]].

This definition is somewhat vague by design. Rather than canonize a fixed set of principles, the nLab adopts a 'pluralist' point of view which recognizes different needs and foundational assumptions among mathematicians who use set theory. Thus, there are various axes to consider when formulating categorical properties one thinks $Set$ should satisfy, including

* first-order vs. higher-order logic

* impredicative vs. [[predicative mathematics]]

* 'variability' vs. 'constancy' (Lawvere)

* classical logic vs. intuitionistic logic

* [[image factorizations]] and [[quotient sets]], i.e. [[sets]] vs [[setoids]].

* [[initial object|initial]] and [[free object|free]] structures, i.e. [[transfinite construction of free algebras]]/[[quotient inductive types]]

* choice principles

* replacement or collection principles

* smallness structures

just to name a few. Quite a bit of axiomatic fine-tuning can enter when one considers the panoply of hypotheses that might appeal to one or another school of intuitionism or constructivism, or various combinatorial or cardinal hypotheses one might attach to ZFC, etc.

## Properties

### Characterization

The category $Set$ has many marvelous properties, which make it a common choice for serving as a '[[foundations|foundation]]' of mathematics.  For instance:

* It is a [[well-pointed category|well-pointed]] {:yes} [[topos]]{:},

  So in particular it is [[locally cartesian closed category|locally cartesian closed]].

* It is {:yes} [[locally small]]{:}.

* It is [[complete category|complete]] and [[cocomplete category|cocomplete]] (in fact, [[locally finitely presentable]]), and therefore $\infty$-[[extensive category|extensive]] (as is any cocomplete topos).{:data complete="yes" co-complete="yes"}

At least assuming [[classical logic]], these properties suffice to characterize $Set$ uniquely up to equivalence among all categories; see [[cocomplete well-pointed topos]].  Note, however, that the definitions of "locally small" and "(co)complete" presuppose a notion of _small_ and therefore a knowledge of what a _set_ (as opposed to a [[proper class]]) is.

The [[core groupoid]] of $Set$ (whose [[morphisms]] are the [[bijections]]) is characterized by the fact that

* $Core(Set)$ is the [[discrete object classifier]] in the category [[Grpd]] of [[groupoids]] and [[functors]], playing a similar role in classifying discrete groupoids in $Grpd$ that the set of truth values $\Omega$ does in classifying subsets in $Set$. The morphism $F:I \rightarrow Set$ is an [[indexed family]] of sets and $I$ is an index groupoid.

As a [[topos]], $Set$ is also characterized by the fact that

* $Set$ is the [[terminal object]] in the [[category]] of [[Grothendieck toposes]] and [[geometric morphism]]s. The terminal morphism $\Gamma\colon E \to Set$ from any other topos $E$ is the [[global section]]s functor.

It is usually assumed that $Set$ satisfies the [[axiom of choice]] and has a [[natural numbers object]].  In Lawvere's theory [[ETCS]], which can serve as a foundation for much of mathematics, $Set$ is asserted to be a well-pointed topos that satisfies the axiom of choice and has a natural numbers object.  It follows that it is automatically "locally small" and "complete and cocomplete" relative to the notion of "smallness" defined in terms of itself (actually, this is true for any topos).

Conversely, $\Set$ in [[constructive mathematics]] cannot satisfy the axiom of choice (since this implies [[excluded middle]]), although constructivists might accept [[COSHEP]] (that $Set$ has [[projective object|enough projectives]]).  In [[predicative mathematics]], $\Set$ is not even a topos, although most predicativists would still agree that it is a [[pretopos]], and predicativists of the constructive school would even agree that it is a [[locally cartesian closed category|locally cartesian closed]] pretopos.

### Size

Above we considered $Set$ to be the category of _all_ sets, so that in particular $Set$ itself is a [[large category]].  Authors who assume a [[Grothendieck universe]] as part of their [[foundations]] often define $Set$ to be the category of *small* sets (those contained in the universe).  One often then writes $SET$ for the category of *large* sets, which is the [[universe enlargement]] of $Set$.

### Opposite category and Boolean algebras
 {#OppositeCategory}

+-- {: .num_prop}
###### Proposition

The [[power set]]-[[functor]]

$$
  \mathcal{P} \;\colon\; Set \to Bool^{op}
$$

is a [[faithful functor]] which in its [(eso+full, faithful) factorization](http://ncatlab.org/nlab/show/%28eso%2Bfull%2C+faithful%29+factorization+system) induces an [[equivalence of categories]] between $Set$ and the [[opposite category]] of that of [[complete atomic Boolean algebras]].

=--

See for instance [van Oosten, theorem 2.4](#vanOosten)

+-- {: .num_remark}
###### Remark

Restricted to [[FinSet]] this equivalence restricts to an equivalence with finite Boolean algebras. See at _[[Stone duality]]_ for more on this.

=--

+-- {: .num_remark}
###### Remark

In constructive mathematics, $\mathcal{P}$ defines an equivalence of $\Set$ with the opposite category of that of [[complete lattice|complete]] [[atom|atomic]] [[Heyting algebras]]. In fact, for any elementary [[topos]] $\mathcal{E}$, the [[power object]] functor defines an equivalence of $\mathcal{E}$ with the opposite category of that of internal complete atomic Heyting algebras. (This phrase can be interpreted using the [[internal language]] of $\mathcal{E}$.)

=--

## Related categories

* [[Set in homotopy type theory]]

* A variant of $Set$ where [[functions]] are generalized to [[relations]] is _[[Rel]]_.

[[!include categories of categories - contents]]

## References

* [[Saunders MacLane]], §IV.10 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#vanOosten} [[Jaap van Oosten]], _Basic category theory_ ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))

On [[Set in homotopy type theory]]:

* {#RijkeSpitters13} [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics)  ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

category: category

[[!redirects Sets]]
[[!redirects SET]]
[[!redirects category of sets]]
[[!redirects the category of sets]]


set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

# Sets
* table of contents
{: toc}

## What should a set be?

The concept of _set_ appears in several different guises in [[mathematics]], and particularly in [[category theory]].

It is common to use [[foundations of mathematics]] in which 'set' is an undefined term; this is [[set theory]] as a foundation.  In a [[pure set|pure]] [[material set theory]] like [[ZFC]], every object is a set.  Even in a [[structural set theory|structural approach]] such as [[ETCS]], it is common for every object to be a [[structured set]] in some way or another.

Even if sets are not at the bottom of your foundations, still they are probably close.  For instance, in [[type theory]] sometimes sets are defined as [[predicates]] on [[types]], or as [[setoids]].  In [[homotopy type theory]], sets are defined as [[truncated object|0-truncated]] homotopy types; see below.

Material set theory conflates two notions of sets, which were elegantly (but not first) described by [[Mathieu Dupont]] in [a blog post](http://math.breckes.org/2009/03/the-two-meanings-of-the-word-%E2%80%9Cset%E2%80%9D-in-mathematics/) as 'set¹' and 'set²', which we will here call 'abstract set' and 'concrete set'.  In the latter case (set²), we have some fixed [[universe of discourse]] (say, consisting of all real numbers), and a __concrete set__ is a set of elements of this universe (so a set of real numbers, such as $\{5\}$, $\{ x | x \gt 2 \}$, or $\empty$).  In the former case (set¹), an __abstract set__ is a purely abstract concept, an unstructured (except perhaps for the [[equality relation]]) collection of unlabelled elements.  Arguably (this argument goes back at least to [[Bill Lawvere|Lawvere]]), [[Georg Cantor|Cantor]]\'s original concept of [[cardinal number]] (Kardinalzahl) was the abstraction from a concrete set to its underlying abstract set.

Here we adopt a structural approach, in which a __set__ is an abstract set, a (mostly) unstructured [[collection]].  For concrete sets, see [[subset]]; if the fixed universe of discourse is an abstract set $S$, then a concrete set will be a subset of $S$.  In material set theory (as usually conceived), the fixed universe of discourse is an abstract set (or [[proper class]]) $V$, an abstract set (internal to the theory) is an element of $V$, and a concrete set is a sufficiently [[small set|small]] subset of $V$; the concepts may be identified because $V$ comes with an [[bijection|isomorphism]] to its class of small subsets.

## What is a set?

We still need to clarify exactly what sort of collection a set is.  Although most foundations leave this unspecified, we may (perhaps circularly, and perhaps controversially) define it in various ways, as follows:

### In category theory

+-- {: .un_defn #AsCategory}
###### Definition

A __set__ is a [[category]] (or instead an $\infty$-[[infinity-groupoid|groupoid]] or even an $\infty$-[[infinity-category|category]]) that is [[small category|small]], [[discrete category|discrete]], and [[skeletal category|skeletal]]. (This definition needs explanation since it seems to be recursive, as "smallness" refers to what a set is.)
=--
Each of these three adjectives describes a different aspect of what makes something a 'set', and they serve different purposes, which should not be conflated.

That a set is (in the category-theoretic sense) [[discrete category|discrete]] is the most important point; nobody would call a category a 'set' merely because it is small and skeletal.  (This clause is also the reason why it makes no difference whether we start from categories, $\infty$-groupois, or $\infty$-categories; discreteness immediately limits us to $0$-[[0-groupoid|groupoids]].)  The discreteness of a set reflects its lack of internal structure; while a category may have much structure relating its objects, the only structure remaining in a discrete category is whether any two given objects are [[isomorphism|isomorphic]] (and the fact that isomorphism is an [[equivalence relation]]); if (and only if) they are, then they are considered __[[equality|equal]]__ as elements of the set.

That a set is [[small category|small]] allows there to be a collection of 'all' sets; this collection is not itself small, but we still have a [[large category]] of all sets, [[Set]].  A category that is merely discrete and skeletal may be called a __[[class]]__ instead.  But notice that the difference between a set and a class is a rather technical one; a class is just like a set, only (possibly) larger.  Indeed, not everyone would agree with the requirement that a set must be small; although that is probably the most common way of talking to deal with size issues, it is not the only way.  Another way, used by no less an authority than _[[Categories Work]]_, is to posit the existence of a strongly [[inaccessible cardinal]] $\kappa$ and define a __[[small set]]__ to be a set whose [[cardinality]] is less than $\kappa$.

That a set is [[skeletal category|skeletal]] is arguably the least important requirement; in fact, it violates the [[principle of equivalence]].  A category that is merely small and discrete may be called a __[[setoid]]__ instead.  However, if you forbid yourself from referring to [[equality]] of elements of the setoid (which are the objects of the small discrete category) --- or, equivalently, interpret "equality" to mean "isomorphism" therein --- then you cannot distinguish the setoid from a set; each is merely a (small) collection of elements with an equivalence relation (called 'equivalence' in the setoid and 'equality' in the set, in both cases corresponding to isomorphism in a small discrete category).  While [[size issues]] are real and cannot be ignored completely, it is possible to adopt foundations in which the issue of skeletality simply does not appear.

### In homotopy type theory {#InHoTT}

As [[homotopy type theory]] may be viewed as a theory of $\infty$-[[infinity-groupoid|groupoids]], it falls under the preceding section; but since the language of HoTT is important to learn, let us look at it explicitly:
+-- {: .num_defn}
###### Definition

A **set** is a [[type]] in which there is an operation connecting any two parallel [[identity type|paths]] by a 2-path:

    Definition is_set A := forall (x y : A) (p q : x == y), p == q

(There are numerous other equivalent definitions; see [[h-set]].)
=--
Because this operation is defined internally, it acts on paths as well as points and implies that, for any two points of $A$, the [[identity type|type of paths]] between them is [[contractible space|contractible]] as soon as it is [[inhabited]].

Regarding types as [[∞-groupoids]], this definition takes care only of the discreteness requirement.  Regarding smallness, HoTT is usually formulated in [[Martin-Löf type theory]] with [[universes]] ([[type of types]]).  If we have chosen one particular universe as the universe of "small" things, then "setness" could be restricted to types belonging to that universe.

Finally, skeletality is mostly meaningless in HoTT because paths are the only notion of (propositional) [[equality]] which we have to reason with.  (There is also the notion of "definitional" equality, but as we cannot assert or prove statements of the form "$x$ and $y$ are definitionally equal" inside the theory, this is mostly not relevant for the purposes of setness.)

## Related concepts

* [[Set]]

* [[Bishop set]], [[setoid]]

* [[definable set]], [[constructible set]]

* [[recursive subset]]

* [[finite set]], [[hereditarily finite set]]

* [[quantum set]]

[[!include oidification - table]]

[[!include homotopy n-types - table]]

## References

### General

Formalization of sets in [[homotopy type theory]] (via [[h-sets]]):

* [[Egbert Rijke]], [[Bas Spitters]], *Sets in homotopy type theory*, Mathematical Structures in Computer Science **25**  5 "From type theory and homotopy theory to Univalent Foundations of Mathematics" (2015) 1172-1202
[[doi:10.1017/S0960129514000553](https://doi.org/10.1017/S0960129514000553), [arXiv:1305.3835](http://arxiv.org/abs/1305.3835)]

### History

An early definition of "set" appears in

* Bernard Bolzano, _Paradoxien des Unendlichen_ (1847)

where in section 4 it says in translation

> There are wholes which, although they contain the same parts A, B, C, D, ..., nevertheless present themselves as different when seen from our point of view or conception (this kind of difference we call 'essential'), e.g. a complete and a broken glass viewed as a drinking vessel.  [...]  A whole whose basic conception renders the arrangement of its parts a matter of indifference (and whose rearrangement therefore changes nothing essential from our point of view, if only that changes), I call a set.

[[!redirects set]]
[[!redirects sets]]

[[!redirects abstract set]]
[[!redirects abstract sets]]


Set in homotopy type theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The [[syntax]] for [[h-set|sets in]] [[homotopy type theory]] behaves a lot like the [[internal language]] of a [[topos]].

## Internal ##

Chapter 10 in the [[HoTT book]] and [RijkeSpitters](#RijkeSpitters) make this precise. [[Set]] is a [[ΠW-pretopos|$\Pi$W-pretopos]]. Assuming the resizing axiom, [Set]] is actually a [[topos]]. These arguments don't quite work in the absence of [[type universes]], because they use so-called large elimations --- recursively defined functions into a universe. But one can augment [[higher inductive type|(H)]][[inductive type|ITs]] in a universeless theory by special rules for large elims --- in the HIT case, replacing paths in the universe by the equivalences that would correspond to them by univalence -- and then the arguments go through.

## Semantics ##

If all types are sets, we cannot have very many univalent universes. Moreover, not all 1-toposes even contain non-univalent universes (beyond Prop). So we need a version of dependent type theory with a separate judgment "A is a type" that is not identified with "A:U for some universe U". Then we need to assume function extensionality explicitly, since we don't have univalence to derive it from, and we can assume a univalent Prop with resizing. At this point (and without any (H)ITs) we have a theory that's almost just a notational variant of iHOL, hence corresponds very closely to elementary 1-toposes. (In this case the coherence issues are solvable, because the only universe Prop is already strict.) Adding ordinary inductive types corresponds to requiring the topos to have a NNO -- it's known that all inductive types can be constructed from a NNO in a topos.

However, now there is a problem on the other side: not every elementary topos with NNO admits *all* large elims. For instance, with a large elim over nat you can construct [beth-omega](https://en.wikipedia.org/wiki/Beth_number#Beth_omega), which is known not to exist in all toposes (e.g. models of Zermelo set theory). One can  construct large elims "categorically" out of something else by using replacement and separation axioms, however, it's not clear how to express in type theory.

Moreover, with HITs the problem is even worse: even replacement and separation don't suffice. Using HITs we can construct uncountable regular cardinals, whereas (possibly assuming some large cardinal property) there are toposes satisfying replacement and separation (namely, certain models of ZF) in which all uncountable cardinals are singular. (we should check whether this construction requires large elims, but at the least, even without large elims, it's certainly not clear how one might show that every elementary topos admits HITs).

## See also

* [[model of type theory in an (infinity,1)-topos]].

* [[Set]]

### References ###

* {#RijkeSpitters} [[Egbert Rijke]] and [[Bas Spitters]], _Sets in homotopy type theory_ $[$[arXiv:1305.3835](http://arxiv.org/abs/1305.3835)$]$ special issue "From type theory and homotopy theory to univalent foundations" of MSCS.

[[!redirects semantics of Set in homotopy type theory]]


set theoretic Yang-Baxter equation

Set theoretic Yang-Baxter equation is the [[quantum Yang-Baxter equation]] in the Cartesian monoidal category of sets. Thus we talk about set-theoretic solutions of (quantum) Yang-Baxter equation.

Pioneering work

* T. Gateva-Ivanova, [[M. Van den Bergh]], _Semigroups of I-type_, J. Algebra __206__ (1998) 97-112

based on some observations from

* Tatiana Gateva-Ivanova, _Noetherian properties of skew polynomial rings with binomial relations_, Trans. Amer. Math. Soc. __343__ (1994) 203-219, _Skew polynomial rings with binomial relations_, J. Algebra __185__ (1996) 710-753

eventually lead to modern theory of [[brace]]s and skew-braces.

* [[Pavel Etingof]], [[Travis Schedler]], Alexandre Soloviev, _Set-theoretical solutions to the quantum Yang-Baxter equation_,  Duke Math. J. __100__(2) (1999) 169-209 [doi](10.1215/S0012-7094-99-10007-X)
* W. Rump, _Braces, radical rings, and the quantum Yang-Baxter equation_, J. Algebra __307__ (2007), no. 1, 153-170 [doi](https://doi.org/10.1016/j.jalgebra.2006.03.040)
* Tatiana Gateva-Ivanova, [[Shahn Majid]], _Matched pairs approach to set theoretic solutions of the Yang–Baxter equation_, J. Algebra __319__:4, 15 (2008) 1462-1529 [doi](https://doi.org/10.1016/j.jalgebra.2007.10.035)
* F. Cedó, E. Jespers, J. Okniński, _Braces and the Yang–Baxter equation_, Commun. Math. Phys. __327__, 101–116 (2014)  [doi](https://doi.org/10.1007/s00220-014-1935-y)
* Aryan Ghobadi, _Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat_, J. Algebra __586__ (2021) 607-642 [doi](https://doi.org/10.1016/j.jalgebra.2021.07.006)

A dynamical version of set theoretic quantum Yang-Baxter is introduced using category theory in

* Ryosuke Ashikaga, Youichi Shibukawa, _Dynamical reflection maps_, [arXiv:2209.10079](https://arxiv.org/abs/2209.10079)
* Noriaki Kamiya, Youichi Shibukawa, _Dynamical Yang-Baxter maps associated with homogeneous pre-systems_, J. Gen. Lie Theory Appl. __5__ (2011)

category: algebra
[[!redirects set-theoretic Yang-Baxter equation]]

[[!redirects set theoretic quantum Yang-Baxter equation]]


set theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

\tableofcontents

## Idea

A __set theory__ is a theory of [[sets]].

## Naïve vs axiomatic set theory

__Naïve set theory__ is the basic algebra of the [[subsets]] of any given set $U$, together with a few levels of [[power sets]], say up to
 $\mathcal{P}\mathcal{P}\mathcal{P}U$ and possibly no further.  Often students see this first for the set of [[real numbers]] as $U$ (although in fact one could start with the set of [[natural numbers]] and go one level further for an equivalent theory).  One could also use [[function sets]] instead of power sets as the basic set-forming operation (especially for a [[predicative mathematics|weakly predicative]] theory in [[constructive mathematics]]).

Once you start thinking very much about the nature of sets in general (rather than merely _using_ naïve set theory), it quickly becomes clear that one must be careful about how one can and cannot form sets.  [[Georg Cantor]] is credited with being the first to think about sets this deeply; although he did not propose a system of general rules for valid set-making operations, he recognised that some sets were 'inconsistent'.  An __axiomatic set theory__ is a set theory which carefully states the rules (or '[[axioms]]') which sets are assumed to obey.

[[Gottlob Frege]] has perhaps the first axiomatic set theory, but it was found (by [[Bertrand Russell]]) to be logically trivial; see [[Russell's paradox]]. [[Charles Sanders Peirce]] presented what could be regarded as the first consistent axiomatic set theory in [Peirce 1885](#Peirce1885) (see [Brady 2000](#Brady2000)). The more widely known consistent set theory is [[Ernst Zermelo]]'s set theory in 1905, which forms the basis for the axiomatic set theory used today.

## Foundational vs definitional set theory

There are two ways to go about doing axiomatic set theory.  The more ambitious is to develop a __foundational set theory__: set theory as [[foundations]] for all of mathematics.  This is what Frege proposed (although he failed through inconsistency) and which Zermelo achieved.

A variation of Zermelo\'s system (developed by Fraenkel and Skolem and called Zermelo--Fraenkel set theory or [[ZFC]]) is the orthodox foundations today, although it needs to be supplemented by [[Grothendieck universes]] (or something along those lines) to handle modern [[category theory]], and set theorists often consider further strengthenings of it through [[large cardinal]] [[axioms]] (of which the existence of Grothendieck universes is an example, just the tip of the iceberg).

It is also possible to make a __definitional set theory__, in which one defines sets in terms of some more primitive concept. Examples include

* [[class theory]], where a set may be defined as a small [[class]].

* [[formal category theory]], where a set may be defined as a [[discrete category]].

* [[dependent type theory]], where a set may be defined as those [[types]] which are [[h-sets]].

* [[preset theory]], where a set may be defined as a preset equipped with an [[equivalence relation]].

In [[computer science]], a foundation based on the [[lambda-calculus]] is sometimes seen; in these terms, the concept of _[[list]]_ is more natural than set, with the difference being that sets have a coarser notion of equality.

## Constructive vs classical set thoery

There are also a difference in the set theory, depending on whether the set theory is built in [[classical logic]] or in [[constructive logic]]. In the latter, the set theory is called **[[constructive set theory]]**. In order to make sure that classical logic isn't implied by the axioms, certain axioms, such as the [[axiom of choice]], cannot be accepted in classical logic.

## Material vs structural set theory
   {#MaterialSetTheory}

On the $n$Lab we like to distinguish between two types of set theory, especially in foundations:

*  In a __[[material set theory]]__ (also called a _membership-based set theory_), the elements of a set exist independently of that set.  As such, it makes sense to take two completely unrelated sets and ask if two of their members are [[equal]], and sometimes the answer will be Yes.  Frequently in material set theory one takes everything to be a [[pure set]], so that elements of sets are themselves sets.  Therefore, any two sets may be meaningfully compared to ask if they are equal or if one is a member of the other.

*  A __[[structural set theory]]__, on the other hand, looks more like [[type theory]].  Here, the elements of a set have no existence or structure apart from their identity as elements of that set.  In particular, they are not themselves sets, and cannot be elements of any *other* set, at least not without invoking some explicit [[type casting|type-casting]] operator (which here could simply be a [[function]] from one set to the other).  Similarly, elements of different sets cannot be compared to each other (without type-casting them to become elements of the same set).

Among category theorists, it\'s popular to state the axioms of a structural set theory by specifying elementary properties of [[Set|the category of sets]]; the orthodoxy here (to the extent that there is one) is probably [[Bill
Lawvere]]\'s [[ETCS]], which suffices for most everyday uses but must be supplemented to handle some esoteric parts of modern mathematics.  Another structural set theory, which is stronger than ETCS and less closely tied to category theory, is [[SEAR]].

In contrast, [[ZFC]] is an example of a material set theory.  From a model of either kind of set theory we can construct a model of the other, so the two are, broadly speaking, equivalent; for example, SEARC (SEAR with the [[axiom of choice]]) is equivalent in this way to ZFC, while ETCS is equivalent to a weak ('bounded') variation of ZFC.  A more precise statement is that the two kinds of theories form categories related by the [[material-structural adjunction]].

## Category theory on set theory

Category theory can provide a common [[model theory]] to compare various set theories.  Although only structural set theories like ETCS treat the elementary properties of the category $Set$ of sets as fundamental, one can ask for any set theory what properties $Set$ satisfies and compare them in those terms.  At the very least, $Set$ should be a [[pretopos]].

There is also [[algebraic set theory]], in which a material set theory is interpreted in the [[internal logic]] of some [[ambient category]], often called a "category of classes".  See also [[stack semantics]].

## Related concepts

* [[foundational axiom]]

* [[type theory]]

* [[constructive set theory]]

* [[algebraic set theory]]

* [[descriptive set theory]]

* [[internal logic of set theory]]

* [[set-theoretic multiverse]]

* [[foundations of mathematics]]

  * [[New Foundations]]

  * [[univalent foundations of mathematics]]

## Literature

### Historical

* {#Peirce1885} [[Charles Sanders Peirce]], *On the Algebra of Logic: A Contribution to the Philosophy of Notation*, American Journal of Mathematics **7** (1885)

* [[Felix Hausdorff]], *[[Grundzüge der Mengenlehre]]*, Leipzig: Veit (1914) [:ISBN:978-0-8284-0061-9] Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [[ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)]

* {#Brady2000} Geraldine Brady (2000), From Peirce to Skolem: A Neglected Chapter in the History of Logic, North-Holland/Elsevier Science BV, Amsterdam, Netherlands.

### General

* [[Kenneth Kunen]], _Set theory,an introduction to independence proofs_, North-Holland.

* {#LawvereRosebrugh} [[William Lawvere]], [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , Cambridge UP  2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))

* [[Andreas Blass]], [[Yuri Gurevich]], *Why Sets?*, Bull. Europ. Assoc. Theoret. Comp. Sci. **84** (2004) 139-156.  [[doi:10.1007/978-3-540-78127-1_11](https://doi.org/10.1007/978-3-540-78127-1_11), [pdf](https://web.eecs.umich.edu/~gurevich/Opera/172.pdf), [pdf](http://www.math.lsa.umich.edu/~ablass/set.pdf)]

  > (with commentary on [[ETCC]])

* {#Shulman18} [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

### In homotopy type theory
 {#ReferencesInHomotopyTypeTheory}

Formalization of set theory in [[homotopy type theory]] (via [[h-sets]]) is discussed in

* [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_ ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

* [[Univalent Foundations Project]], section 3 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

* Jérémy Ledent, _Modeling set theory in homotopy type theory_ (2014) ([pdf](http://perso.ens-lyon.fr/jeremy.ledent/internship_report.pdf))

* Håkon Robbestad Gylterud, Elisabeth Bonnevier, _Non-wellfounded sets in homotopy type theory_ ([arXiv:2001.06696](https://arxiv.org/abs/2001.06696))

See also at [[homotopy type theory FAQ]] _[What is HoTT? -- For set theorists](homotopy+type+theory+FAQ#WhatIsHoTTForSetTheorists)_.

[[!redirects set theory]]
[[!redirects set theories]]