filtered complex

> This needs to be merged with _[[filtered chain complex]]_

### Idea

It's easy to say what chain complex and homology mean (that is, these notions are definable); where things get tricky is, when calculating them, to figure out what the modules and differentials, kernels and images actually _are_.  Sometimes there's extra structure, e.g. a further hierarchy beyond the usual grading, that lets us figure these things out one layer at a time.  Then we have to glue the layers back together, and that's one place a [[spectral sequence]] is handy

### Definition

For a [[poset]] $I$ and an [[abelian category]] $A$, an $I$-**filtered complex** is a functor $F:I\to Mon(Ch(A))$, from $I$ to monomorphisms of chain complexes in $A$.  Roughly, this boils down to

* A Complex $d:C\to C$, $d^2=0$
* with a submodules $F_i \lt C$ for $i$ an object of $I$
* such that $F_i \lt F_{j} $ for $i\lt j$ in $I$
* such that $ d F_i \lt F_i $.

(You may have noticed this isn't the usual notation for functors.  It's traditional.)

The most frequent examples have $I=\mathbb{N}\cup\{\infty\}$, $I=-\mathbb{N}$, or $I=\mathbb{Z}\cup\{\infty\}$, with their usual total orderings; in this connection see [[spectral sequence of a filtered complex]]

Usually $C$ is a _graded_ complex, with $d_j:C_j\to C_{j-1}$, and in this case we ask
$$d_j:F_i\cap C_j \to F_i\cap C_{j-1}.$$
(If you prefer cohomology differentials, read $+$ for $-$.)

### Associated Graded complex

In the special case of a discrete totally-ordered filtration, one defines the associated graded complex $G_i(F) = F_{i+1}/F_i$ with differential induced by $d[x] = [d x]$; again, if $(C,d)$ is graded, we have a bigraded complex with components $G_i\cap C_j$ and differential of bidegree $(\pm 1, 0)$.

### References

Any book introducing spectral sequences.


filtered derived category

## Idea

Given an [[abelian category]], the _filtered derived category_ is the category of [[chain complexes]] of [[filtered objects]] up to [[quasi-isomorphism]].

## Definition

Given an [[abelian category]] $A$, one may consider the [[additive category]] $Fil(A)$ of [[filtered objects]] in $A$, whose objects are pairs $(a, F^*)$ with $a$ an object of $A$ and $F^*$ a [[filtration]] on $a$.
Let $Comp(Fil(A))$ denote the category of [[chain complexes]] in $Fil(A)$.
One defines a morphism in $Comp(Fil(A))$ to be a **quasi-isomorphism** if it induces [[quasi-isomorphisms]] on each degree of the [[associated graded objects]].
The **filtered derived category** of $A$,
  $$ D_{fil}(A) = Comp(Fil(A))[qis^{-1}] $$
is the [[localization]] of $Comp(Fil(A))$ at the class of [[quasi-isomorphisms]].

## See also

* [[derived category]]
* [[filtered chain complex]]

## References

* [[Stacks Project]], [Tag 03T9](http://stacks.math.columbia.edu/tag/03T9).

[[!redirects filtered derived categories]]
[[!redirects filtered quasi-isomorphism]]
[[!redirects filtered quasi-isomorphisms]]


filtered group

## Definition

A [[filtered object]] in the [[category of groups]].

## Related concepts

* [[Malcev group]]

* [[complete Hopf algebra]]


filtered homotopy colimit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[homotopy colimits]] of [[filtered diagrams]]

## Characterization

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

In a [[combinatorial model category]], for every sufficiently large  [[regular cardinal]] $\kappa$ the following holds:

* $\kappa$-[[filtered colimits]] preserve weak equivalences;

* hence $\kappa$-[[filtered colimits]] are already [[homotopy colimits]].

=--

This appears as ([Dugger 00, prop 7.3](#Dugger00)).

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

The point is to choose $\kappa$ such that all domains and codomains of the generating cofibrations are $\kappa$-[[compact object]]. This is possible since by assumption that $C$ is a [[locally presentable category]] all its objects are [[small objects]], hence each a $\lambda$-[[compact object]] for some cardinal $\lambda$. Take $\kappa$ to be the maximum of these.

Let $F, G : J \to C$ be $\kappa$-filtered diagrams in $C$ and $F \to G$ a [[natural transformation]] that is degreewise a weak equivalence. Using the functorial factorization provided by the [[small object argument]] this may be factored as $F \to H \to G$ where the first transformation is objectwise an acyclic cofibration and the second objectwise an acyclic fibration, and by functoriality of the factorization this sits over a factorization

$$
  \lim_\to F \stackrel{\simeq}{\hookrightarrow} \lim_\to H \stackrel{}{\to}\lim_\to G
  \,.
$$

It remains to show that the second morphism is a weak equivalence. But by our factorization and by [[category with weak equivalences|2-out-of-3]] applied to our componentwise weak equivalences, we have that all its components $H(j) \to G(j)$ are acyclic fibrations.

At [[small object]] it is described in detail how $\kappa$-smallness of an object $X$ implies that morphisms from $X$ into a $\kappa$-filtered colimit lift to some component of the colimit

$$
  \array{
    \cdots&\to&H(j-1) &\to& H(j) &\to& H(j+1) &\to& \cdots
    \\
    &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow
    \\
    &&X& \stackrel{\forall f}{\to} &\lim_\to H
  }
  \,.
$$

So given a diagram

$$
  \array{
    X &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow
    \\
    Y &\to& \lim_\to G
  }
$$

we are guaranteed, by the $\kappa$-[[small object|smallness]] of $X$ and $Y$ that we established above, a lift

$$
  \array{
    X &\to& H(j) &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} && \downarrow^{\in \mathrlap{\in rlp(I)}}
    &&
    \downarrow
    \\
    Y &\to& G(j) &\to& \lim_\to G
  }
$$

into some component at $j \in J$ and hence a lift

$$
  \array{
    X &\to& H(j) &\to& \lim_\to H
    \\
    \downarrow^{\mathrlap{\in I}} &
    \nearrow
    & \downarrow^{\in \mathrlap{\in rlp(I)}}
    &&
    \downarrow
    \\
    Y &\to& G(j) &\to& \lim_\to G
  }
  \,.
$$

Thereby $\lim_\to H \to \lim_\to G$ is in $rlp(I) \subset W$.

=--

+-- {: .num_cor}
###### Corollary

In the situation of prop \ref{CharacterizationOfFilteredHomotopyColimits},
since finite [[homotopy limits]] are given, after [[fibrant resolution]], by [[finite limits]], it follows from the ordinary commutativity of [[filtered colimits]] with finite limits that also filtered homotopy colimits commute with finite homotopy limits.
=--

## Related concepts

* [[filtered (∞,1)-colimit]]

## References

* {#Dugger00} [[Daniel Dugger]], _Combinatorial model categories have presentations_ ([arXiv:math/0007068](http://arxiv.org/abs/math/0007068))

[[!redirects filtered homotopy colimits]]


filtered Lie algebra

## Definition

A [[filtered object]] in the [[category of Lie algebras]].

## Related concepts

* [[Malcev Lie algebra]]

* [[complete Hopf algebra]]


filtered limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Most of the earliest instances of [[limits]] and [[colimits]] used in mathematics were for [[diagrams]] indexed by the [[partially ordered set]] of [[natural numbers]], which we now call _[[sequential limit|sequential]] (co)limits_.  Many of the nice features of these (co)limits apply more generally to _[[codirected limits]]_ and _[[directed colimits]]_, where the indexing category is a (co)-[[directed set]] (much as [[sequences]] in [[topology]] generalise fruitfully to [[nets]]).  A _filtered (co)limit_ is a further generalisation of this, essentially removing the requirement that the indexing category be a [[poset]] while preserving the directedness aspect in a [[categorification|categorified]] way.

Another very important class of early limits and colimits involved situations that generalised [[intersections]] and [[unions]].  If one is looking at a [[family of subsets]] of some [[set]], then one can close it up under finite intersections and/or unions (if they are not already included) and use it to index diagrams.  For instance, the family of [[continuous functions]] defined on [[open neighbourhoods]] of some point in a [[topological space]] will have this property.  It was noticed that these limits and colimits behaved very nicely and a closer look showed that it was the _(co)filtering_ nature of the indexing category that was the key.  This also leads us to filtered (co)limits.

So, a _filtered colimit_ is a [[colimit]] over a [[diagram]] from a [[filtered category]], and a _cofiltered limit_ (sometimes called a filtered limit) is a [[limit]] over a [[diagram]] from a [[cofiltered category]]. Taken in a suitable category such as [[Set]], **a colimit being filtered is equivalent to its commuting with [[finite limits]]**.

More generally, for $\kappa$ a [[regular cardinal]], a _$\kappa$-filtered colimit_ is one over a $\kappa$-filtered category (and dually), and when taken with values in [[Set]] these are precisely the colimits that commute with $\kappa$-[[small limit]]s.

## Definition

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

A __filtered colimit__ or **finitely filtered colimit** is a [[colimit]] of a [[functor]] $F\colon D \to C$ where $D$ is a [[filtered category]].

For $\kappa$ a [[regular cardinal]] a **$\kappa$-filtered colimit** is one over a $\kappa$-filtered category.

Similarly, a __cofiltered limit__ is a [[limit]] of a functor $F\colon D \to C$ where $D$ is a [[cofiltered category]], or equivalently of a [[contravariant functor]] $F\colon D \to C$ (that is a functor $F\colon D^{op} \to C$) where $D$ is a filtered category.

=--

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

A cofiltered limit may also be called a __filtered limit__ (although this can be unclear); the respective terms __filtered [[direct limit]]__ and __filtered [[inverse limit]]__ are also popular.

=--

A [[functor]] that preserves all finitely filtered colimits is called a _[[finitary functor]]_ .

## Properties
 {#Properties}

### $\kappa$-Filtered colimits commute with $\kappa$-small limits

The following is the crucial property of filtered colimits: that they [[commutativity of limits with colimits|commute]] with [[finite limits]].

\begin{remark}
For $C$ and $D$ two [[diagram]] categories and

$$
  F \,\colon\, C \times D \longrightarrow Set
$$

a [[diagram]], there is a canonical morphism

$$
  \lambda
  \,\colon\,
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \;
  \underset{\underset{D}{\longleftarrow}}{\lim}
  \,
  F
  \longrightarrow
  \underset{\underset{D}{\longleftarrow}}{\lim}
  \;
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \,
  F
$$

from the [[colimit]] over $C$ of the [[limit]] over $D$ to the limit over $D$ of the colimit over $C$ of $F$:

$\lambda$ is given by a [[cone]], whose components

$$
  \lambda_d
  \,\colon\,
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \;
  \underset{\underset{D}{\longleftarrow}}{\lim}
  \,
  F
    \longrightarrow
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \,
  F(-,d)
$$

are in turn given by a [[cocone]] with components

$$
  (\lambda_d)_c
  \,\colon\,
  \underset{\underset{D}{\longleftarrow}}{\lim}
  \,
  F(c,-)
    \longrightarrow
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \,
  F(-,d)
  \,.
$$

This finally take to have as components

$$
  \underset{\underset{D}{\longleftarrow}}{\lim}
  \,
  F(c,d)
    \longrightarrow
  F(c,d)
    \longrightarrow
  \underset{\underset{C}{\longrightarrow}}{\lim}
  \,
  F(c,d)
  \,.
$$

One checks that this indeed implies that all the components are natural and gives the existence of  the original morphism.

Notice that in general $\lambda$ is _not_ an [[isomorphism]].
\end{remark}

\begin{definition}
We say the [[limit]] ${\lim_\leftarrow}_D F(-,-)$ **[[commutativity of limits with colimits|commutes]]** with the colimit ${\lim_\to}_C F(-,-)$ if the morphism $\lambda$ above is an [[isomorphism]]

$$
  {\lim_\to}_C {\lim_\leftarrow}_D F
   \stackrel{\simeq}{\longrightarrow}
  {\lim_\leftarrow}_D {\lim_\to}_C  F
  \,.
$$
\end{definition}

\begin{proposition}
\label{FilteredColimitsCommuterWithFiniteLimits}
In [[Set]], filtered colimits [[commutativity of limits with colimits|commute]] with [[finite limits]].

In fact, [[filtered categories]] $C$ are precisely those shapes of [[diagram]] categories such that colimits over them commute with all finite limits in Sets.

More generally, for $\kappa$ a [[regular cardinal]], then $\kappa$-filtered colimits commute with $\kappa$-small limits.
\end{proposition}

Proof of the first part is in [MacLane (1971), §IX.2 Thm. 1 (p. 211)](#MacLane71), [Borceux (1994), theorem I2.13.4](#Borceux). For more discussion see also [BJTS 14](#BJTS14),
and see also *[[limits and colimits by example]]*.

\begin{remark}
It is not true that filtered colimits and finite limits commute in *any* category $C$ which has them. A simple example is where $C$ is the [[poset]] of [[closed subspaces]] of the [[one-point compactification]] $\mathbb{N} \cup \{\infty\}$ of the [[discrete space]] of [[natural numbers]]. If $A = \{\infty\}$ and $B_i$ ranges over finite subsets of $\mathbb{N}$, then $A \times colim_i B_i = \{\infty\} \cap (\mathbb{N} \cup \{\infty\}) = \{\infty\}$, but $colim_i A \times B_i = colim_i \{\infty\} \cap B_i = colim_i \varnothing = \varnothing$.
\end{remark}

According to 1.5 and 1.21 in [[LPAC]], a category has $\kappa$-[[directed colimits]] precisely if it has $\kappa$-filtered ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones. A proof of this result, following Adamek & Rosicky, may be found [[directed colimits imply filtered colimits|here]].

The fact that directed colimits suffice to obtain all filtered ones may be regarded as a convenience similar to the fact that all [[colimits]] can be constructed from [[coproducts]] and [[coequalizers]].  Of course, a [[duality|dual]] result holds for [[codirected limits]].

### Flat functors and points of presheaf toposes

Let $C$ be a small category. A functor $F: C \to Set$ is **[[flat functor|flat]]** if it is a filtered colimit of [[representable functor]]s.

Equivalently, a [[module]] $F: C \to Set$ is flat if and only if the [[tensor product of functors|tensor product]]

$$- \otimes_C F: Set^{C^{op}} \to Set$$

is [[left exact functor|left exact]]. One may prove as a corollary that if $C$ is [[finitely complete category|finitely complete]], $F$ is flat if and only if it is [[left exact functor|left exact]] (preserves finite limits). Since this tensor product is automatically a left adjoint, we have the following basic result:

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

For $C$ a [[small category]], the category of [[point of a topos|topos points]] of the [[presheaf topos]] $Set^{C^{op}}$ (i.e., [[geometric morphism]]s $Set \to Set^{C^{op}}$ and natural transformations between them) is equivalent to the category of flat modules on $C$.

=--

### Description in Set, Grp, Top and alike

Elements in filtered colimits in [[Set]] and [[Grp]] are given as classes of equivalences, so called [[germs]]. Filtered limits in [[Set]] and [[Top]] are given as families of compatible elements, so called [[threads]].

### More

(More was/is to be written here, including an application to [[geometric realization]], relation to [[Diaconescu's theorem]], and perhaps more.)

## Related concepts

* [[filtered category]], [[compact object]]

* [[sifted colimit]], [[sifted (∞,1)-colimit]]

* [[filtered (∞,1)-colimit]], [[filtered homotopy colimit]]

* [[Kolmogorov product]]

Filtered colimits are also important in the theory of [[locally presentable category|locally presentable]] and [[accessible category|accessible]] categories.  See also [[pro-object]] and [[ind-object]].

## References

* {#MacLane71} [[Saunders MacLane]], §IX.2 in: *[[Categories for the Working Mathematician]]*, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux} [[Francis Borceux]], Section 2.13 in: _[[Handbook of Categorical Algebra]] vol 1_

Also

* {#BJTS14} [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]

[[!redirects filtered limit]]
[[!redirects filtered limits]]
[[!redirects cofiltered limit]]
[[!redirects cofiltered limits]]
[[!redirects filtered colimit]]
[[!redirects filtered colimits]]
[[!redirects filtered inverse limit]]
[[!redirects filtered inverse limits]]
[[!redirects cofiltered inverse limit]]
[[!redirects cofiltered inverse limits]]
[[!redirects filtered direct limit]]
[[!redirects filtered direct limits]]


filtered object

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

#Contents#
* table of contents
{:toc}

## Definition

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

Given a [[category]] $\mathcal{C}$,  then a **filtered object** is an [[object]] $X$ of $\mathcal{C}$ equipped with a _filtration_:

A **descending filtration** or **decreasing filtrations** of $X$ is a sequence of [[morphisms]] (often required to be [[monomorphisms]]) of the form

  $$
    \cdots
     \longrightarrow
    X_{n+1}
     \longrightarrow
    X_n
     \longrightarrow
    X_{n-1}
      \longrightarrow
    \cdots
      \longrightarrow
    X
  $$

An **ascending filtration** or **increasing filtration** of $X$ is of the form

  $$
    X
      \longrightarrow
    \cdots
      \longrightarrow
    X_{n-1}
      \longrightarrow
    X_n
      \longrightarrow
    X_{n+1}
      \longrightarrow
    \cdots
  $$

=--

(In more generality, it is also possible to index using any [[ordered group|ordered abelian group]].)

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

A decreasing filtration $\{X_s\}_s$ of $X$ (def. \ref{Filtration}) is called

* **exhaustive** if $\underset{\longrightarrow}{\lim}_s X_s \simeq X$ ($X$ is the [[colimit]] of the filter stages)

* **Hausdorff** if $\underset{\longleftarrow}{\lim}_s X_s \simeq 0$ (the [[limit]] of the the filter stages is [[initial object|initial]] ([[zero object|zero]] in the case of an [[abelian category]]))

and for a filtration of [[abelian groups]]:

* **complete** if $\underset{\longleftarrow}{\lim}^1_s X_n = 0$ (also the first [[derived functor|derived]] [[limit]] ([[lim^1]]) vanishes)

=--

([Boardman 99, def. 2.1](#Boardman99), see also [Rognes 12, section 2.1](#Rognes12))

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

Beware that for decreasing filtrations (def. \ref{Filtration}) often exhaustiveness (def. \ref{ExhaustiveHausdorffAndCompleteFiltrations}) is understood by default, and often it is even assumed by default that $X_s = X$ for $s \leq 0$.

=--

## Properties

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

If a decreasing filtration (def. \ref{Filtration}) of [[abelian groups|abelian]] [[subgroups]]

$$
  \cdots \hookrightarrow A_{n+1} \hookrightarrow A_n \hookrightarrow A_{n-1} \hookrightarrow \cdots \hookrightarrow A
$$

is exhaustive and complete Hausdorff (def. \ref{ExhaustiveHausdorffAndCompleteFiltrations}) then $A$ may be reobtained from the subquotients of the filtering as the [[limit]]/[[colimit]]

$$
  \begin{aligned}
    A
      & \simeq
    \underset{\longleftarrow}{\lim}_s (A/A_s)
    \\
    &
      \simeq
    \underset{\longleftarrow}{\lim}_s
    (\underset{\longrightarrow}{\lim}_t
     ( A_t / A_s ))
  \end{aligned}
  \,.
$$

=--

([Boardman 99, prop. 2.5](#Boardman99))

## Related concepts

[[!include filtered objects -- contents]]

## References

* {#Boardman99} [[Michael Boardman]], section I.2 of _Conditionally convergent spectral sequences_, 1999 ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf))

* {#Rognes12} [[John Rognes]], _The Adams spectral sequence_ (following [Bruner 09](Adams+spectral+sequence#Bruner09)), 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

[[!redirects filtered objects]]

[[!redirects filtration]]
[[!redirects filtrations]]

[[!redirects filtering]]
[[!redirects filterings]]

[[!redirects exhaustive filtering]]
[[!redirects exhaustive filterings]]

[[!redirects Hausdorff filtering]]
[[!redirects Hausdorff filterings]]

[[!redirects complete filtering]]
[[!redirects complete filterings]]

[[!redirects exhaustive filtration]]
[[!redirects exhaustive filtrations]]

[[!redirects Hausdorff filtration]]
[[!redirects Hausdorff filtrations]]

[[!redirects complete filtration]]
[[!redirects complete filtrations]]

[[!redirects complete Hausdorff filtration]]
[[!redirects complete Hausdorff filtrations]]


filtered object in an (∞,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[(∞,1)-category theory]] the typical notion of [[filtered object]] in [[category theory]] simplifies: instead of a [[sequential diagram]] of [[monomorphisms]], it is just any [[tower diagram]].

## Definition

Let $\mathcal{C}$ be an [[(∞,1)-category]].

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

A _filtering_ on an [[object]] $X \in \mathcal{C}$ is a [[sequential diagram]] $X_\bullet \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

$$
  \cdots
  X_{n-1} \to X_n \to X_{n+1} \to \cdots
$$

such that

$$
  X \simeq \underset{\rightarrow}{\lim} X_\bullet
$$

is the [[sequential colimit|sequential]] [[homotopy colimit]] of the tower.

Dually a _cofiltering_ of $X$ is a tower $X_\bullet$ such that

$$
  X \simeq \underset{\leftarrow}{\lim} X_\bullet
$$

is the [[homotopy limit]].

=--

This appears as ([[Higher Algebra|Higher Algebra, def. 1.2.2.9]]).

## Applications

### Spectral sequence

If $\mathcal{C}$ is a [[stable (∞,1)-category]] with [[sequential limits]]/[[sequential colimits]] and with a [[t-structure]], then every filtering/cofiltering on $X$ induces a [[spectral sequence of a filtered stable homotopy type]] which converges to the [[homotopy groups]] of $X$.

The [[spectral sequence of a filtered stable homotopy type]] associated with the chromatic tower (regarded as a [[filtered object in an (infinity,1)-category]]) is the _[[chromatic spectral sequence]]_ ([Wilson 13, section 2.1.2](#Wilson13))

## Related concepts

* [[filtered chain complex]]

* [[filtered spectrum]]

* [[tower of homotopy fibers]]

* [[persistent homotopy theory]]

[[!include Lurie spectral sequences -- table]]

\linebreak

[[!redirects filtered objects in an (∞,1)-category]]

\linebreak

[[!include filtered objects -- contents]]

[[!redirects filtered objects in an (∞,1)-category]]

[[!redirects filtered object in an (infinity,1)-category]]
[[!redirects filtered objects in an (infinity,1)-category]]

[[!redirects filtered object of an (∞,1)-category]]
[[!redirects filtered objects of an (∞,1)-category]]

[[!redirects filtered object of an (infinity,1)-category]]
[[!redirects filtered objects of an (infinity,1)-category]]


filtered objects -- contents

**[[filtered objects]]**

* [[filtered topological space]]

  * [[spectral sequence of a tower of fibrations]]

* [[filtered vector space]]

* [[filtered chain complex]]

  * [[spectral sequence of a filtered complex]]

  * [[Hodge filtration]]

* [[filtered ring]]

* [[filtered object in an (∞,1)-category]]

* [[filtered stable homotopy type]]

  * [[spectral sequence of a filtered stable homotopy type]]

**[[associated graded objects]]**

* [[associated graded vector space]]

* [[associated graded ring]]


filtered ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A __filtered ring__ (resp. filtered [[associative algebra|algebra]]) is a [[monoid object]] in the [[category]] of filtered abelian groups
(resp. [[filtered vector space]]s).

One considers positive and negative filtrations, as well as $\mathbb{Z}$-filtrations.

To-do list: complete filtrations, associated graded ring, symbol map, Poisson structure on the associated graded algebra if the latter is commutative.

## Examples

A major example is the [[universal enveloping algebra]] of any [[Lie algebra]].

## Related concepts

See also [[Lazard's criterion]] and [[microlocalization]].

\linebreak

[[!include filtered objects -- contents]]

## References

* wikipedia [filtration (mathematics)](http://en.wikipedia.org/wiki/Filtration_%28mathematics%29), [completion (ring_theory)](http://en.wikipedia.org/wiki/Completion_%28ring_theory%29)

[[!redirects filtered algebra]]
[[!redirects filtered algebras]]
[[!redirects filtered rings]]


filtered spectrum

#Contents#
* table of contents
{:toc}

## Idea

A _filtered spectrum_ is a [[filtered object in an (∞,1)-category]] in the [[stable (∞,1)-category of spectra]].  See there for more.

## Related concepts

* [[filtered chain complex]]

* [[tower of homotopy fibers]]

[[!include Lurie spectral sequences -- table]]

[[!redirects filtered spectra]]


filtered topological space

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

# Contents
* table of contents
{: toc}

## Definition

A **filtered topological space** $X_*$ is a [[filtered object]] in [[Top]], hence

1. a [[topological space]] $X=X_\infty$

2. equipped with a [[sequence]] of [[subspaces]]

   $$X_*:= \quad X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X_\infty. $$

A filtered space $X_*$ is called a **[[connected filtered space]]** if it satisfies the following condition:

$(\phi)_0$: The function $\pi_0X_0  \to \pi_0 X_r$ induced by  inclusion is surjective for all $r \geq 0$; and, for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$   for all $r \gt i$  and $ v \in X_0$.

There are two other forms of this condition which are useful under different circumstances.

## Examples

1. A [[CW-complex]] $X$ with its filtration by [[simplicial skeleton|skeleta]] $X^n$.

1. A [[configuration space of points]] is filtered by number of points.

2. The [[free construction|free]] [[topological monoid]] $F X$ on a [[topological space]] $X$ filtered by the length of words. Given a [[pointed topological space]] $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter filtered space is known as the _[[James construction]]_ $J(X,x)$ ([James 55](#James55)).

   The [[James construction]] $J(X,x)$ may be constructed [[homotopy theory|homotopy-theoretically]] ([Brunerie 13](#Brunerie13), [Brunerie 17](#Brunerie17)). Recall that, for $K$ a finite [[simplicial complex]], for $(X,A)$ a pair of spaces, its _polyhedral product_ $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.

   +-- {: .num_defn}
   ###### Definition
   For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows.
   Set $\fil_0$ as $\{x\}$.
   For $k\ge 1$, require that the following square is homotopy pushout:
   $$
     \array{
    &&&&
        (X,x)^{\partial \Delta[k-1]}
        &&&&
         \\
         &
         && inc \swarrow
          &
          & \searrow
         &&
        \\
   && X^k &&&& fil_{k-1}
   \\
         &
         && {}_{p_k}\searrow
          &
          & \swarrow_{j_k}
         &&
        \\
   &&&&
        fil_k
        &&&&
     }
   $$
   where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps
   $$(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}$$
   for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex.
   =--

   +-- {: .num_prop #TypeOfJames}
   ###### Proposition
    For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.
   =--

3. A similar example to the last using free groups instead of free monoids.

4. A similar example to the last using free groupoids on topological graphs.

5. A similar example to the last using the universal topological groupoid $U_\sigma(G)$ induced from a topological groupoid $G$ by a continuous function $f: Ob(G) \to Y$ to a space $Y$.

Examples of [[connected filtered spaces]] are:

1. The skeletal filtration of a CW-complex.

2. The word length filtration of the James construction for a space with base point such that $\{x\} \to X$ is a closed cofibration.

3. The filtration $(B C)_*$ of the classifying space of a crossed complex, filtered using skeleta of $C$.

This condition occurs in the [[higher homotopy van Kampen theorem]] for [[crossed complex]]es.

## Properties

We thus see that filtered spaces arise from many geometric and algebraic situations, and see also [[stratified space]]s).
It is  therefore interesting that one can define strict higher homotopy groupoids for filtered spaces more easily than for spaces themselves.

Note also that it is standard to be able to replace, using mapping cylinders, a sequence of maps $Y_n \to Y_{n+1}$ by a sequence of inclusions.

## Related concepts

* [[stratified space]]

[[!include filtered objects -- contents]]

## References

The [[James construction]] is due to

* {#James55} [[Ioan James]], _Reduced product spaces_, Annals of Mathematics, Second Series, 62: 170–197 (1955)

Review of the [[James construction]] includes

* [[Dylan Wilson]], _James construction_, 2017 ([pdf](http://www.math.uchicago.edu/~dwilson/pretalbot2017/james-construction.pdf))

* Wikipedia, _[James reduced product](https://en.wikipedia.org/wiki/James_reduced_product)_

Discussion of the [[James construction]] via [[homotopy type theory]] includes the following

* {#Brunerie13} [[Guillaume Brunerie]], _The James Construction and $\pi_4(S^3)$_, talk at the Institute of Advanced Studies on March 27, 2013 ([recording](https://video.ias.edu/univalent/1213/0327-GuillaumeBrunerie))

* {#Brunerie17} [[Guillaume Brunerie]], _The James construction and $\pi_4(S^3)$ in homotopy type theory_ ([arXiv:1710.10307](https://arxiv.org/abs/1710.10307))

[[!redirects filtered topological space]]
[[!redirects filtered topological spaces]]
[[!redirects filtered space]]
[[!redirects filtered spaces]]


filtered vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _filtered vector space_ is a [[filtered object]] in a [[category]] [[Vect]] of [[vector spaces]].

## Related concepts

[[!include filtered objects -- contents]]

[[!redirects filtered vector spaces]]


filtrality

# Contents
* table of contents
{: toc}

## Idea

For an object $X$ of a [[coherent category]] $C$, consider $\mathcal{B}(X)$, the Boolean algebra of complemented subobjects of $X$, and then $\mathcal{F}(\mathcal{B}(X))$, the set of all non-empty [[filters]] of $\mathcal{B}(X)$, partially ordered by reverse inclusion.

There is a canonical (bounded) [[lattice]] homomorphism

$$\phi_X: Sub(X) \to \mathcal{F}(\mathcal{B}(X))$$

taking a subobject $A \hookrightarrow X$ to the filter of complemented subobjects over it. Then $X$ is said to be **filtral** if $\phi_X$ is an isomorphism.

Furthermore, the category $C$ is said to be **filtral** if each of its objects is covered by a filtral one, that is, for every $Y$ in $C$ there is a [[regular epimorphism]] with $X$ filtral ([Marra-Reggio 18, Sec. 4](#MarraReggio)).

## Examples

* The category of [[compacta]]

(Presumably because free compact Hausdorff spaces $\beta X$ are filtral.)

## References

* {#MarraReggio} Vincenzo Marra, Luca Reggio, _A characterisation of the category of compact Hausdorff spaces_, ([arXiv:1808.09738](https://arxiv.org/abs/1808.09738))


filtrant category > history

< [[filtrant category]]
[[!redirects filtrant category -- history]]


filtration by support

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[filtered object|filtered]] [[space]] entities like [[quasicoherent sheaves]] over $X$ get induced a filtering by their [[support]] on the [[strata]] of $X$.

## Examples

* The [[height filtration]] on the [[moduli stack of formal groups]] induces a filtration by support. The corresponding [[spectral sequence of a filtered stable homotopy type]] appears

## Related concepts

[[!include Lurie spectral sequences -- table]]

## References

Section 2.1.3 of

* [[Dylan Wilson]] _Spectral Sequences from Sequences of Spectra: Towards the
Spectrum of the Category of Spectra_ lecture at [2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php) ([pdf](https://ncatlab.org/nlab/files/DylanWilsonOnANSS.pdf))
 {#Wilson13}

[[!redirects filtrations by support]]


filtrations on algebraic K-theory

This article is about [[filtrations]] on [[algebraic K-theory]].

## See also

* [[algebraic K-theory]]
* [[lambda-rings]]
* [[Quillen-Brown-Gersten spectral sequence]]
* [[Thomason hypercohomology spectral sequence]]
* [[Friedlander-Suslin spectral sequence]]

## References

* [[Daniel Quillen]], _Higher algebraic K-theory: I_, Higher K-Theories, Lecture Notes in Mathematics, vol 341, pp. 85-147, [doi](http://dx.doi.org/10.1007/BFb0067053).

* [[Henri Gillet]], [[Christophe Soulé]], _Filtrations on higher algebraic K-theory_, [web](http://www.math.uiuc.edu/K-theory/0327/).

* [[Charles A Weibel]], _A Brown-Gersten spectral sequence for the K-theory of varieties with isolated singularities_, Advances in Mathematics, Volume 73, Issue 2, February 1989, Pages 192-203.

* [[Paul Balmer]], _Niveau spectral sequence and failure of Gersten-type conjecture, on singular schemes_, [web](http://www.math.uiuc.edu/K-theory/0849/).

* [[Paul Balmer]], _Supports and filtrations in algebraic geometry and modular representation theory_, [doi](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.531.257).

* _When do the $\gamma$-filtration and codimension filtration of K-theory agree?_, [MO/106021](http://mathoverflow.net/questions/106021/when-do-the-gamma-filtration-and-codimension-filtration-of-k-theory-agree).


final (infinity,1)-functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **final $(\infty,1)$-functor** (also called a *cofinal $(\infty,1)$-functor*, but see Rem \ref{CofinalTerminologyWarning}) is the [[vertical categorification]] of the notion of *[[final functor]]* from [[category theory]] to [[(∞,1)-category]]-theory:

An [[(∞,1)-functor]] $p \colon K' \to K$ is final precisely if precomposition with $p$ [[preserved colimit|preserves]] [[(∞,1)-colimits]]:
if $p$ is final then for $F \colon K \to C$ any [[(∞,1)-functor]] we have
$$
  \lim_\to (K \stackrel{F}{\to} C)
  \simeq
  \lim_{\to} ( K' \stackrel{p}{\to} K \stackrel{F}{\to} C),
$$
when either of these [[(∞,1)-colimits]] exist.

## Terminology
 {#Terminology}

The terminology conventions surrounding the topic of final functors are a bit of a mess. Often

1. “final map” is used to refer to $\infty$-functors such that restriction along them preserves [[(infinity,1)-colimit|$\infty$-colimiting]] [[cocones]].

1. “initial map” is used for functors for which restriction preserves [[(infinity,1)-limit|$\infty$-limiting]] [[cones]].

A mnemonic rule for this terminology is that [[final objects]] are picked out by [[final functors]] and likewise for [[initial objects]].

But beware that this convention is not universal, for instance [[Lurie]]'s work is a notable exception to this rule. Also beware of the following:

\begin{remark}\label{CofinalTerminologyWarning}
Beware that the term “cofinal (∞,1)-functor” can mean either a functor for which the precomposition functor preserves colimits (as in Lurie's *[[Higher Topos Theory]]*) or limits (as in Cisinski's *[[Higher Categories and Homotopical Algebra]]*).

Given that the two main sources for [[quasicategories]] assign opposite meanings to this term, it is best to avoid its usage altogether.

Further adding to the confusion is that some sources, like [[Francis Borceux|Borceux]]'s *[[Handbook of Categorical Algebra]]* use the term “final functor” for a functor for which the precomposition functor preserves limits, in contrast to the majority of the literature.
Such usage, fortunately, is marginal.

In more detail:

* [[Joyal]]'s *[[The Theory of Quasi-Categories and its Applications]]*:

  colimit-preserving: _final_;

  limit-preserving: _initial_ (page 171);

* [[Cisinski]]'s *[[Higher Categories and Homotopical Algebra]]*:

  colimit-preserving: _final_ (Definition .4.1.8);

  limit-preserving: _cofinal_ (Definition 4.4.13);

* [[Riehl]]–[[Verity]]'s *[[Elements of ∞-Category Theory]]*:

  colimit-preserving: _final_;

  limit-preserving: _initial_ (Definition 2.4.5).

* [[Lurie]]'s *[[Higher Topos Theory]]*:

  colimit-preserving: _cofinal_ (Definition 4.1.1.1);

  limit-preserving: (no terminology introduced);

* [[Lurie]]'s *[[Higher Algebra]]*:

  colimit-preserving: _left cofinal_;

  limit-preserving: _right cofinal_ (page 13);

* [[Lurie]]'s *[[Kerodon]]*:

  colimit-preserving: _right cofinal_ ([Tag 02N1](https://kerodon.net/tag/02N1));

  limit-preserving ([[initial (∞,1)-functors]]): _left cofinal_.

See also the warning in the article *[[final functor]]* ([here](final+functor#WarningOnTerminology)).
\end{remark}

## Definition

+-- {: .num_defn}
###### Definition
**(final morphism of simplicial set)**

A morphism $p : S \to T$ of [[simplicial set]]s is **final** if for every [[fibrations of quasi-categories|right fibration]] $X \to T$ the induced morphism of simplicial sets

$$
  sSet_{/T}(T,X) \to sSet_{/T}(S,X)
$$

is a [[homotopy equivalence]].

=--

So in the [[overcategory]] $sSet/T$ a final morphism is an object such that morphisms out of it into any right fibration are the same as morphisms out of the [[terminal object]] into that right fibration.

$$
  \left\{
    \array{
      T &&\to&& X
      \\
      & {}_{\mathllap{=}}\searrow && \swarrow_{}
      \\
      && T
    }
  \right\}
  \;\;
  \simeq
  \left\{
    \array{
      S &&\to&& X
      \\
      & {}_{\mathllap{p}}\searrow && \swarrow_{}
      \\
      && T
    }
  \right\}
  \,.
$$

This definition is originally due to [[Andre Joyal]]. It appears as [[Higher Topos Theory|HTT, def 4.1.1.1]].

This is equivalent to the following definition, in terms of the [[model structure for right fibrations]]:

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

The morphism $p : S \to T$ is final precisely if the terminal morphism $(p \to *) =   \left(
    \array{
      S &&\to&& T
      \\
      & {}_{\mathllap{}}\searrow && \swarrow_{=}
      \\
      && T
    }
  \right)$ in the [[overcategory]] $sSet_T$ is a weak equivalence in the [[model structure for right fibrations]] on $sSet_T$.

=--

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

This is [[Higher Topos Theory|HTT, prop. 4.1.2.5]].

=--

+-- {: .num_cor}
###### Corollary

If $T$ is a [[Kan complex]] then $p : S \to T$ is final precisely if it is a weak equivalence in the standard [[model structure on simplicial sets]].

=--

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

This is [[Higher Topos Theory|HTT, cor. 4.1.2.6]].

=--

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

Suppose $F : C \to D$ is a functor between ordinary categories. If the induced morphism $N(F) : N(C) \to N(D)$ is a final (∞,1)-functor, then $F$ is a [[final functor]] in the sense of ordinary category theory.

However, the converse need not be true. For example, while the inclusion of the parallel pair $[1] \rightrightarrows [0]$ in $\Delta^{op}$ is final in the sense of ordinary category theory, it is not a final (∞,1)-functor.

=--

## Properties {#Properties}

+-- {: .num_prop}
###### Proposition
**(preservation of undercategories and colimits)**

A morphism $p : K' \to K$ of [[simplicial set]]s is final precisely if for every [[quasicategory]] $C$

*  and for every morphism $\bar F : K^{\triangleright} \to C$ that exhibits a [[colimit in a quasi-category|colimit co-cone in]] $C$, also $(K')^\triangleright \stackrel{p}{\to} K^{\triangleright} \stackrel{\bar F}{\to} C$ is a colimit co-cone.

and equivalently precisely if

* ... and for every $F : K \to C$ the morphism

  $$
    F/C \to (F\circ p)/C
  $$

  of [[over quasi-category|under quasi-categories]] induced by composition with $p$ is an equivalence of [[(∞,1)-categories]].

=--

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

This is [[Higher Topos Theory|HTT, prop. 4.1.1.8]].

=--

The following result is the $(\infty,1)$-categorical analog of what is known as **Quillen's Theorem A**.

+-- {: .num_theorem #Recognition}
###### Theorem
**(recognition theorem for final $(\infty,1)$-functors)**

A morphism $p : K \to C$ of [[simplicial sets]] with $C$ a [[quasi-category]] is final precisely if for each object $c \in C$ the [[comma category|comma]]-object $c/p \coloneqq c/C \times_C K$ is weakly [[contractible]].

=--

More explicitly, the comma object in question here is the [[pullback]]

$$
  \array{
    c/p &\to& c/C
    \\
    \downarrow && \downarrow
    \\
    K &\stackrel{p}{\to}& C
    \mathrlap{\,.}
  }
$$

where $c/C$ is the [[over quasi-category|under quasi-category]] under $c$.

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

This is due to [[Andre Joyal]]. A proof appears as [[Higher Topos Theory|HTT, prop. 4.1.3.1]].

=--

The following says that up to equivalence, the cofinal maps of simplicial sets are the right [[anodyne morphisms]]

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

A map of simplicial sets is cofinal precisely if it factors as a right anodyne map followed by a trivial fibration.

=--

This is ([Lurie, cor. 4.1.1.12](#Lurie)).

## Examples

### General
 {#General}

+-- {: .num_example}
###### Example

The inclusion $\ast \to \mathcal{C}$ of a [[terminal object in an (∞,1)-category|terminal object]] is final.

=--

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

By theorem \ref{Recognition} the inclusion of the point is final precisely if for all $c \in \mathcal{C}$, the [[(∞,1)-categorical hom-space]] $\mathcal{C}(c,\ast)$ is contractible. This is the definition of $\ast$ being terminal.

=--

+-- {: .num_example}
###### Example

A (weak) localization $f: \mathcal{C} \to \mathcal{D}$ is both initial and final.

=--

This appears, for example, as ([Cisinski, 7.1.10](#Cisinski)).

### Cofiber products in co-slice categories
 {#CofiberProductsInCosliseCategories}

+-- {: .num_example }
###### Example

Consider the inclusion of the [[walking structure|walking]] [[span]]-category, into the result of adjoining an [[initial object]] $t$:

\[
  \label{SpanCategoryAndInitialObjectAdjoined}
  \Big\{
  \array{
     x
     &\longleftarrow&
     b
     &\longrightarrow&
     y
  }
  \Big\}
  \;\;
  \overset{\phantom{AAAA}}{\hookrightarrow}
  \;\;
  \left\{
  \array{
     && t
     \\
     & \swarrow & \downarrow & \searrow
     \\
     x
     &\longleftarrow&
     b
     &\longrightarrow&
     y
  }
  \right\}
\]

One readily sees that for each object on the right, its comma category over this inclusion has contractible nerve, whence Theorem \ref{Recognition} implies that this inclusion is a final $\infty$-functor.

As an application of the finality of (eq:SpanCategoryAndInitialObjectAdjoined), observe that for $\mathcal{C}$ an [[(∞,1)-category]] and $T \in \mathcal{C}$ an object, [[(∞,1)-colimits]] in the [[under-(∞,1)-category]]

$$
  \mathcal{C}^{T/}
  \overset{\;\;U\;\;}{\longrightarrow}
  \mathcal{C}
$$

are given by the $\infty$-colimit in $\mathcal{C}$ itself of the given [[cone]] of the original diagram, with tip $X$ (by [this Prop.](over+quasi-categories#LimitsInSliceAreLimitsInUnderlyingCategoryOverCoconeDiagram)): For

$$
  F \;\colon\; \mathcal{I} \longrightarrow \mathcal{C}^{T/}
$$

a small [[diagram]], we have

$$
  U \big(
    \underset{\longrightarrow}{\lim}\, F
  \big)
  \;\simeq\;
  \underset{\longrightarrow}{\lim}\, \big( T/U(F) \big)
$$

(when either $\infty$-colimit exists).

Now for $\mathcal{I}$ the [[walking structure|walking]] [[span]] diagram on the left of (eq:SpanCategoryAndInitialObjectAdjoined), this means that [[homotopy cofiber products]] in $\mathcal{C}^{T/}$ are computed as $\infty$-colimits in $\mathcal{C}$ of diagrams of the shape on the right of (eq:SpanCategoryAndInitialObjectAdjoined). But since the inclusion in (eq:SpanCategoryAndInitialObjectAdjoined) is final, these are just homotopy cofiber products in $\mathcal{C}$.

Explicitly: Given

$$
  \array{
     T &=& T &=& T
     \\
     {}^{\mathllap{
       \phi_X
     }}
     \big\downarrow
     &&
     {}^{\mathllap{
       \phi_B
     }}
     \big\downarrow
     &&
     {}^{\mathllap{
       \phi_Y
     }}
     \big\downarrow
     \\
     X
     &
       \underset{
       f
       }{\longleftarrow}
     &
     B
     &
       \underset{
         g
       }{
         \longrightarrow
       }
     &
     Y
  }
$$

regarded as a [[span]] in $\mathcal{C}^T$, hence with underlying objects

$$
  U\big(  (X,\phi_X) \big) \;=\; X
  \,,
  \;\;\;\;\;\;
  U\big(  (B,\phi_B) \big) \;=\; B
  \,,
  \;\;\;\;\;\;
  U\big(  (Y,\phi_Y) \big) \;=\; Y
  \,,
$$

we have:

$$
  U
  \Big(
    \;
    (X,\phi_X)
    \underset{
      (B,\phi_B)
    }{\coprod}
    (Y,\phi_Y)
    \;
  \Big)
  \;\;\;\simeq\;\;\;
  X \underset{B}{\coprod} Y
  \,.
$$

In particular, if $(B,\phi_B) \;\coloneqq\; (T,id_T) $ is the [[initial object]] in $\mathcal{C}^{T/}$, in which case the cofiber product is just the coproduct

$$
  (X,\phi_X)
  \coprod
  (Y,\phi_Y)
  \;\;=\;\;
  (X,\phi_X)
  \underset{
    (T,id_T)
  }{\coprod}
  (Y,\phi_Y)
$$

we find that the coproduct in the co-slice category is the co-fiber product under the given tip object in the underlying category

$$
  U
  \Big(
    \;
    (X,\phi_X)
    \coprod
    (Y,\phi_Y)
    \;
  \Big)
  \;\;\;\simeq\;\;\;
  X \underset{T}{\coprod} Y
  \,.
$$

=--

### On categories of simplices
 {#OnCategoriesOfSimplices}

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

For $K \in $ [[sSet]] a [[simplicial set]], write $\Delta_{/K}$ for its [[category of elements]], called here the *[[category of simplices]]* of the simplicial set:

an [[object]] of $\Delta_{/K}$ is a morphism of simplicial sets of the form $\Delta^n \to K$ for some $n \in \mathbb{N}$ (hence an $n$-[[simplex]] of $K$) and a morphism is a [[commuting diagram]]

$$
  \array{
    \Delta^{n_1}&&\to&& \Delta^{n_2}
    \\
    & \searrow && \swarrow
    \\
    && K
  }
  \,.
$$

Moreover, write

$$
  \Delta_{/K}^{nd} \hookrightarrow \Delta_{/K}
$$

for the non-full subcategory on the non-degenerate simplices.

=--

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

When the simplicial set $K$ is non-singular, i.e. every face of a non-degenerate simplex is still non-degenerate, then the category $\Delta_{/K}^{nd}$ is a [[poset]] and it coincides with the [[barycentric subdivision]] of $K$.

=--

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

Suppose $K$ is a non-singular simplicial set. Then the inclusion

$$
  N(\Delta_{/K}^{nd}) \hookrightarrow N(\Delta_{/K})
$$

is a cofinal morphism of [[quasi-categories]].

=--

This appears as ([Lurie, variant 4.2.3.15](#Lurie)).

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

For every simplicial set $K$ there exists a cofinal map

$$
  N(\Delta_{/K}) \to K
  \,.
$$

=--

This is ([Lurie, prop. 4.2.3.14](#Lurie)).

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

If the simplicial set $K$ is singular, it is not in general true
that the inclusion $N(\Delta_{/K}^{nd}) \hookrightarrow N(\Delta_{/K})$
is final. For a counter-example, see ([Hovey, 9](#Hovey)).

=--

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

For every simplicial set $K$, evaluation at the initial vertex

$$
  N(\Delta_{/K})^{op} \to K
  \,.
$$

is both initial and final.

=--

This appears as ([Shah, 12.2](#Shah)) and also follows from the fact that this map is a weak localization ([Cisinski, 7.3.15](#Cisinski)).

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

This can be used to establish a [[Bousfield-Kan formula]] for [[homotopy colimits]]; see ([Shah, 12.3](#Shah)).

=--

## Related concepts

* [[final functor]]

* [[homotopy final functor]]

* [[initial (∞,1)-functor]]

[[!include properties of functors -- contents]]

## References

* {#Lurie} [[Jacob Lurie]], Section 4.1 of: _[[Higher Topos Theory]]_

* [[Dan Dugger]], Section 6 ofL _A primer on homotopy colimits_ ([pdf](http://pages.uoregon.edu/ddugger/hocolim.pdf))

* {#Hovey} [[Mark Hovey]], Item 9 of: _Errata to Model Categories_ ([pdf](http://hopf.math.purdue.edu/Hovey/model-err.pdf))

* {#Shah} [[Jay Shah]], *Parametrized higher categories and higher algebra: Expose II- Indexed homotopy limits and colimits*

* {#Cisinski} [[Denis-Charles Cisinski]], *Higher Categories and Homotopical Algebra*

[[!redirects cofinal (infinity,1)-functor]]
[[!redirects cofinal (∞,1)-functor]]
[[!redirects cofinal (infinity,1)-functors]]
[[!redirects cofinal (∞,1)-functors]]
[[!redirects final (infinity,1)-functor]]
[[!redirects final (∞,1)-functor]]
[[!redirects final (infinity,1)-functors]]
[[!redirects final (∞,1)-functors]]
[[!redirects initial (infinity,1)-functor]]
[[!redirects initial (∞,1)-functor]]
[[!redirects initial (infinity,1)-functors]]
[[!redirects initial (∞,1)-functors]]
[[!redirects homotopy final functor]]
[[!redirects homotopy final functors]]
[[!redirects homotopically final functor]]
[[!redirects homotopically final functors]]
[[!redirects homotopy cofinal functor]]
[[!redirects homotopy cofinal functors]]
[[!redirects homotopically cofinal functor]]
[[!redirects homotopically cofinal functors]]
[[!redirects homotopy initial functor]]
[[!redirects homotopy initial functors]]
[[!redirects homotopically initial functor]]
[[!redirects homotopically initial functors]]

[[!redirects final infinity1-functor]]


final functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[functor]] $F \colon C \to D$ is **final**, if restricting [[diagrams]] along $F$ does not change their [[colimit]].

Dually, a functor is **initial** if pulling back diagrams along it does not change the [[limits]] of these diagrams.

Beware that this property is pretty much unrelated to that of a functor being an [[initial object]] or [[terminal object]] in the [[functor category]] $[C,D]$.  The terminology comes instead from the fact that an object $d\in D$ is initial (resp. terminal) just when the corresponding functor $d \colon 1\to D$ is initial (resp. final).

\begin{remark}\label{WarningOnTerminology}
**(warning on terminology)**
\linebreak
In older references (and also some others like *[[Higher Topos Theory|HTT]]*), final functors are sometimes called *cofinal*, the terminology having been imported from [[order theory]] (cf. *[[cofinality]]*).  However, this is confusing in [[category theory]] because usually the prefix "co-" denotes [[formal dual|dualization]].  In at least one place ([Borceux](#fb)) this non-dualization was treated as a dualization and the word "final" used for the *dual* concept, but in general it seems that the consensus is to use "final" for what used to be called "cofinal", and "initial" for the dual concept (since "co-final" would be ambiguous).  For example, [[Peter Johnstone|Johnstone]] in *[[Sketches of an Elephant]]* says (before Proposition B2.5.12 ):

> Traditionally, final functors were called 'cofinal functors'; but this use of 'co' is potentially misleading as it has nothing to do with dualization — it is derived from the Latin 'cum' rather than 'contra' — and so it is now generally omitted.

See also the warning at *[[final (infinity,1)-functor|final $(\infty,1)$-functor]]* ([here](final+infinity1-functor#CofinalTerminologyWarning)).
\end{remark}

## Definition

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

A [[functor]] $F : C \to D$ is **final** if for every [[object]] $d \in D$ the [[comma category]] $(d/F)$ is (non-empty and) [[connected category|connected]] (the non-emptiness condition is redundant since connected categories are non-empty by convention).

A [[functor]] $F : C \to D$ is **initial** if the [[opposite category|opposite]] $F^{op} : C^{op} \to D^{op}$ is final, i.e. if for every [[object]] $d \in D$ the [[comma category]] $(F/d)$ is [[connected category|connected]].

=--

## Properties

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

Let $F : C \to D$ be a [[functor]]

The following conditions are equivalent.

1. $F$ is final.

1. {#AnItem} For all functors $G : D \to Set$ the natural [[function]] between [[colimits]]

   $$
     \lim_\to G \circ F \to \lim_{\to} G
   $$

   is a [[bijection]].

1. For all categories $E$ and all functors $G : D \to E$ the natural [[morphism]] between [[colimit]]s

   $$
     \lim_\to G \circ F \to \lim_{\to} G
   $$

   is an [[isomorphism]].

1. For all functors $G : D^{op} \to Set$ the natural [[function]] between [[limit]]s

   $$
     \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}
   $$

   is a [[bijection]].

1. For all categories $E$ and all functors $G : D^{op} \to E$ the natural [[morphism]]

   $$
     \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}
   $$

   is an [[isomorphism]].

1. For all $d \in D$

   $$
     {\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq *
     \,.
   $$

1. for all functors $G: D \to E$, the canonical map

   $$
     Nat(G,\Delta-)\to Nat(GF,\Delta-) : E \to Set
   $$

   between the functors of [[cocones]] on $G$ and $GF$ is an [[isomorphism]]. (Here, the $\Delta$'s denote the functors sending objects to constant functors, and $Nat$ stands for the set of natural transformations.)

=--

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

If $F : C \to D$ is final then $C$ is connected precisely if $D$ is.

=--

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

If $F_1$ and $F_2$ are final, then so is their composite $F_1 \circ F_2$.

If $F_2$ and the composite $F_1 \circ F_2$ are final, then so is $F_1$.

Final functors are stable under [[pushout]].

The [[coproduct]] of final functors in the [[arrow category]] $\mathbf{Cat}^{\mathbf{2}}$ is a final functor.

If $F_1$ is a [[full and faithful functor]] and the composite is final, then both functors seperately are final.

=--

The first two statements of Proposition \ref{Stability} in fact follow
from the stability properties of orthogonal factorization systems:

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

Final functors and [discrete fibrations](http://ncatlab.org/nlab/show/discrete+fibration) form an [orthogonal factorization system](https://ncatlab.org/nlab/show/orthogonal+factorization+system) called the [[comprehensive factorization system]].

=--

## Generalizations

### Exact squares

The characterization of final functors is a special case of the characterization of [[exact squares]].

### Final enriched functors
 {#FinalityForWeightedLimits}

Finality for [[enriched functors]] with respect to [[weighted colimits]] is discussed in [Kelly 1982 §4.5](#Kelly82).

See also discussion of finality specifically for [[coends]] (initiality for [[weighted limits]] and [[ends]]) by:

* [[Tim Campion]]: *Cofinality for Coends* (2020) [[MO:q/353876](https://mathoverflow.net/q/353876/381), [MO:a/354097](https://mathoverflow.net/a/354097/381)]

### Final $\infty$-functors

The generalization of the notion of final functor from [[category theory]] to [[(∞,1)-category|(∞,1)]]-[[higher category theory]] is described at

* [[final (∞,1)-functor]].

## Examples
 {#Examples}

\begin{example}
  \label{InclusionOfATerminalObjectIsFinal}
**([[subcategory|inclusion]] of a [[terminal object]] is [[final functor]])** \linebreak
If $D$ has a [[terminal object]] then the functor $F : {*} \to D$ that picks that terminal object is final: for every $d \in D$ the [[comma category]] $d/F$ is equivalent to $*$.  The converse is also true: if a functor $*\to D$ is final, then its image is a terminal object.

In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.

More generally, a functor whose domain is a [[discrete category]] is final if and only if it is a right adjoint.

\end{example}

\begin{example}\label{FinalFunctorBetweenGroupoids}
  A functor between [[groupoids]] is final iff it is [[essentially surjective functor|essentially surjective]] and [[full functor|full]].
\end{example}
(eg. [Cigoli 2018, Prop. 1.1](#Cigoli18))

+-- {: .num_example }
###### Example

Every [[right adjoint|right]] [[adjoint functor]] is final.

=--

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

Let $(L \dashv R) : C \to D$ be a pair of [[adjoint functors]].To see that $R$ is final, we may for instance check that for all $d \in D$ the comma category $d / R$ is non-empty and connected:

It is non-empty because it contains the [[unit of an adjunction|adjunction unit]] $(L(d), d \to R L (d))$. Similarly, for

$$
  \array{
    && d
    \\
    & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{g}}
    \\
    R(a) &&&& R(b)
  }
$$

two objects, they are connected by a zig-zag going through the unit, by the [[adjoint functor#UniversalArrows|universal factorization property]] of adjunctions

$$
  \array{
    && d
    \\
    & \swarrow &\downarrow& \searrow
    \\
    R(a) &\stackrel{R \bar f}{\leftarrow}& R L (d)& \stackrel{R(\bar g)}{\to} & R(b)
  }
  \,.
$$

=--

+-- {: .num_example}
###### Example

The inclusion $\mathcal{C} \to \tilde \mathcal{C}$ of any category into its [[idempotent completion]] is final.

=--

See at _[[idempotent completion]]_ in the section on _[Finality](Karoubi+envelope#Finality)_.

+-- {: .num_example #CoconeUnderCospan}
###### Example

The inclusion of the [[cospan]] [[diagram]] into its [[cocone]]

$$
  \left(
    \array{
       a
       \\
       \downarrow
       \\
       c
       \\
       \uparrow
       \\
       b
    }
  \right)
  \hookrightarrow
  \left(
    \array{
       a
       \\
       \downarrow & \searrow
       \\
       c &\longrightarrow & p
       \\
       \uparrow & \nearrow
       \\
       b
    }
  \right)
$$

is initial.

=--

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

By the characterization ([here](overcategory#LimitsInSliceViaLimitsOfCoconedDiagram)) of limits in a [[slice category]], this implies that [[fiber products]] in a [[slice category]] are computed as fiber products in the underlying category, or in other words that [[dependent sum]] to the point preserves fiber products.

=--

\begin{example}\label{FirstPairOfFaceMapsFinalInOppositeSimplexCategory}
  For $\Delta^{op}$ the [[opposite category|opposite]] of the [[simplex category]], the non-full [[subcategory]] inclusion of the lowest two face maps
  $$
    \big(
      [1]
      \underoverset
        {d_0}
        {d_1}
        {\rightrightarrows}
      [0]
    \big)
    \;\xrightarrow{\;\;\;\;}\;
    \Delta^{op}
  $$
  is a final functor.

  It follows that the [[colimit]] over a [[simplicial diagram]] is equivalently the [[coequalizer]] of the lowest two face maps.
\end{example}

(e.g. [Riehl 14, Exp. 8.3.8](#Riehl14))

## Related concepts

* [[cofinal diagram]]

* [[homotopy final functor]]

* [[final (∞,1)-functor]]

[[!include properties of functors -- contents]]

## References

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 2.5 of: _[[Categories and Sheaves]]_

* [[Saunders Mac Lane]], Section IX.3 of: _[[Categories for the Working Mathematician]]_

* {#fb} [[Francis Borceux]], Section 2.11 of: _[[Handbook of Categorical Algebra]] 1, Basic category theory_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)

  > (this says "final functor" for the version under which limits are invariant)

* {#Riehl14} [[Emily Riehl]], Section 8.3 of: _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

* [[Paolo Perrone]], [[Walter Tholen]], *Kan extensions are partial colimits*, Applied Categorical Structures, 2022. ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531))

  > (this says "confinal functor" for the version under which colimits are invariant)

In [[enriched category theory]]:

* {#Kelly82} [[Max Kelly]], §4.5 in: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

In [[internal category]] theory

for [[internal functors]] between [[internal groupoids]] in [[exact categories]]:

* {#Cigoli18} [[Alan S. Cigoli]], *A characterization of final functors between internal groupoids in exact categories*, Theory and Applications of Categories **33** 11 (2018) 265-275.  [[arXiv:1711.10747](https://arxiv.org/abs/1711.10747), [tac:33-11](http://www.tac.mta.ca/tac/volumes/33/11/33-11abs.html)]

[[!redirects cofinal functor]]
[[!redirects cofinal functors]]
[[!redirects final functor]]
[[!redirects final functors]]
[[!redirects initial functor]]
[[!redirects initial functors]]


final lift

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

# Final and initial lifts
* table of contents
{: toc}

## Definition

Let $U\colon C\to D$ be a [[functor]].

+-- {: .un_defn}
###### Definition
A **$U$-structured sink** is a [[sink]] of the form $\{f_i \colon U(A_i) \to Y\}$ in $D$.
=--

Note that like all sinks, a $U$-structured sink is not necessarily assumed to be [[small category|small]].

A **lift** of $Y$ along a $U$-structured sink $\{f_i\colon U(A_i)\to Y\}$ is an object $B$ of $C$, equipped with a sink $\{\phi_i\colon A_i \to B\}$ in $C$ and a morphism $h\colon Y\to U(B)$ in $D$, such that $U(\phi_i) = h \circ f_i$ for each $i$. A **morphism of lifts** is, of course, a morphism $\xi\colon B\to B'$ of $C$ such that the sink $\{\phi'_i\colon A_i\to B'\}$ factors through the sink $\{\phi_i\colon A_i\to B\}$ as $\phi_i'=\xi\circ\phi_i$, and such that the morphism $h'\colon Y\to U(B')$ factors through the morphism $h\colon Y\to U(B)$ as $h'=p(\xi)\circ h$.

Note that if $U$ is faithful, then it suffices to demand merely that the sink $\{\phi_i\colon A_i \to B\}$ exists, rather than giving it as part of the structure.

+-- {: .un_defn}
###### Definition
A **semi-final lift** of $\{f_i \colon U(X_i) \to Y\}$ is a lift $B$ of $Y$ admitting a unique morphism of lifts to any other lift of $Y$.  If the morphism $h\colon Y\to U(B)$ is an isomorphism, then the semi-final lift is called a **final lift**.  If $h$ is an identity, we call $B$ a **strictly final lift**.
=--

If objects of $C$ are regarded as objects of $D$ equipped with [[stuff, structure, property|structure]], for a (strictly) final lift we say that $B$ is the **final structure** or **strong structure** on $Y$ induced by the sink.  Note that if $U$ is an [[isofibration]], then any final lift may be made into a strictly final one.

The dual concept, which applies to cosinks ("sources"), is called a (perhaps semi- or strictly) **initial lift**, an **initial structure** or a **weak structure**.

## Examples

* If $D$ is the [[terminal category]], then a $U$-structured sink is simply a family of objects $A_i$ of $C$, a lift is just a cocone $A_i\to B$, a morphism of lifts is a morphism of cocones, hence a semi-final lift is simply a [[colimit]] in $C$.

* An empty $U$-structured sink is just an object $Y$ of $D$, a lift of it is a morphism $Y\to U(B)$, and a morphism of lifts is a morphism $B\to B'$ so that $Y\to U(B')$ factors as $Y\to U(B)\to U(B')$. Hence, a semi-final lift of such a sink is a [[free object]] with unit morphism $h\colon Y\to U(B)$ of $D$.  Thus $U$ admits semi-final lifts of empty sinks precisely when it has a [[left adjoint]].  Similarly, it admits final lifts of empty sinks precisely when it has a [[fully faithful functor|fully faithful]] left adjoint (i.e. if it admits [[discrete objects]]).

* A singleton $U$-structured sink is just a morphism of the form $f\colon U(X) \to Y$.  A strictly final lift of such a sink is precisely an [[opcartesian arrow]] lying over $f$.  Thus $U$ admits strictly final lifts of singleton structured sinks precisely when it is a [[Grothendieck opfibration]] (and final lifts of such sinks precisely when it is a [[Street opfibration]]).

* A [[topological concrete category]] is a functor that admits final lifts of *all* (not necessarily small) structured sinks.  This turns out to be equivalent to admitting initial lifts of all structured cosinks.  The most famous example is then [[initial topology|initial topologies]] and [[final topology|final topologies]] for $U\colon Top \to Set$.

* More generally, a [[solid functor]] is one that admits *semi-final* lifts of all structured sinks.

* If $U$ has both a left and right adjoint, of which one (and hence also the other) is fully faithful, and $C$ is cocomplete, then $U$ admits final lifts of all *small* structured sinks.  See [[adjoint triple]] for a proof.  Dually, if $C$ is complete in this situation, then $U$ admits initial lifts of all small structured cosinks.

## Remarks

* (Semi-)final lifts can be generalized to *(semi-)final extensions*, which are to (semi-)final lifts as [[Kan extensions]] are to [[colimits]].

* In [[Higher Topos Theory]] (section 4.3.1) the corresponding notion of (strictly) *final lift* for [[(∞,1)-categories]] is called a *$U$-colimit*.

[[!redirects final lift]]
[[!redirects final lifts]]
[[!redirects semi-final lift]]
[[!redirects semi-final lifts]]
[[!redirects semifinal lift]]
[[!redirects semifinal lifts]]
[[!redirects strictly final lift]]
[[!redirects strictly final lifts]]
[[!redirects final structure]]
[[!redirects final structures]]
[[!redirects strong structure]]
[[!redirects strong structures]]

[[!redirects initial lift]]
[[!redirects initial lifts]]
[[!redirects semi-initial lift]]
[[!redirects semi-initial lifts]]
[[!redirects semiinitial lift]]
[[!redirects semiinitial lifts]]
[[!redirects strictly initial lift]]
[[!redirects strictly initial lifts]]
[[!redirects initial structure]]
[[!redirects initial structures]]
[[!redirects weak structure]]
[[!redirects weak structures]]


final pullback complement

# Final pullback complements

* table of contents
{: toc}

## Idea

A *final pullback complement* is a "universal completion of a pair of [[arrows]] to a [[pullback]]".  Intuitively, it is a way to "remove part of an [[object]] of a [[category]]" while retaining information about how that part is "connected" to other parts.  Final pullback complements play an important role in some kinds of [[span rewriting]].

## Definition

Given [[morphisms]] $m:C\to A$ and $g:A\to D$ in a [[category]], a **pullback complement** is a pair of arrows $f:C\to B$ and $n:B\to D$ such that the square
$$ \array{ C & \to & A \\ \downarrow & & \downarrow \\ B & \to & D }$$
[[commutative square|commutes]] and is a [[pullback]].  This is of course the dual of a [[pushout complement]].

A morphism of pullback complements is a map $B\to B'$ making the obvious diagrams commute.  A **final pullback complement** is a [[terminal object]] in the category of pullback complements.  Since they have a universal property, final pullback complements are unique up to unique isomorphism when they exist.

## Relation to exponentials

Recall from [[distributivity pullback]] that a **pullback around $(g,m)$** is a diagram
$$\array{ X & \xrightarrow{p} & C & \xrightarrow{m} & A\\
  ^q\downarrow &&&& \downarrow^g\\
  Y && \xrightarrow{r} && D}
$$
in which the outer rectangle is a pullback, and that a **distributivity pullback** around $(g,m)$ is a terminal object in the category of pullbacks aroung $(g,m)$.  This makes the following evident:

+--{: .un_prop}
###### Proposition
If a distributivity pullback around $(g,m)$ exists and has the property that $p$ is an [[isomorphism]], then it is also a final pullback complement.
=--

A distributivity pullback has precisely the universal property of an [[exponential object]] of $m$ along $g$.  Thus, whenever such an exponential $\Pi_g m$ exists and the [[counit]] $g^* \Pi_g m \to m$ is an isomorphism, a final pullback complement exists.  This is the case in particular in a [[locally cartesian closed category]] if $g$ and $m$ are both [[monomorphisms]].

## Relation to pushout complements

In an [[adhesive category]], any pushout complement of a monomorphism is also a final pullback complement.

## Related pages

* [[pushout complement]]
* [[span rewriting]]
* [[distributivity pullback]]

## References

* Andrea Corradini, Tobias Heindel, Frank Hermann, and Barbara König, *Sesqui-pushout rewriting*, 2006, [springerlink](https://link.springer.com/chapter/10.1007/11841883_4), [pdf](http://www.ti.inf.uni-due.de/publications/koenig/icgt06b.pdf).
 {#CHHK}

[[!redirects final pullback complements]]


final subcategory

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

A [[subcategory]] $i : C \hookrightarrow D$ is called **final** (sometimes **cofinal**) if the injection [[functor]] $i$ is a [[final functor]].

[[!redirects final subcategories]]
[[!redirects cofinal subcategory]]
[[!redirects cofinal subcategories]]


FinCat

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

* table of contents
{: toc}

## Idea

$FinCat$ is the [[2-category]] of [[finite categories]] (that is, categories with finitely many objects and finitely many morphisms). $FinCat$ is a [[2-pretopos]] which does not have [[coinserters]], [[coinverters]], or [[coequifiers]]. Also, it does not admit groupoid reflections.


FinDimVect

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[finite dimensional vector spaces]] and [[linear functions]] between them.

A [[compact closed monoidal category]]

## Properties
 {#Properties}

### Splitting lemma
 {#SplittingLemma}

The [[splitting lemma]] says that ever [[short exact sequence]] of [[vector spaces]] [[split exact sequence|splits]]
so that (in [[categorification]] of the [[rank-nullity theorem]]) every linear map $f \,\colon\, D \to C$ is equivalent to

$$
  f
  \;\colon\;
  ker(f) \oplus im(f)
  \xrightarrow{
     (0 , \iota_{im(f)})
  }
  C
  \,,
$$

where

* $ker(f) \xrightarrow{\iota_{ker(f)}} D$ is the [[kernel]]

* $im(f) \xrightarrow{\iota_{im(f)}} C$ is the [[image]]

of $f$.

### Fiber (co)products
 {#FiberCoProducts}

The [[cartesian product]] in $FinDimVect$ is a [[biproduct]] given by [[direct sum of vector spaces]].

More generally, the [[fiber product]] of a [[pair]] of [[linear maps]] is given by the [[direct sum]] of their [[kernels]] and of the [[intersection]] of their [[images]]:

\begin{tikzcd}[
  sep=20pt
]
  & &&
  \def\arraystretch{.9}
  \begin{array}{c}
  \mathrm{ker}(f_1) \oplus \mathrm{ker}(f_2)
  \\
  \oplus
  \\
  \mathrm{im}(f_1) \cap_{_B} \mathrm{im}(f_2)
  \end{array}
  \ar[
    d,
    phantom,
    "\simeq"{rotate=-90}
  ]
  \\[-10pt]
  &
  &&
  V_1 \times_{{}_B} V_2
  \ar[dll]
  \ar[drr]
  \ar[
    dd,
    phantom,
    "\scalebox{.6}{(pb)}"{description, pos=.2}]
  \\
  \mathrm{ker}(f_1)
  \oplus
  \mathrm{im}(f_1)
  \ar[r, phantom, "{\simeq}"]
  \ar[drr, "{ (0,\mathrm{id}) }"{description}]
  &[-10pt]
  V_1
  \ar[
    drr,
    end anchor={[yshift=+3pt]},
    "{ f_1 }"{description}
  ]
  &&
  &&
  V_2
  \ar[
    dll,
    end anchor={[yshift=+3pt]},
    "{ f_2 }"{description}
  ]
  &[-10pt]
  \ar[l, phantom, "{\simeq}"]
  \mathrm{im}(f_2) \oplus \mathrm{ker}(f_2)
  \ar[dll, "{ (\mathrm{id}, 0) }"{description}]
  \\
  &
  &
  \mathrm{im}(f_1)
  \ar[r, hook]
  &
  B
  &
  \mathrm{im}(f_2)
  \ar[l, hook']
\end{tikzcd}

The [[coproduct]] in the [[slice category]] $FinDimVect_{/B}$ is given (by [general facts](over+category#LimitsAndColimits)) as the [[coproducts]], hence the [[direct sum]], of the domains, equipped with the induced maps to the base.

Applying [[(-1)-truncation]] to this fiber-wise coproduct of a [[pair]] of linear [[monomorphisms]] yields the [[linear span]] in $B$ of the two subspaces:

\begin{tikzcd}[sep=20pt]
  V_1
  \ar[ddr, hook]
  \ar[r, hook]
  &
  V_1 \oplus V_2
  \ar[d, ->>]
  &
  V_2
  \ar[ddl, hook']
  \ar[l, hook']
  \\[-10pt]
  &
  \mathrm{Span}_B(V_1, V_2)
  \ar[d, hook]
  \\[+10pt]
  &
  B
\end{tikzcd}

In summary this means that the [[internal logic]] of [[slice category|slices]] of $FinDimVect$ is a [Birkhoff-vonNeumann quantum logic](quantum+logic#BirkhoffVonNeumannQuantumLogic).

## Related concepts

* [[Vect]]

* [[sFinVect]]

* [[quantum information theory]]$\;$[[quantum information theory in terms of dagger-compact categories|in terms of dagger-compact categories]]

* [[linear logic]]

## References

Discussion of [[linear algebra]] in $FinDimVect$ as [[categorical semantics]] for [[linear logic]]:

* {#Murfet14} [[Daniel Murfet]], *Logic and linear algebra: an introduction* [[arXiv:1407.2650](http://arxiv.org/abs/1407.2650)]

[[!redirects FinDimVec]]

[[!redirects FinVect]]
[[!redirects FinVec]]


fine model structure on topological G-spaces

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#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[topological group]], there exists a [[model category]]-[[mathematical structure|structure]] on the [[category]] of [[topological G-spaces]] whose [[weak equivalences]] and [[fibrations]] are those morphisms whose [[underlying]] [[continuous functions]] between $H$-[[fixed loci]], for all [[closed subgroups]] $H \subset_{clsd} G$, are weak equivalences or fibrations, respectively, in the [[classical model structure on topological spaces]], hence [[weak homotopy equivalences]] or [[Serre fibrations]], respectively. Hence the weak equivalences are the [[equivariant weak homotopy equivalences]].

In the case that $G$ is a [[compact Lie group]], the corresponding [[homotopy theory]] coincides with that of [[G-CW complexes]] [[localization|localized]] at $G$-[[equivariant function|equivariant]] [[homotopy equivalences]].

For general $G$, [[Elmendorf's theorem]] asserts that the fine equivariant model structure is [[Quillen equivalence|Quillen equivalent]] to the [[model category of simplicial presheaves]] on the [[orbit category]] of $G$.

All this makes the fine model structure serve as a foundation for [[equivariant homotopy theory]] and for [[equivariant cohomology]] in its refined form subsuming [[Bredon cohomology]].

This is in contrast to the "coarse" or *[[Borel model structure]]* whose [[weak equivalences]] are simply the underlying [[weak homotopy equivalences]] (which need not restrict to weak homotopy equivalences on all [[fixed loci]]). The coarse Borel model structure instead presents the [[slice (infinity,1)-category|slice]] homotopy theory over the [[classifying space]] $B G$. The intrinsic [[cohomology]] of this coarse [[equivariant homotopy theory]] is just "Borel equivariant", hence computes cohomology of [[Borel constructions]].

While one may, therefore, think of the fine model structure as exhibiting "genuine" equivariance (e.g. [Guillou, May & Rubin 13, p. 14-15](#GuillouMayRubin13)), beware that the term "genuine equivariant homotopy theory" has come to be adopted for something yet a little richer, namely to [[equivariant stable homotopy theory]] whose [[G-spectra]] are in addition equipped with "transfer maps".

However, when the [[closed subgroups]] of $G$ that enter the definition of the fine model structure are taken to be [[compact groups]], then it is not wrong to speak of *[[proper equivariant homotopy theory]]* (conflating two usages of the term "proper", but in a sensible way).

## Definition

Throughout, we write

* [[TopologicalSpaces|TopSp]] for the [[conventient category of topological spaces|convenient category]] of [[compactly generated weak Hausdorff spaces]];

* $TopSp_{Qu}$ for its [[classical model structure on topological spaces]] ([here](classical+model+structure+on+topological+spaces#ModelStructureOnCompactlyGeneratedTopologicalSpaces));

* $G \,\in\, Grp(CptSmthMfd) \xrightarrow{\;} Grp(TopSp)$ for the [[underlying]] [[topological group]] of a [[compact Lie group]]

  (most of the following statements hold for general [[topological groups]], at least if some further qualifications are added)

* $G Act(TopSp)$ for the [[category]] of [[continuous function|continuous]] $G$-[[group action|actions]], hence for the category of [[topological G-spaces]] with [[continuous function|continuous]] [[equivariant functions]] between them.

\begin{proposition}\label{FineModelStructureOnGSpaces}
**(fine model structure on $G$-spaces)**
\linebreak
  There is a [[model category]]-[[mathematical structure|structure]]
$G Act\big(TopSp_{Qu}\big)_{fine}$ on [[topological G-spaces]] whose [[weak equivalences]] and [[fibrations]] are those [[morphisms]] $f \,\colon\, X \xrightarrow{\;} Y$ such that for each [[closed subgroup]] $H \,\underset{clsd}{\subset}\, G$ their ([[corestriction|co-]])[[restriction]] $f^H \,\colon\, X^H \xrightarrow{\;} Y^H$ to the  $H$-[[fixed loci]] is, respectivelhy, a [[weak equivalence]] or [[fibration]] in the [[classical model structure on topological spaces]], hence a [[weak homotopy equivalence]] or [[Serre fibration]].
\end{proposition}

([Dwyer & Kan 1984, 1.2](#DwyerKan84))

## Properties

### Cofibrant generation, enrichment and properness

\begin{prop}\label{BasicPropertiesOfTheFineModelStructure}
  The model category $G Act\big( TopSp_{Qu}\big)_{fine}$ (Prop. \ref{FineModelStructureOnGSpaces}) is:

1. {#CofibrantGeneration} [[proper model category|proper]] and [[cofibrantly generated model category]] with generating (acyclic) cofibrations the images under forming [[products]] ([[k-ification|k-ified]] [[product topological spaces]]) of [[coset spaces]] $G/H$ with the classical generating cofibrations ([here](classical+model+structure+on+topological+spaces#TopologicalGeneratingCofibrations) and [here](classical+model+structure+on+topological+spaces#TopologicalGeneratingAcyclicCofibrations)):

   \[
     \label{GeneratingCofibrations}
     \begin{aligned}
     I_{G Top}
       &\;\coloneqq\;
     \big\{
       G/H \times S^{n-1} \xhookrightarrow{\;} G/H \times D^n
     \big\}_{n \in \mathbb{N}, H \underset{clsd}{\subset} G}
     \\
     I_{G Top}
       &\;\coloneqq\;
     \big\{
       G/H \times D^n \xhookrightarrow{\;} G/H \times D^n \times I
     \big\}_{n \in \mathbb{N}, H \underset{clsd}{\subset} G}
     \end{aligned}
   \]

   ([Guillou 2006, Prop. 3.12](#Guillou06); [Fausk 2008, Prop. 2.11](#Fausk08); [Stephan 2013, Prop. 2.6](#Stephan13))

1. in addition an [[enriched model category]] over [[classical model structure on topological spaces|$TopSp_{Qu}$]] with [[hom-objects]] given by the $G$-[[fixed loci]] of the [[conjugation action]] on the [[mapping spaces]], hence such that
   \[
     \label{EnrichmentOverTopologicalSpaces}
     Maps(-,-)^G
       \;\colon\;
     G Act\big(TopSp_{Qu}\big)_{fine}^{op}
       \times
     G Act\big(TopSp_{Qu}\big)_{fine}^{op}
      \xrightarrow{\;}
     TopSp_{Qu}
   \]
   is a [[Quillen bifunctor]].

   ([Guillou, May & Rubin 2013, Thm 3.7](#GuillouMayRubin13); [Schwede 2018, Prop. B.7](#Schwede18); [DHLPS 2019, Prop. 1.1.3 (i-ii) ](#DHLPS19))

\end{prop}

\begin{remark}\label{SpecializationToBorelModelStructure}
**(specialization to [[Borel model structure]])**
\linebreak
The direct analog of Prop. \ref{FineModelStructureOnGSpaces}, Prop. \ref{BasicPropertiesOfTheFineModelStructure} holds for any choice of family of closed subgroups of $G$. In the case that the family contains only the [[trivial group]] $1 \subset G$ the result is the topological [[Borel model structure]].
\end{remark}

\begin{corollary}\label{GCWComplexesAreCofibrant}
  Every [[G-CW complex]] (being, by definition, a special [[cell complex]] in the generating cofibrations (eq:GeneratingCofibrations)) is a [[cofibrant object]] in the fine equivariant model structure.
\end{corollary}

The $TopSp_{Qu}$ enrichment of Prop. \ref{BasicPropertiesOfTheFineModelStructure} in fact underlies a model enrichment of $G Act(TopSp_{Qu})_{fine}$ over itself:

\begin{prop}\label{MonoidalModelCategoryStructure}
**([[cartesian monoidal category|cartesian]] [[monoidal model category]] structure)** \linebreak
  The model category $G Act\big( TopSp_{Qu}\big)_{fine}$ (Prop. \ref{FineModelStructureOnGSpaces}) is a [[cartesian monoidal category|cartesian]] [[monoidal model category]] in that it satisfies the [[pushout-product axiom]] with respect to [[Cartesian product]] of ([[compactly generated topological space|cgwh]]) $G$-spaces.
\end{prop}
(Observe that the [[unit axiom]] is automatic, by [this Prop.](monoidal+model+category#CaseOfCofibrantTensorUnit), since the [[tensor unit]] -- which here is the [[point space]] equipped with the [[trivial action]] -- is clearly a [[G-CW complex]], $\ast \simeq G/G$, and hence [[cofibrant object|cofibrant]], by Cor. \ref{GCWComplexesAreCofibrant}.)
\begin{proof}
Prop. \ref{MonoidalModelCategoryStructure} seems to have been [[folklore]] statement, based on the fact that the [[equivariant triangulation theorem]] implies that [[products]] of [[coset spaces]] $G/H_2 \,\times\, G/H_2$ admit an [[G-CW complex]]-structure (crucially using here that $G$ is assumed to be a [[Lie group]], so that its [[coset spaces]] have the structure of [[smooth manifolds]] with [[smooth function|smooth]] [[group actions]].) The required argument to make this into a proof of monoidal model category structure is spelled out as [DHLPS 2019, Prop. 1.1.3 (iii)](#DHLPS19), there in the further generality of [[proper equivariant homotopy theory]]. (Under the above assumption that $G$ is not just a Lie group but a [[compact Lie group]], the classes of "$\mathcal{Com}$-cofibrations" and of "$G$-cofibrations" in [DHLPS 19, Def. 1.1.2](#DHLPS19) agree, since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]).
\end{proof}

Prop. \ref{MonoidalModelCategoryStructure} immediately implies (by [this general Prop.](monoidal+model+category#InternalHomQuillenAdjunction)):
\begin{proposition}
  \label{InternalHomQuillenAdjunction}
**([[internal hom]] [[Quillen adjunction]])**
\linebreak
  For $X \,\in\, G Act\big(TopSp_{Qu}\big)_{fine}$ a [[cofibrant object]], the [[functor]] which assigns [[mapping spaces]] out of $X$ equipped with the [[conjugation action]], is a [[right Quillen functor]], hence makes a [[Quillen adjunction]] together with the [[functor]] of taking the [[product]] with $X$ (the [[k-ification|k-ified]] [[product topological space]]) equipped with the [[diagonal action]]:
\[
  G Act\big( TopSp_{Qu}\big)_{fine}
    \underoverset
    {\underset{Maps(X,-)}{\longrightarrow}}
    {\overset{X \times (-)}{\longleftarrow}}
    {\bot_{\mathrlap{Qu}}}
  G Act\big( TopSp_{Qu}\big)_{fine}
  \mathrlap{\,.}
\]
\end{proposition}

## Related concepts

* [[Borel model structure]]

* [[equivariant homotopy theory]]

* [[Elmendorf's theorem]]

## References

The model structure itself was first discussed in:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], Section 1.2 of: _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

Further properties, such as cofibrant generation, properness, and topological enrichment and are established in:

* {#Guillou06} [[Bert Guillou]], Prop. 3.12 in: _A short note on models for equivariant homotopy theory_, 2006 ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf), [[GuillouEquivariantHomotopy.pdf:file]])

* {#Fausk08} [[Halvard Fausk]], Prop. 2.11 in: *Equivariant homotopy theory for pro-spectra*,  Geom. Topol. 12 (2008) 103-176 ([doi:10.2140/gt.2008.12.103](http://dx.doi.org/10.2140/gt.2008.12.103), [arXiv:math/0609635](https://arxiv.org/abs/math/0609635))

* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], Theorem 1.6 in: _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [doi:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))

* {#Stephan13} [[Marc Stephan]], _On equivariant homotopy theory for model categories_, Homology Homotopy Appl. 18(2) (2016) 183-208 ([arXiv:1308.0856](http://arxiv.org/abs/1308.0856), [doi:10.4310/HHA.2016.v18.n2.a10](https://dx.doi.org/10.4310/HHA.2016.v18.n2.a10))

* {#Schwede18} [[Stefan Schwede]], Prop. B.7 in: _[[Global homotopy theory]]_, New Mathematical Monographs **34** Cambridge University Press, 2018 ([doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161), [arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

In addition, the monoidal model category structure is made explicit (in the generality of [[proper equivariant homotopy theory]]) in:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], Prop. 1.1.3 in: _Proper equivariant stable homotopy theory_, Memoirs of the AMS ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

For more see [the references](Elmendorf's+theorem#References) at *[[Elmendorf's theorem]]*.


fine sheaf

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## Definition

Let $X$ be a [[paracompact space|paracompact]] [[Hausdorff space]]. A [[sheaf]] $F$ of [[groups]] over $X$ is __fine__ if for every two disjoint closed subsets $A,B\subset X$, $A\cap B = \emptyset$, there is an [[endomorphism]] of the sheaf of groups $F\to F$ which restricts to the identity in a neighborhood of $A$ and to the $0$ endomorphism in a neighborhood of $B$. Every fine sheaf is [[soft sheaf|soft]].

A slightly different definition is given in [Voisin 2002](#Voisin02) (Vol I, Def. 4.35):

A __fine sheaf__ $\mathcal{F}$ over $X$ is a sheaf of $\mathcal{A}$-[[modules]], where $\mathcal{A}$ is a sheaf of [[rings]] such that, for every [[open cover]] $U_i$ of $X$, there is a [[partition of unity]] $1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering.

Another definition is given by [Godement 1958](#Godement58)
(the paragraph after Theorem II.3.7.2):
a sheaf $F$ is __fine__ if the [[internal hom]] $Hom(F,F)$ is a [[soft sheaf]].

## Properties

* For any fine sheaf $J$ over a paracompact Hausdorff space $M$, the [[sheaf cohomology]] groups of positive degree $j$ are all trivial

$$
H^j(M,J) = 0.
$$

* Any sheaf $S$ [[abelian sheaf|of abelian groups]] admits a [[resolution]] by fine sheaves $\{J_i\}$

$$
0\to S\to J_0\to J_1\to \cdots
$$

These resolutions can be used to compute the sheaf cohomology groups of $S$ in terms of the induced maps of sections

$$
d_i^*: \Gamma(M,J_i)\to\Gamma(M,J_{i+1}).
$$

as

$$
H^q(M,S) \cong (\text{ker}d^*_{q})/(\text{im}d^*_{q-1})
$$

(see Section 3 of [Gunning (1966)](#Gunning66).).

## Related concepts

* [[sheaf]]

* [[abelian sheaf cohomology]]

  * [[soft sheaf]]

  * [[flabby sheaf]]

## References

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

* {#Gunning66} [[Robert C. Gunning]], *Lectures on Riemann Surfaces*, Princeton University Press (1966) [[pdf](http://ronan.terpereau.perso.math.cnrs.fr/Master_Class_2023_Dijon/GUNNING_Lectures%20on%20Riemann%20Surfaces.pdf)]

* {#Voisin02} [[Claire Voisin]], _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__ (2002/3) [[doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344)]

category: sheaf theory


fine structure constant

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## Idea

In [[quantum electrodynamics]] the square of the [[coupling constant]] $g$ of the [[electron-photon interaction]] is called the _fine structure constant_

$$
  \alpha = \tfrac{1}{4 \pi} g^2
  \,.
$$

Historically the term originates from the fact via [[perturbative QFT]] effects such as the [[Lamb shift]], this constant governs tiny corrections ("fine structure") to the [[energy]] spectrum of the [[electron]] in the [[hydrogen atom]].

## References

See also:

* Wikipedia, _[Fine structure constant](https://en.wikipedia.org/wiki/Fine-structure_constant)_

* Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio, *Tenth-Order QED Contribution to the Electron $g-2$ and an Improved Value of the Fine Structure Constant*, Phys. Rev. Lett. 109, 111807 ([arXiv:1205.5368](https://arxiv.org/abs/1205.5368), [doi:10.1103/PhysRevLett.109.111807](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.111807))

[[!redirects fine structure constant]]


finer topology

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## Definition

+-- {: .num_defn #TopologyFinerCoarser}
###### Definition
**(finer/coarser topologies)**

Let $X$ be a [[set]], and let $\tau_1, \tau_2 \subset P(X)$ be two [[topological space|topologies]] on $X$,
hence two choices of [[open subsets]] for $X$, making it a [[topological space]]. If

$$
  \tau_1 \subset \tau_2
$$

hence if every open subset of $X$ with respect to $\tau_1$ is also regarded as open by $\tau_2$, then
one says that

* the topology $\tau_2$ is _finer_ than the topology $\tau_1$

* the topology $\tau_1$ is _coarser_ than the topology $\tau_2$.

=--

## Examples

+-- {: .num_example #CoDiscreteTopology}
###### Example
**(discrete and co-discrete topology)**

Let $X$ be any [[set]]. Then there are always the following two extreme
possibilities of equipping $X$ with a topology $\tau \subset P(X)$, and hence making it a [[topological space]]:

1. $\tau \coloneqq P(X)$ the set of _all_ subsets;

   this is called the _[[discrete topology]]_ on $X$, it is the [[finer topology|finest topology]] (def. \ref{TopologyFinerCoarser}) on $X$,

   we write $Disc(X)$ for the resulting topological space;

1. $\tau \coloneqq \{ \emptyset, X \}$ the set contaning only the [[empty set|empty]] subset of $X$ and all of $X$ itself;

   this is called the _[[codiscrete topology]]_ on $X$, it is the [[coarser topology|coarsest topology]] (def. \ref{TopologyFinerCoarser}) on $X$

   we write $CoDisc(X)$ for the resulting topological space.

=--

## References

* Wikipedia, _[Comparison of topologies](https://en.wikipedia.org/wiki/Comparison_of_topologies)_

[[!redirects finer topologies]]

[[!redirects coarser topology]]
[[!redirects coarser topologies]]

[[!redirects comparison of topologies]]


FinInj

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## Idea
 {#Idea}

__$FinInj$__ (or __$FinSet_inj$__ or often just **$FI$**) refers to the [[wide subcategory]] of [[FinSet]] with its [[morphisms]] restricted to [[monomorphisms]] ([[injective functions]]).

It [may be characterised](https://github.com/punkdit/categories/blob/master/www.mta.ca/cat-dist/archive/2008/08-12#L1944-L1959) as:

* the free ([[symmetric monoidal category|symmetric]]) [[semicocartesian category]] on a single object

* the free [[symmetric monoidal category]] on a pointed object $I \to X$ (where $I$ is the [[monoidal category|monoidal unit]])

* the free [[monoidal category]] on an object $X$ equipped with an involution $s : X \otimes X \to X \otimes X$ satisfying the braid relations, and a morphism $i : I \to X$ satisfying $s \circ i \otimes X = s \circ X \otimes i$

A [[functor]] from $FinSet_{inj}$ to [[Vect]] (or more generally to some $R$[[Mod]]) is also known as an *[[FI-representation]]*, a kind of accumulated form of [[representation theory of the symmetric group|$Sym(n)$-representations]] as $n \in \mathbb{N}$ ranges (cf. also *[[representation stability]]*).

## Related concepts

* [[Inj]]

* [[FinSet]]

* [[FI-representation]], [[representation stability]]

## References

Introducing the notion of *[[FI-modules]]* ([[FinInj|$FinSet_{inj}$]]-[[representations]]):

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]

[[!redirects FinSetInj]]


finitary functor

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##Idea

According to the usual definition, a [[functor]] $F\colon C \to D$ is __finitary__ if it preserves all [[filtered colimits]].  An important property is that a [[monadic category]] (over [[Set]]) is [[finitary monadic category|finitary]] (in other words, given by a [[Lawvere theory]]) if and only if its [[forgetful functor]] is finitary.

A slightly weaker definition is used in _The [[Joy of Cats]]_; there, $F$ is __finitary__ if it maps every [[directed colimit]] in $C$ to a jointly [[epimorphism|epic]] diagram (but not necessarily a universal one) in $D$.  With this definition, an [[algebraic category]] is [[finitary algebraic category|finitary]] if and only if its [[forgetful functor]] is finitary.  The previous result also holds, since the two definitions are equivalent for monads on [[Set]].

[[!redirects finitary category]]