pullback

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###Context###
#### Limits and colimits
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[[!include infinity-limits - contents]]
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#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In the [[category]] [[Set]] a 'pullback' is a [[subset]] of the [[cartesian product]] of two [[sets]].  Given a [[diagram]] of [[sets]] and [[functions]] like this:

\begin{center}
  \begin{tikzcd}
    A \ar[rd, "f"'] &   & B \ar[ld, "g"] \\
                    & C
  \end{tikzcd}
\end{center}

the 'pullback' of this diagram is the [[subset]] $X \subseteq A \times B$ consisting of pairs $(a,b)$ such that the [[equation]] $f(a) = g(b)$ holds.

A pullback is therefore the [[categorical semantics]] of an _[[equation]]_.

This construction comes up, for example, when $A$ and $B$ are [[fiber bundles]] over $C$: then $X$ as defined above is the [[product]] of $A$ and $B$ in the category of fiber bundles over $C$.  For this reason, a pullback is sometimes called a __fibered product__ (or _fiber product_ or _fibre product_).

In this case, the fiber of $A \times_C B$ over a [[generalized element|(generalized)]] element $x$ of $C$ is the ordinary [[product]] of the fibers of $A$ and $B$ over $x$.  In other words, the fiber product is the product taken fiber-wise.  Of course, the fiber of $A$ at the generalized element $x\colon I \to C$ is itself a fibre product $I \times_C A$; the terminology depends on your point of view.

Note that there are maps $p_A\colon X \to A$, $p_B\colon X \to B$ sending any $(a,b) \in X$ to $a$ and $b$, respectively.  These maps make this [[commuting diagram|square commute]]:

\begin{center}
  \begin{tikzcd}
                    & X \ar[ld, "p_A"'] \ar[rd, "p_B"] \\
    A \ar[rd, "f"'] &   & B \ar[ld, "g"]               \\
                    & C
  \end{tikzcd}
\end{center}

In fact, the pullback is the [[universal property|universal]] solution to finding a commutative square like this.  In other words, given _any_ [[commutative diagram|commutative square]]

\begin{center}
  \begin{tikzcd}
                    & Y \ar[ld, "q_A"'] \ar[rd, "q_B"] \\
    A \ar[rd, "f"'] &   & B \ar[ld, "g"]               \\
                    & C
  \end{tikzcd}
\end{center}

there is a unique function $h\colon Y \to X$ such that

$$ p_A h = q_A $$

and

$$ p_B h = q_B\,.$$

Since this universal property expresses the concept of pullback purely arrow-theoretically, we can formulate it in any category.  It is, in fact, a simple special case of a [[limit]].

## Definition

### In category theory

####As a limit

A **pullback** is a [[limit]] of a [[diagram]] like this:

\begin{center}
  \begin{tikzcd}
    a \ar[rd, "f"'] &   & b \ar[ld, "g"] \\
                    & c
  \end{tikzcd}
\end{center}

Such a diagram is also called a **pullback diagram** or a [[cospan]].  If the [[limit]] exists, we obtain a commutative square

\begin{center}
  \begin{tikzcd}
                    & x \ar[ld, "p_a"'] \ar[rd, "p_b"] \\
    a \ar[rd, "f"'] &   & b \ar[ld, "g"]               \\
                    & c
  \end{tikzcd}
\end{center}

and [[generalized the|the]] object $x$ is also called the **pullback**. It is well defined up to unique [[isomorphism]]. It has the universal property already described above in the special case of the category $Set$.

The last commutative square above is called a __pullback square__.

The concept of pullback is dual to the concept of [[pushout]]: that is, a pullback in $C$ is the same as a pushout in the [[opposite category]] $C^{op}$.

####Nuts and bolts

Let $\mathcal{C}$ be a category, with $f\colon a\to c$ and $g\colon b\to c$ coterminal arrows in $\mathcal{C}$ as below

\begin{center}
  \begin{tikzcd}
    a \ar[rd, "f"'] &   & b \ar[ld, "g"] \\
                    & c
  \end{tikzcd}
\end{center}

A **pullback** of $f$ and $g$ consists of an object $x$ together with arrows $p_a\colon x\to a$ and $p_b\colon x\to b$ such that the following diagram commutes universally

\begin{center}
  \begin{tikzcd}
                    & x \ar[ld, "p_a"'] \ar[rd, "p_b"] \\
    a \ar[rd, "f"'] &   & b \ar[ld, "g"]               \\
                    & c
  \end{tikzcd}
\end{center}

This means that for any other object $x'$ with arrows $p'_a\colon x'\to a$ and $p'_b\colon x'\to b$ such that

\begin{center}
  \begin{tikzcd}
                    & x' \ar[ld, "p'_a"'] \ar[rd, "p'_b"] \\
    a \ar[rd, "f"'] &   & b \ar[ld, "g"]               \\
                    & c
  \end{tikzcd}
\end{center}

commutes, there exists a unique arrow $u\colon x'\to x$ such that

\begin{center}
  \begin{tikzcd}
                    & x' \ar[ld, "p'_a"'] \ar[rd, "p'_b"] \ar[d, "u", dotted] \\
    a & x \ar[l, "p_a"'] \ar[r, "p_b"]  & b               \\
  \end{tikzcd}
\end{center}

commutes.

### In type theory

In [[type theory]] a [[pullback]] $P$ in

\begin{center}
  \begin{tikzcd}
    P \ar[r] \ar[d] & A \ar[d, "f"] \\
    B \ar[r, "g"']  & C
  \end{tikzcd}
\end{center}

is given by the [[dependent sum]] over the [[dependent type|dependent]] [[equality type]]

$$
  P = \sum_{a\colon A} \sum_{b\colon B} (f(a) = g(b))
  \,.
$$

## Properties

+-- {: .num_prop #PullbacksAsEqualizers}
###### Proposition
**(pullbacks as equalizers)

If [[product]]s exist in $C$, then the pullback

\begin{center}
  \begin{tikzcd}
    a \times_c b \ar[r] \ar[d] & a \ar[d, "f"] \\
    b \ar[r, "g"']             & c
  \end{tikzcd}
\end{center}

is equivalently the [[equalizer]]

\begin{center}
  \begin{tikzcd}
    a \times_c b \ar[r] & a \times b \ar[r, shift left, "fp_1"] \ar[r, shift right, "gp_2"'] & c
  \end{tikzcd}
\end{center}

of the two morphisms induced by $f$ and $g$ out of the [[product]] of $a$ with $b$.

=--

As a consequence of the above, any category with binary products and equalizers has pullbacks.  Conversely any category with binary products and pullbacks has equalizers: the equalizer of $f,g:A \to B$ is the pullback of $(1,f) \maps A \to A \times B$ and $(1,g) \maps A \to A \times B$.

+-- {: .num_prop #PullbackPreservesMonomorphisms}
###### Proposition
**(pullbacks preserve monomorphisms and isomorphisms)

Pullbacks preserve [[monomorphisms]] and [[isomorphisms]]:

If

\begin{center}
  \begin{tikzcd}
    a \ar[r] \ar[d, "f^\ast g"'] & b \ar[d, "g"] \\
    c \ar[r, "f"']               & d
  \end{tikzcd}
\end{center}

is a pullback square in some category then:

1. if $g$ is a [[monomorphism]] then $f^\ast g$ is a monomorphism;

1. if $g$ is an [[isomorphism]] then $f^\ast g$ is an isomorphism.

On the other hand that $f^\ast g$ is a monomorphism does not imply that $g$ is a monomorphism.

=--

+-- {: .num_prop}
###### Proposition
**([[pasting law for pullbacks]])**

In any category consider a diagram of the form

\begin{center}
  \begin{tikzcd}
    a \ar[r] \ar[d] & b \ar[r] \ar[d] & c \ar[d] \\
    d \ar[r]        & e \ar[r]        & f
  \end{tikzcd}
\end{center}

There are three commuting squares: the two inner ones and the outer one.

Suppose the right-hand inner square is a pullback, then:

The square on the left is a pullback if and only if the outer square is.

=--

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

Pasting a morphism $x \to a$ with the outer square gives rise to a commuting square over the (composite) bottom and right edges of the diagram.  The square over the cospan in the left-hand inner square arising from $x \to a$ includes a morphism into $b$, which if $b$ is a pullback induces the same commuting square over $d \to e \to f$ and $c \to d$.  So one square is universal iff the other is.
=--

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

The converse implication does not hold: it may happen that the outer and the left square are pullbacks, but not the right square.

=--

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

For instance let $i\colon a \to b$ be a [[split monomorphism]] with [[retract]] $p\colon b \to a$ and consider

\begin{center}
  \begin{tikzcd}
    a \ar[r, "="] \ar[d, "="'] & a \ar[r, "="] \ar[d, "i"] & a \ar[d, "="] \\
    a \ar[r, "i"']             & b \ar[r, "p"']            & a
  \end{tikzcd}
\end{center}

Then the left square and  the outer rectangle are pullbacks but the right square cannot be a pullback unless $i$ was already an [[isomorphism]].

=--

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

On the other hand, in the [[(∞,1)-category]] of [[∞-groupoids]], there is a sort of "partial converse"; see [[homotopy pullback#HomotopyFiberCharacterization]].

=--

### Saturation

The [[saturated class of limits|saturation]] of the class of pullbacks is the class of limits over categories $C$ whose groupoid reflection $\Pi_1(C)$ is trivial and such that $C$ is [[L-finite category|L-finite]].

### Pullback functor
{#PullbackFunctor}

If $f\colon X \to Y$ is a [[morphism]] in a [[category]] $C$ with pullbacks, there is an induced pullback [[functor]] $f^*\colon C/Y \to C/X$, sometimes also called [[base change]].

## Related concepts

* [[pushout]]

[[!include notions of pullback -- contents]]

## References

Early use of the terminology "pullback":

* {#Mitchell65} [[Barry Mitchell]], Section I.7 of: *Theory of categories*, Pure and Applied Mathematics **17**, Academic Press (1965) [[ISBN:978-0-12-499250-4](https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4)]

Textbook accounts:

* {#MacLane} [[Saunders MacLane]], p. 71 in: *[[Categories for the Working Mathematician]]*, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 2.5 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

See also:

* [[Joyal|André Joyal]], _[[joyalscatlab:Cartesian squares]]_

In view of more general [[finite limits]] and [[L-finite limits]]:

* [[Robert Paré]], *Simply connected limits*.  Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, [CMS](http://math.ca/10.4153/CJM-1990-038-6)

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[[!redirects cartesian diagram]]
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[[!redirects fibre product]]
[[!redirects fibre products]]
[[!redirects fibered product]]
[[!redirects fibered products]]
[[!redirects fibred product]]
[[!redirects fibred products]]


pullback bundle

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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[bundle]] $p \colon E \to X$, and a morphism $f \colon Y \to X$, then the _pullback bundle_ $f^\ast E \to Y$ is (if it exists) simply the [[pullback]] of $p$ along $f$, regarded as a [[bundle]] over $Y$.

## Universal bundles

Over a [[paracompact topological space]] every [[vector bundle]] is [[isomorphism|isomorphic]] to a pullback bundle of a [[universal vector bundle]] over a [[classifying space]]. See there for more.

## Related concepts

[[!include notions of pullback -- contents]]

[[!redirects pullback bundles]]

[[!redirects pullback of bundles]]
[[!redirects pullbacks of bundles]]

[[!redirects pullback of fiber bundles]]
[[!redirects pullbacks of fiber bundles]]

[[!redirects pullback of vector bundles]]
[[!redirects pullbacks of vector bundles]]

[[!redirects pullback vector bundle]]
[[!redirects pullback vector bundles]]


pullback in a derivator

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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# Pullbacks in derivators

* table of contents
{: toc}

## Idea

A *pullback in a derivator* is the generalization to the context of a [[derivator]] of the notion of [[pullback]] in ordinary category theory.  Viewing a derivator as the "shadow" of an [[(∞,1)-category]], the notion of pullback therein coincides with the notion of [[homotopy pullback]] in an $(\infty,1)$-category.

## Definition

Let $\square$ denote the category
$$ \array{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d} $$
that is the "free-living [[commutative square]]", and let $L$ be the full subcategory of $\square$ on $b,c,d$, with inclusion $u\colon L\to \square$.

Let $D$ be a [[derivator]] and let $X\in D(\square)$ be a commutative square in $D$.  We say that $X$ is a **pullback square**, or is **cartesian**, if the unit $X\to u_* u^* X$ of the [[adjunction]] $u^*\dashv u_*$ is an isomorphism.  Since $u$ is [[fully faithful]], so is $u_*$, so this is equivalent to saying that there exists some $Y\in D(L)$ such that $X\cong u_* Y$.

If $D$ is merely a prederivator, then we can phrase the same definition by saying that $X$ has the [[free object|universal property]] that $u_* u^* X$ would, if the whole functor $u_*$ existed.

The dual notion, of course, is a **pushout** or **cocartesian** square.

## Properties

### Pasting law {#Pasting}

Using properties of [[homotopy exact squares]], we can prove the "pasting law" for pullback squares in a derivator:

+-- {: .un_lemma}
###### Lemma
Given a diagram
$$ \array{a & \to & b & \to & c \\
\downarrow & & \downarrow & & \downarrow \\
d & \to & e & \to & f } $$
in which the right-hand square $bcef$ is a pullback, then the left-hand square $abde$ is a pullback if and only if the outer rectangle $acdf$ is a pullback.
=--

The following proof should be compared and contrasted with the [standard proof](http://ncatlab.org/nlab/show/pullback#Pasting) for pullbacks in 1-categories, and the [quasi-categorical proof](http://ncatlab.org/nlab/show/%28infinity%2C1%29-pullback#QuasiCatPastingLaw) for pullbacks in $(\infty,1)$-categories.  In particular, note that the statement for derivators is a generalization of both, since both 1-categories and $(\infty,1)$-categories give rise to derivators.

+-- {: .proof}
###### Proof
First of all, by "a diagram" in a derivator, we mean an object of $D(X)$ for some suitable category $X$.  In the above case, $X$ is the category consisting of two commutative squares, as pictured above.  We'll write $abcdef$ for this $X$, and similarly $cdef$ for its lower-right L-shaped subcategory, and so on.  We leave the verification of [[homotopy exact square|homotopy exactness]] of all squares to the reader.

Firstly, since the squares
$$\array{cef & \overset{}{\to} & bcef\\
  \downarrow && \downarrow\\
  cdef & \underset{}{\to} & abcdef} \qquad and \qquad
\array{cdf & \overset{}{\to} & acdf\\
  \downarrow && \downarrow\\
  cdef& \underset{}{\to} & abcdef}$$
are exact, if we start from a $cdef$-diagram and right Kan extend it to a full $abcdef$-diagram, then the right-hand square and outer rectangle must be pullback squares.  Moreover, by composition of adjoints, right Kan extension from $cdef$ to $abcdef$ is equivalent to first extending to $bcdef$ and then to $abcdef$, and since the square
$$\array{bde & \overset{}{\to} & abde\\
  \downarrow && \downarrow\\
  bcdef & \underset{}{\to} & abcdef}$$
is exact, the left-hand square in such an extension must also be a pullback.

Now if we start with an $abcdef$-diagram, say $F$, we can restrict it to a $cdef$-diagram and then right Kan-extend it to a new $abcdef$-diagram.  If $u\colon cdef \to abcdef$ is the inclusion, then this results in $u_\ast u^\ast F$, and we have a canonical natural transformation $\eta \colon F \to u_\ast u^\ast F$ (the unit of the adjunction $u^\ast \dashv u_\ast$).  Since the square
$$\array{cdef & \overset{}{\to} & cdef \\
  \downarrow && \downarrow\\
  cdef & \underset{}{\to} & abcdef}$$
is exact, the counit $u^\ast u_\ast G \to G$ is an isomorphism for any $G$, and in particular for $G=u^\ast F$, from which it follows by the triangle identities that $u^\ast F \to u^\ast u_\ast u^\ast F$ is also an isomorphism --- i.e. the components of $\eta\colon F \to u_\ast u^\ast F$ at $c$, $d$, $e$, and $f$ are isomorphisms.  Now if the right-hand square of $F$ is a pullback, then the restrictions of $F$ and $u_\ast u^\ast F$ to $bcef$ are both pullback squares; hence since the $cef$-components of $\eta$ are isomorphisms, so is the $b$-component.  And if the left-hand square of $F$ is a pullback, then we can play the same game with $abde$ to show that the $a$-component of $\eta$ is an isomorphism, while if the outer rectangle is a pullback, we can play it with $acdf$.  Hence in both of these cases, $\eta$ itself is an isomorphism, since all of its components are --- and thus the remaining square in $F$ is also a pullback, since we have shown that it is so in $u_\ast u^\ast F$.
=--

### Detection Lemma {#Detection}

The following lemma, which detects when squares occurring in a Kan extension are pullbacks or pushouts, is due to [Jens Franke](#Franke); see also [Groth](#Groth).  We state it in terms of pushouts.

+-- {: .num_lemma #CartesianSquareDetection}
###### Lemma
Let $f\colon K\to J$ be any functor and let $i\colon \square \to J$ be injective on objects, with lower vertex $i(1,1) = z$.  Suppose that $z$ is not in the image of $f$, and that the induced functor $\Gamma \to (J \setminus z)/z$ is a nerve equivalence (such as if it has an adjoint).  Then for any derivator $D$ and any $Y\in D(K)$, the square $i^* f_! Y$ is cocartesian.
=--
+-- {: .proof}
###### Proof
Since $f_! = j_! \bar{f}_!$ where $\bar{f}\colon K\to (J\setminus z)$ is induced by $f$ and $j\colon (J\setminus z) \to J$ is the inclusion, it suffices to suppose that $K = (J\setminus z)$.  Now what we want is to prove that the following square is [[homotopy exact square|homotopy exact]]:
$$ \array{ \Gamma &\to & (J\setminus z) \\
  \downarrow && \downarrow\\
  \square &\to & J}$$
Exactness is trivial at all objects of $\square$ except $(1,1)$.  In that case, we paste with another square:
$$ \array{ \Gamma &\to& \Gamma &\to & (J\setminus z) \\
  \downarrow & \swArrow & \downarrow && \downarrow\\
  \ast &\underset{(1,1)}{\to} &\square &\to & J}$$
The left-hand square is a [[comma square]], hence homotopy exact, so it suffices to show that the composite square is homotopy exact.  But the comma object associated to the cospan $\ast \to J \leftarrow (J\setminus z)$ is $(J \setminus z)/z$, and of course this comma square is also exact.  And the composite square factors through this comma square by the functor $\Gamma \to (J \setminus z)/z$ which is assumed a nerve equivalence; hence it is also homotopy exact.
=--

## References

See all references at [[derivator]].  Referred to particularly above are:

* Jens Franke, _Uniqueness theorems for certain triangulated categories with an Adams spectral sequence_, [K-theory archive](http://www.math.uiuc.edu/K-theory/0139/)
{#Franke}

* Moritz Groth, _Derivators, pointed derivators, and stable derivators_ [pdf](http://www.math.uni-bonn.de/~mgroth/groth_derivators.pdf)
{#Groth}

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pullback in cohomology

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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given any kind of [[cohomology theory]], it is [[contravariant functor]] on some [[category]] $\mathcal{C}$ of [[spaces]] of sorts. For example, for [[ordinary cohomology]] or [[Whitehead-generalized cohomology theories]] this functor goes from [[pointed topological spaces|pointed]] [[CW-complexes]] $X$ (or more generally: [[CW-pairs]]) to [[graded abelian groups]] $E^\bullet(X)$.

The value of this functor on any morphism $f \;\colon\; X \to Y$ is called the _pullback in $E$-cohomology_ $f^\ast \;\colon\; E^\bullet(Y) \to E^\bullet(X)$.

Notice that, a priori, this is not related to the notion of [[pullback]] in the sense of a [[cospan]]-shaped [[limit]] in some [[category]], though for good enough "geometric cycles" for $E$-cohomology the notions may actually agree. For example, pullback in $G$-[[non-abelian cohomology]] is given by forming the [[pullback bundles]] of the $G$-[[principal bundles]] which are classified by the given cohomology classes.

## Related concepts

* [[long exact sequence in cohomology]]

  [[connecting homomorphism]]

[[!include notions of pullback -- contents]]


pullback of a differential form

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For $f \colon X \to Y$ a [[smooth function]] between [[smooth manifold]], and for $\omega \in \Omega^n(Y)$ a [[differential n-form]], there is the _pullback_ $n$-form $f^* \omega \in \Omega^n(X)$.

### In terms of push-forward of vector fields

If differential forms are defined as linear duals to [[vectors]] then pullback is the dual operation to [[pushforward of a vector field]]

$$
  f^* \omega(v_1, \cdots, v_n) = \omega(f_* v_1, \cdots, f_* v_n)
  \,.
$$

### In terms of coordinate expression

Differential forms may be defined by [[Yoneda extension]] from differential forms on [[Cartesian spaces]] (see at _[[geometry of physics -- differential forms]]_).

For $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^k$ [[Cartesian spaces]] and $f \;\colon\; X \longrightarrow Y$ a [[smooth function]] between them, and on [[differential 1-forms]]

$$
  \omega = \sum_{i = 1}^k \omega_i \mathbf{d}x^i
$$

the pullback operation $f^\ast$ is given by

$$
  f^* \mathbf{d}x^i
  \;\coloneqq\;
  \sum_{j = 1}^{\tilde k} \frac{\partial f^i}{\partial \tilde x^j} \mathbf{d}\tilde x^j
$$

and hence

$$
  f^* \omega = f^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right)
  \coloneqq
  \sum_{i = 1}^k \left(f^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial f^i }{\partial \tilde x^j}  \mathbf{d} \tilde x^j
  \,,
$$

where

* $f^* \omega_i$ is the pullback of functions defined by

  $$
    (f^* \omega_i)(x) = \omega_i(f(x)) \;\;\;\forall x \in X
  $$

* the function

  $$
    \frac{\partial f^i}{\partial \tilde x^j} \colon \mathbb{R}^{\tilde k} \to \mathbb{R}
  $$

  is the [[partial derivative]] of the $k$-th [[coordinate]] component of $f$ along the $j$the coordinate.

## Properties

### Compatibility with the de Rham differential

+-- {: .num_prop #CompatibilityWithDeRhamDifferential}
###### Proposition
**(compatiblity with the [[de Rham differential]])**

Pullback of differential forms commutes with the [[de Rham differential]]:

$$
  f^* \circ \mathbf{d}_Y = \mathbf{d}_X \circ f^*
  \,.
$$

Hence it constitutes a [[chain map]] between the [[de Rham complexes]]

$$
  f^* \colon \Omega^\bullet(Y) \to \Omega^\bullet(X)
  \,.
$$

=--

### Sheaf of differential forms

Under pullback differential forms form a [[presheaf]] on the catories [[CartSp]] and [[SmthMfd]], in fact a [[sheaf]] with respect to the standard [[open cover]]-[[coverage]].

## Related concepts

* [[change of integration variables]]

[[!include notions of pullback -- contents]]

## References

A standard reference is

* Bott, Tu, _Differential forms in algebraic topology_.

See also for instance

* [[Theodore Frankel]], Section 2.7 of _[[The Geometry of Physics - An Introduction]]_

[[!redirects pullbacks of a differential form]]

[[!redirects pullback of differential forms]]
[[!redirects pullbacks of differential forms]]

[[!redirects pullback form]]
[[!redirects pullback forms]]

[[!redirects pullback of functions]]
[[!redirects pullbacks of functions]]

[[!redirects pullback of a function]]
[[!redirects pullbacks of functions]]


pullback of a distribution

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

#Contents#
* table of contents
{:toc}

## Idea

The _pullback_ of a [[distribution]] $u \in \mathcal{D}'(X)$ along a [[smooth function]] $f\colon Y \to X$ is supposed to like the [[composition]] $u \circ f$ if $u$ is thought of as a [[generalized function]], such that it is exactly this in the case that $u$ actually is an ordinary function.

## Definition

In general, the pullback of distributions along a smooth function $f$ is meant to be a unique extension to distributions of the operation which on [[bump functions]] is given by pre-composition with $f$. There are various different conditions that are sufficient for this being well defined.

If the map $f$ we are pulling back along is a [[submersion]], the all distributions pull back:

+-- {: .num_prop #PullbackOfDistributionAlongSubmersion}
###### Proposition
**(pullback of any distribution along a [[submersion]])**

If $X_1, X_2 \subset \mathbb{R}^n$ are two [[open subsets]] of [[Euclidean space]], and if

$$
  f \;\colon\; X_1 \overset{}{\longrightarrow} X_2
$$

is a [[submersion]] (i.e. its [[differential]] is a [[surjective function]] $d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2$ for all $x \in X_1$), then there is a unique [[continuous linear functional]]

$$
  f^\ast \;\colon\; \mathcal{D}'(X_2) \longrightarrow \mathcal{D}'(X_1)
$$

between spaces of distributions ([this def.](distribution#DistributionOnOpenSubsetOfEuclideanSpace)) which extends the [[pullback of functions]] in that on a [[non-singular distribution]] represented by a [[bump function]] $b$ it is given by pre-[[composition]]

$$
  f^\ast b = b \circ f
  \,.
$$

This is hence called the _pullback of distributions_.

If $f$ happens to be a [[diffeomorphism]] with [[inverse function]] $f^{-1}$ then $f^\ast u$ for $u \in \mathcal{D}'(X_2)$ is explicitly given by

$$
  \langle  f^\ast u , b \rangle
  \;=\;
  \langle u, \frac{1}{det(D f)}  b \circ f^{-1} \rangle
$$

where $det(D f)$ denotes the [[Jacobian determinant]] (the [[determinant]] of the [[derivative]] of $f$).

=--

([Hörmander 90, theorem 6.1.2](#Hoermander90), [Melrose 03, sections 4.17, 4.19 and 4.21](#Melrose03))

If $f$ is not a submersion, then pullback is still defined on those distributions whose [[wave front set]] does not intersect the [[conormal bundle]] of $f$:

+-- {: .num_defn #PullbackOfDistributionsWhoseWaveFrontDoesNotIntersectNormalBundle}
###### Proposition
**(pullback if [[wave front set]] is [[disjoint subset|disjoint]] from [[conormal bundle]])**

Given a [[smooth function]] $f \colon X \to Y$, then there is a unique [[continuous linear functional]] $f^\ast$ from the space of those distributions on $Y$ whose [[wave front set]] does not intersect the [[conormal bundle]] of $f$

$$
  f^\ast
   \;\colon\;
  \left\{
    u \in \mathcal{D}'(Y) \;\vert\; WF(u) \cap N^\ast_f = \emptyset
  \right\}
    \longrightarrow
  \mathcal{D}'(X)
$$

such that

1. on [[non-singular distributions]] $u_g$ corresponding to [[smooth functions]] $g \colon Y \to \mathbb{C}$ it acts by [[precomposition]] with $f$:

   $$
     f^\ast (u_g) = u_{g \circ f}
   $$

1. for $u$ a distribution in the domain on the left, the [[wave front set]] of its pullback is inside the pullback of its wave front set

  $$
    \label{WaveFrontSetOfPullbackDistributionIsInsidePullbackOfWaveFrontSet}
    WF(f^\ast u)
    \subset
    f^\ast WF(u)
    \coloneqq
    \left\{
      (x, (d f(x))^\ast k)
      \;\vert\;
      ( f(x), \eta )
      \in
      WF(u)
    \right\}
  $$

=--

([Hörmander 90, theorem 8.2.4](#Hoermander90))

## Properties

* for continuity see at _[[Hörmander topology]]_

## Examples

+-- {: .num_example #RestrictionOfDistributions}
###### Example
**(restriction of distributions)**

Let $i \;\colon\; Y \hookrightarrow X$ be an [[open subset]] inclusion. This is clearly a [[submersion]], in fact a [[local diffeomorphism]], and hence prop. \ref{PullbackOfDistributionAlongSubmersion} applies. The resulting pullback operation

$$
  i^\ast \;\colon\; \mathcal{D}'(Y) \longrightarrow \mathcal{D}'(X)
$$

is also called _restriction of distributions_ ([Hörmander 90, first lines of section 2.2](#Hoermander90)). For $b \in C^\infty_c(Y)$ a [[bump function]] on $Y$, the restriction $i^\ast u$ of a distribution $u \in \mathcal{D}'(X)$ acts by

$$
  \langle i^\ast u, b\rangle = \langle  u, i_\ast b \rangle
  \,,
$$

where $i_\ast b \in C^\infty_{cp}(X)$ is the result of [[extension|extending]] $b$ by zero to all of $X$.

=--

## Related concepts

* [[change of integration variables]]

* [[composition of distributions]]

* [[scaling degree of a distribution]]

* [[extension of distributions]]

* [[support of a distribution]]

[[!include notions of pullback -- contents]]

## References

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#Melrose03} [[Richard Melrose]], sections 4.17, 4.19 and 4.21 of _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

* [[Sergiu Klainerman]], chapter 3, section 4 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

[[!redirects pullbacks of a distribution]]

[[!redirects pullback of distributions]]
[[!redirects pullbacks of distributions]]

[[!redirects restriction of a distribution]]
[[!redirects restrictions of a distribution]]

[[!redirects restriction of distributions]]
[[!redirects restrictions of distributions]]


pullback power

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

#Contents#
* table of contents
{:toc}

## Idea

The [[formal dual]] to the [[pushout product]] (frequently considered in the context of [[enriched model category]] theory) does not have a widely established name, but plausibly deserves to be called the _pullback powering_ operation. Note that it sometimes called _pullback hom_.

More precisely, a pushout product is defined with respect to a functor of the form $E_1\times E_2 \to E_3$, while a pullback power is defined with respect to a functor of the form $E_2^{op} \times E_3\to E_1$ or $E_1^{op}\times E_3 \to E_2$, of the sort that would be the right adjoints in a [[two-variable adjunction]].

Pullback powers and pushout products are related to [[factorization systems]] by the [[Joyal-Tierney calculus]].

## Definition

Let $\mathcal{C}$ be [[category]] with [[finite limits]] and let

$$
  [-,-]
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
  \longrightarrow
   \mathcal{C}
$$

a [[functor]] (out of the [[product category]] of the [[opposite category]] of $\mathcal{C}$ with $\mathcal{C}$ itself). Then for

$$
  g \;\colon\; X \to Y
$$

and

$$
  f \;\colon\; A \to B
$$

two [[morphisms]] in $\mathcal{C}$, their _pullback powering_ $g^f$ is the morphism

$$
   [B,X] \stackrel{[i , p]}{\to} [A,X] \times_{[A,Y]} [B,Y]
$$

into the evident [[fiber product]] on the right.

## Related concepts

* [[pullback power axiom]]

* [[powering]]

* [[Joyal-Tierney calculus]]

* [[enriched model category]]

[[!redirects pullback-power]]
[[!redirects pullback-powers]]

[[!redirects pullback powering]]
[[!redirects pullback powerings]]

[[!redirects pullback-powering]]
[[!redirects pullback-powerings]]

[[!redirects pullback powers]]

[[!redirects Leibniz cotensor]]

[[!redirects pullback hom]]
[[!redirects pullback-hom]]


pullback powering > history

see _[[pullback power]]_


pullback-stable colimit

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

#Contents#
* automatic table of contents goes here
{:toc}

## Definition

Let $C$ be a [[category]] with [[pullbacks]] and [[colimits]] of some [[small category|shape]] $D$.  We say that colimits of shape $D$ are __stable by [[base change]]__, or __stable under pullback__, or that these colimits are __universal__ if for every functor $G : D \to C$ and for all [[pullback]] diagrams of the form

$$
  \array{
    (colim_D G) \times_Z Y &\to& colim_D G
    \\
    \downarrow && \downarrow
    \\
    Y &\underset{f}{\to} & Z
  }
$$

the canonical morphism

$$
  \label{IsoForPullbackStabilityOfColimis}
  \underset{d \in D}{colim} (G(d) \times_Z Y)
    \stackrel{\simeq}{\to}
  (\underset{d \in D}{colim} G(d)) \times_Z Y
$$

is an [[isomorphism]].  This is equivalent to saying that every pullback [[functor]] $f^*: C/Z \to C/Y$ [[preserved limit|preserves]] $D$-colimits.  Similar definitions can be given for [[higher categories]].

To see that this definition is equivalent to the universality condition given at [[van Kampen colimit#universality_and_descent|van Kampen colimit]], observe that, given a [[natural transformation]] $\alpha' : F' \Rightarrow G'$, the diagram

$$
  \array{
    F'(*) &\overset{\alpha'_*}{\to}& colim_D G
    \\
    ^\id\downarrow && \downarrow^\id
    \\
    F'(*) &\underset{\alpha'_*}{\to} & colim_D G
  }
$$

is a degenerate [[pullback]] square, hence there is a canonical isomorphism

$$
  colim_{d \in D}(G(d) \times_{colim_D G} F'(*)) \simeq F'(*).
$$

But if $\alpha'$ is [[equifibered natural transformation|equifibered]], we have $G(d) \times_{colim_D G} F'(*) \simeq F(d)$, hence we get the desired isomorphism $F'(*) \simeq colim_D F$.

Conversely, given a pullback diagram as above, let $F' = f^* \circ G'$ (viewing $G'$ as a functor $D \to C/Z$ and remembering that colimits in $C/Z$ are [[over category#ColimitInSliceAreReflectedByColimitsInPlainCategory|computed]] as colimits in $C$) and $\alpha' : F' \Rightarrow G'$ the natural transformation induced by the pullback projections, which is [[equifibered natural transformation|equifibered]] as a consequence of the [[pasting law for pullbacks]]. Then $f^* \circ G'$ is a colimiting cocone, which is to say that $f^*$ preserves $colim_D G$.

## Examples

### Toposes

The stability of all colimits is one of [[Giraud's axioms]] that characterize [[Grothendieck toposes]] in the [[category theory|1-categorical context]] and Grothendieck-Rezk-Lurie [[(∞,1)-toposes]] in the [[higher category theory|higher categorical context]].  The fact that colimits are stable in toposes can be seen from the characterization of toposes as left-exact reflective subcategories of presheaf categories as follows:

* First observe that colimits are stable in $C =$ [[Set]].
* Now colimits are stable for $C =$ a [[presheaf]] category $[S^{op},Set]$, since colimits in such $C$ are computed objectwise in $Set$.  (See [[limits and colimits by example]].)
* Finally, stability of colimits is preserved in reflective subcategories, since the reflector preserves both colimits and pullbacks.

For [[(∞,1)-toposes]], this is [[Higher Topos Theory|HTT, theorem 6.1.0.6 (3) ii)]]

### Non-toposes

More generally, colimits are stable in any [[locally cartesian closed category]], since in that case the pullback functors $f^*$ all have right adjoints.  Conversely, if $C$ is cocomplete with all stable colimits, and the [[adjoint functor theorem]] applies to all its slice categories, then it is locally cartesian closed.

Colimits are also stable in any [[regular category|regular]] infinitary [[extensive category]].

But colimits are not stable in, for instance, $C = $ [[Ab]].

## Relation to commutativity and distributivity

Although stability of colimits appears as a sort of "commutativity" between colimits and pullbacks, it is not literally an instance of [[commutativity of limits and colimits]].  It is an *example* of the latter if the [[colimit]] over $D$ of the diagram constant on a single object (such as $Y$) is that single object.  For ordinary colimits in [[category theory]] this is a mild condition,  requiring $D$ to be a [[connected category]]; but in [[higher category theory]] this becomes an ever stronger condition; for [[colimits in an (infinity,1)-category]] it means that the [[infinity-groupoid]] generated by $D$ is [[contractible homotopy type]] (see [this corollary](infinity-limit#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself)).

It *is* generally true that (eq:IsoForPullbackStabilityOfColimis) is an example of _[[distributivity of limits over colimits]]_; see [there](distributivity+of+limits+over+colimits#PullbackStableColimits).

## Related concepts

* [[van Kampen colimit]]
* [[pullback-stability]]

## References

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ (Section 6.1.1)

[[!redirects universal colimit]]
[[!redirects universal colimits]]
[[!redirects stable colimit]]
[[!redirects stable colimits]]
[[!redirects pullback-stable colimit]]
[[!redirects pullback-stable colimits]]


punctured neighborhood

\tableofcontents

## Definition

In [[classical mathematics]], let $(X, \mathcal{O}(X))$ be a [[topological space]]. A **punctured neighborhood** of an element $x \in X$ is the subset of a [[neighborhood]] $U$ of $x$ consisting of all elements in $U$ which are not equal to $x$, or equivalently, the relative [[complement]] $U / \{x\}$.

In [[constructive mathematics]], let $(X, \mathcal{O}(X))$ be a [[topological space]] with a [[tight apartness relation]] $\#$. An element $x \in X$ is strictly apart from a subset $U \subseteq X$ if for all elements $y \in X$, $y \in U$ implies that $x \# y$. A **punctured neighborhood** of an element $x \in X$ is the subset of a [[neighborhood]] $U$ of $x$ consisting of all elements in $U$ which are strictly apart from $x$.

The tight apartness relation is needed in constructive mathematics because in the [[real numbers]], a neighborhood around a real number $c$ is an [[open interval]] $(c - \epsilon, c + \epsilon)$ where $\epsilon$ is a positive real number, and a punctured neighborhood is the set of real numbers $x$ in the open interval $(c - \epsilon, c + \epsilon)$ where $0 \lt {|x - c|}$. The tight apartness relation $x \# c$ holds precisely if and only if $0 \lt {|x - c|}$.

## See also

* [[neighborhood]]

* [[limit point]]

* [[limit of a function]]

[[!redirects punctured neighborhood]]
[[!redirects punctured neighborhoods]]

[[!redirects punctured neighbourhood]]
[[!redirects punctured neighbourhoods]]


punny title

Many mathematics titles (not to mention the names for notions) like to involve a game of double meanings, repetition, wit, connotations, seeming absurdity, self-referencing, self-mockery and the like. I propose creating list of some known examples of such titles in areas of our interest. Maybe I should have titled this page *Title of titles* to keep the spirit on.
+-- {: .query}
In that vein, I just introduced the term 'punny', although it is a cliché.
=--

* [[Pavol Ševera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_, [arXiv/math.SG/0105080](http://arxiv.org/abs/math.SG/0105080)

* A. [[Grothendieck]], _Hodge's general conjecture is false for trivial reasons_, Topology 8: 299--303 (1969).

* Arne Strøm, _The homotopy category is a homotopy category_, Arch. Math. (Basel) 23 (1972), 435--441 [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/Strom.pdf)

* V. A. Hinich, V. V. Schechtman, _On homotopy limit of homotopy algebras_, $K$-theory, arithmetic and geometry (Moscow, 1984--1986), 240--264, Lecture Notes in Math., 1289, Springer, Berlin, 1987.

* [[V. Hinich]], _Homological algebra of homotopy algebras_, Comm. Algebra 25 (1997), no. 10, 3291--3323.

* [[Charles Rezk|C. Rezk]], _A model for the homotopy theory of homotopy theories_, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973--1007, [doi](http://dx.doi.org/10.1090/S0002-9947-00-02653-2).

* [[Julie Bergner|J. Bergner]], _Three models for the homotopy theory of homotopy theories_,

* P. Balmer, G. Tabuada, _The Mother of all isomorphism conjectures via DG categories and derivators_, [arXiv:0810.2099](http://arXiv.org/abs/0810.2099)

* [[Jean-Pierre Serre|J. P. Serre]], _Gèbres_ (in French; Engl. [[gebras|Gebras]]) Enseign. Math. (2) 39 (1993), no. 1-2, 33--85.

* [[Sasha Beilinson]], _Determinant gerbils_, a talk at MPI Bonn (a gerbil is a kind of animal, the pun is on [[gerbes]])

* Micah Goldblum, Jonas Geiping, Avi Schwarzschild, Michael Moeller, Tom Godstein, _Truth or backpropaganda? An empirical investigation of deep learning theory_, [arxiv:/1910.00359](https://arxiv.org/abs/1910.00359) accepted conference paper at ICLR 2020 (the notion and the word formation of "backpropaganda" are akin to the central ingredient of the modern "deep learning" algorithms, _backpropagation_)

Several more candidates of similar character are listed in the responses to MO question [most-memorable-titles](http://mathoverflow.net/questions/44326/most-memorable-titles).
See also MathOverflow [parodies-of-abstruse-mathematical-writing](http://mathoverflow.net/questions/199926/parodies-of-abstruse-mathematical-writing).

This one is a little bit out of the category but has a point:

* Martin A. Schwartz, _The importance of stupidity in scientific research_ (essay) Journal of Cell Science 2008 121: 1771 [doi](https://dx.doi.org/10.1242/jcs.033340)

[[!redirects pun title]]

category: reference


Puppe exact category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
#### Regular and exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
=--
=--

# Puppe-exact category
* table of contents
{: toc}

## Idea

Loosely speaking the notion of **Puppe exact categories** is that of [[abelian categories]] without the requirement of [[additive category|additivity]]. The remaining [[exact category|exactness]] properties still permit a considerable amount of [[homological algebra]].

## Definition

A [[well-powered category|well-powered]] category $\mathcal{C}$ is called _Puppe exact_ (or _p-exact_ for short) iff

* $\mathcal{C}$ has a [[zero object]].

* $\mathcal{C}$ has [[kernels]] and [[cokernels]].

* every [[mono]] is a kernel and every [[epi]] is a cokernel.

* every morphism has an [[factorization system|epi-mono factorization]].

## Related entries

* [[abelian category]], [[semi-abelian category]]

* [[exact category]]

* [[homological category]]

## History

The notion of Puppe exact categories was introduced by [Puppe (1962)](#Puppe1962) as a weakening of the notion of abelian categories. Their [[homological algebra]]  was studied by [Mitchell (1965)](#Mitchell1965). They were extensively used by [[Marco Grandis]] in his work on distributive homological algebra in the 1980s.

Grandis worked on further generalizations that he called *semi-exact* or *[[homological category|homological]]*. This family differs from the concepts around [[semi-abelian categories]] studied in particular in the [2004 monograph](#BorceuxBourn2004) by [[Francis Borceux]] and [[Dominique Bourn]] with respect to the existence of finite limits in the latter. [Grandis (2010)](#Grandis2010) proposed viewing the difference as akin to the difference between projective and affine spaces.

## References

The original articles:

* {#Puppe1962} [[Dieter Puppe]], _Korrespondenzen in abelschen Kategorien_ , Math. Ann. **148** (1962) pp.1-30. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002290812))

* {#Mitchell1965} [[Barry Mitchell]], _Theory of categories_, Pure and Applied Mathematics **17**, Academic Press, 1965.

See also:

* {#BorceuxBourn2004} [[Francis Borceux]], [[Dominique Bourn]], [[Mal'cev, protomodular, homological and semi-abelian categories]], Mathematics and Its Applications __566__, Kluwer, 2004.

* [[Francis Borceux]] , [[Marco Grandis]], _Jordan-Hölder, modularity and distributivity in non-commutative algebra_, JPAA **208** (2007), pp.665-689. ([doi](https://doi.org/10.1016/j.jpaa.2006.03.004))

* [[Aurelio Carboni]], [[Marco Grandis]], _Categories of projective spaces_, JPAA **110** (1996), pp.241-258. ([web](https://www.sciencedirect.com/science/article/pii/0022404995001115))

* [[Peter Freyd]], A. Scedrov, _Categories, Allegories_, North-Holland Amsterdam 1990.

* [[Marco Grandis]], _Transfer functors and projective spaces_, Math. Nachr. **118** (1984) pp.147-165. ([doi](https://doi.org/10.1002/mana.19841180111))

* {#Grandis2010} [[Marco Grandis]], _Homotopy spectral sequences_, J. Homotopy Relat. Struct. **5** (2010), pp.213-252.  ([arXiv:1007.0632](http://arxiv.org/abs/1007.0632), [pdf](http://tcms.org.ge/Journals/JHRS/xvolumes/2010/n1a12/v5n1a12.pdf))

[[!redirects p-exact category]]


pure morphism

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

# Pure morphisms
* table of contents
{: toc}

## In category theory

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

Given a [[regular cardinal]] $\kappa$, a [[morphism]] $f: A\to B$ in a [[category]] $C$ is **$\kappa$-pure** (or **$\kappa$-universally injective**) if for every [[commutative square]]

$$
  \array{
    P
    &
    \overset{g}
     {\longrightarrow}
    &
    Q
    \\
    \mathllap{^u}
    \big\downarrow
    &&
    \big\downarrow
    \mathrlap{^v}
    \\
    A
    &
    \underset{f}
     {\longrightarrow}
    &
    B
  }
$$

in which $P$ and $Q$ are $\kappa$-[[presentable objects]], the morphism $u \colon P\to A$ factors through $g$, i.e. there is some $h: Q\to A$ with $u = h\circ g$.

=--

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

Notice that the above definition does not require that also the morphism $v$ is factored, hence it does _not_ express a [[lifting property]].

=--

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

* In a $\kappa$-[[accessible category]] $C$ every $\kappa$-pure morphism is a [[monomorphism]], hence exhibits a [[pure subobject]].

* In a [[locally presentable category|locally $\kappa$-presentable category]] $\kappa$-pure morphisms are, moreover, [[regular monomorphisms]],

* and in fact coincide with the $\kappa$-[[directed colimits]] of [[split monomorphisms]] in the [[arrow category]] $C^2 = Arr(C)$;

* more generally this characterization holds in all $\kappa$-[[accessible categories]] admitting [[pushouts]].

=--

([Adámek-Hub-Tholen 1996](#AHT96)).

## In ring theory and for schemes

Consider [[unital ring|unital]], possibly [[commutative ring|commutative]], [[rings]] and their [[modules]].

\begin{definition}
\label{PureMorphismOfRings}
Given such a ring $R$, a [[homomorphism]] $f \colon M\to M'$ of left $R$-modules is *pure* if [[tensor product of modules|tensoring]] the [[exact sequence]] of left $R$-modules

$$
  0
   \to
  Ker f
   \longrightarrow
  M
    \overset{f}
     {\longrightarrow}
  M'
   \longrightarrow
  Coker f
   \to
  0
$$

with any right $R$-module $N$ (from the left) yields a [[exact sequence]] of [[abelian groups]].
\end{definition}

\begin{remark}
[[Grothendieck]] has proved that [[faithfully flat morphisms]] of commutative [[schemes]] are of [[effective descent]] for the categories of [[quasicoherent module|quasicoherent]] $\mathcal{O}$-modules. But this was not entirely optimal, as there is in fact a more general class than faithfully flat morphisms which satisfy the effective descent. For a local case of commutative rings,
[Joyal & Tierney (1984)](#JoyalTierney84) have then proved (unpublished) that the [[effective descent morphisms]] for modules are precisely the pure morphisms of rings (or dually of affine schemes)  accotding to Def \ref{PureMorphismOfRings}. The result can be extracted also from their Memoirs volume on [[Galois theory]]. [Janelidze & Tholen (2004)](#JanelidzeTholen04) have reproved this theorem as a [[corollary]] of a result for [[noncommutative rings]] obtained using Beck's [[comonadicity theorem]].
\end{remark}

## Related concepts

* [[pure subobject]]

## References

#### General references

* The [Stacks Project](https://stacks.math.columbia.edu), 28.10. Radicial and universally injective morphisms (of schemes) [tag/01S2](https://stacks.math.columbia.edu/tag/01S2), 34.4. Descent for universally injective morphisms, [tag/08WE](https://stacks.math.columbia.edu/tag/08WE)

#### Descent along pure morphisms

The following paper was the first with the result on that pure morhisms are of the effective descent but the proof has been omitted:

* Jean-Pierre Olivier, _Descente par morphismes purs_, C. R. Acad. Sci. Paris Sér. A-B __271__ (1970) A821–A823

The result is proved (within a larger context)

* {#JoyalTierney84} [[André Joyal]], [[Myles Tierney]], _An extension of the Galois theory of Grothendieck_, Mem. Amer. Math. Soc. __309__ (1984) vol. 51, pages vii+71

Clean proofs are by Mesablishvili and by Janelidze,

* Bachuki Mesablishvili, _Pure morphisms of commutative rings are effective descent morphisms for modules -- a new proof_, Theory and Appl. of Categories __7__, 2000, No. 3, 38-42, [tac](http://www.tac.mta.ca/tac/volumes/7/n3/7-03abs.html)

* Bachuki Mesablishvili, _Pure morphisms are effective for modules_, Applied Categorical Structures 21 (2013), 801–809.  [arXiv](https://arxiv.org/abs/1206.3439), [doi](https://doi.org/10.1007/s10485-012-9283-6).

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge Univ. Press 2003; [errata pdf](http://maths.swan.ac.uk/staff/tb/corinerr.pdf)

* {#JanelidzeTholen04}
[[George Janelidze]], [[Walter Tholen]], _Facets of descent III: monadic descent for rings and algebras_,  Appl. Categ. Structures __12__ 5-6 (2004) 461-477 [[MR2005i:13019](http://www.ams.org/mathscinet-getitem?mr=2107397), [doi](http://dx.doi.org/10.1023/B:APCS.0000049312.36783.0a)]

Much attention on the descent along pure morphism is in a recent survey

* Yves André, Luisa Fiorot, _On the canonical, fpqc, and finite topologies on affine schemes. The state of the art_, in Ann. Sc. Norm. Super. Pisa Cl. Sci. [arXiv:1912.04957](https://arxiv.org/abs/1912.04957)

#### Ziegler spectrum and connections to model theory

The [[Ziegler spectrum]] of indecomposable pure injectives has been introduced in

* M. Ziegler, _Model theory of modules_, Ann. Pure Appl. Logic 26 (1984) 149 – 213.

A textbook account is in

* [[Mike Prest]], _Purity, spectra and localisation_, Enc. Math. Appl. __121__, Camb. Univ. Press 2011, 798 pages; publishers book [page](http://www.cambridge.org/gb/knowledge/isbn/item2327409/?site_locale=en_GB)

The relationship between the [[Ziegler spectrum]] of (the category of modules over) a ring and the Ziegler spectrum of its derived category is studied in

* G. Garkusha, M. Prest, _Triangulated categories and the Ziegler spectrum_, Algebras & Repres. Theory 8, 499–523 (2005) [doi](https://doi.org/10.1007/s10468-005-8147-2)

Other articles:

* M. Prest, P. Rothmaler, M. Ziegler, _Absolutely pure and flat modules and "indiscrete" rings_, J. Alg. __174__:2 (1995) 349-372 [doi](https://doi.org/10.1006/jabr.1995.1131)

* Christian U. Jensen, Helmut Lenzing, _Model theoretic algebra: with particular emphasis on fields, rings, modules_, Algebra, Logic and Applications __2__, Gordon and Breach 1989.

* I. Herzog, _The Ziegler spectrum of a locally coherent Grothendieck category_, Proc. London Math. Soc. __74__:3 (1997) 503-558 [doi](https://doi.org/10.1112/S002461159700018X)

#### Other categoey theoretic articles

* Ivo Herzog, _Pure-injective envelopes_, Journal of Algebra and Its Applications 2(4) (2003), 397-402 [pdf](http://lima.osu.edu/people/iherzog/env.pdf)

* {#AHT96} [[Jiří Adámek]], H. Hub, [[Walter Tholen]], _On pure morphisms in accessible categories_,  J. Pure Appl. Alg. __107__, 1 (1996), pp 1-8, doi

* Michel Hébert, _Purity and injectivity in accessible categories_, doi

* W.W. Crawley-Boevey, _Locally finitely presented additive categories_, Communications in Algebra __22__(5)(1994), 1641-1674.

* Rosanna Laking, _Purity in compactly generated derivators and t-structures with Grothendieck hearts_, Math. Zeitschrift, [doi](https://10.1007/s00209-019-02411-9) (2019).

[[!redirects pure morphisms]]
[[!redirects purity]]
[[!redirects universally injective morphism]]


pure motive

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

#Contents#
* table of contents
{:toc}

## Idea

[[Grothendieck]]  [[conjecture|conjectured]] that every [[Weil cohomology theory]] factors uniquely through some [[category]], which he called the category of _[[motives]]_.  For [[smooth variety|smooth]] [[projective varieties]] (over some [[field]] $k$) such a category was given by Grothendieck himself, called the category of _pure [[Chow motives]]_.  For general smooth varieties the category is still conjectural, see at _[[mixed motives]]_.

## Construction

Fix some [[adequate equivalence relation]] $\sim$ (e.g. [[rational equivalence]]).  Let $Z^i(X)$ denote the group of $i$-[[codimension|codimensional]] [[algebraic cycles]] and let $A^i_\sim(X)$ denote the [[quotient]] $Z^i(X)/\sim$.

### Category of correspondences

Let $Corr_\sim(k)$, the category of [[correspondences]], be the [[category]] whose [[objects]] are [[smooth variety|smooth]] [[projective varieties]] and whose [[hom-sets]] are the [[direct sum]]

$$
  Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y)
  \,,
$$

where $(X_i)$ are the [[irreducible components]] of $X$ and $n_i$ are their respective [[dimensions]].  The [[composition]] of two [[morphisms]] $\alpha \in Corr(X,Y)$ and $\beta \in Corr(Y,Z)$ is given by
$$ p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta)) $$
where $p_{XY}$ denotes the [[projection]] $X \times Y \times Z \to X \times Y$ and so on, and $.$ denotes the [[intersection product]] in $X \times Y \times Z$.

There is a canonical [[contravariant functor]]

$$
  h \colon SmProj(k) \to Corr_\sim(k)
$$

from the category of smooth projective varieties over $k$ given by [[mapping]] $X \mapsto X$ and a morphism $f : X \to Y$ to its [[graph of a function|graph]], the [[image]] of its [[graph morphism]] $\Gamma_f : X \to X \times Y$.

The [[category of correspondences]] is [[symmetric monoidal category|symmetric monoidal]] with $h(X) \otimes h(Y) \coloneqq h(X \times Y)$.

We also define a category $Corr_\sim(k, R)$ of correspondences with [[coefficients]] in some [[commutative ring]] $R$, by [[tensor product|tensoring]] the morphisms with $R$; this is a $R$-[[linear category]] [[additive category|additive]] [[symmetric monoidal category|symmetric monoidal]] category.

### Category of effective pure motives

The [[Karoubi envelope]] (pseudo-abelianisation) of $Corr_\sim(k, R)$ is called the category of **effective pure motives** (with coefficients in $R$ and with respect to the equivalence relation $\sim$), denoted $Mot^eff_\sim(k, R)$.

Explicitly its objects are [[pairs]] $(h(X), p)$ with $X$ a smooth projective variety and $p \in Corr(h(X), h(X))$ an [[idempotent]], and [[morphisms]] from $(h(X), p)$ to $(h(Y), q)$ are morphisms $h(X) \to h(Y)$ in $Corr_\sim$ of the form $q \circ \alpha \circ p$ with $\alpha \in Corr_{\sim}(h(X), h(Y))$.

This is still a [[symmetric monoidal category|symmetric monoidal]] category with $(h(X), p) \otimes (h(Y), q) = (h(X \times Y), p \times q)$.  Further it is a [[Karoubian category|Karoubian]], $A$-[[linear category|linear]] and [[additive category|additive]].

The image of $X \in SmProj(k)$ under the above functor

$$
  h \colon SmProj(k) \to Corr_\sim(k,A) \to Mot^{eff}_\sim(k,R)
$$

is the **the motive of $X$**.

### Category of pure motives

There exists a motive $\mathbf{L}$, called the **[[Lefschetz motive]]**, such that the motive of the [[projective line]] decomposes as

$$h(\mathbf{P}^1_k) = h(\Spec(k)) \oplus \mathbf{L}$$

To get a [[rigid category]] we formally invert the Lefschetz motive and get a category

$$
  Mot_\sim(k, R)
  \coloneqq
  Mot^{eff}_\sim(k,R)[\mathbf{L}^{-1}]
  \,,
$$

the **category of pure motives** (with coefficients in $R$ and with respect to $\sim$).

This is a [[rigid category|rigid]], [[Karoubian category|Karoubian]], [[symmetric monoidal category]].  Its objects are triples $(h(X), p, n)$ with $n \in \mathbf{Z}$.

### Category of pure Chow motives

When the relation $\sim$ is [[rational equivalence]] then $A^*_\sim$ are the [[Chow groups]], and $Mot_\sim(k) = Mot_{rat}(k)$ is called the category of **pure [[Chow motives]]**.  This category has the advantage that it is universal for [[Weil cohomology theory|Weil cohomology theories]]: that is, every Weil cohomology factors uniquely through it.

### Category of pure numerical motives

When the relation $\sim$ is [[numerical equivalence]], then one obtains
_[[numerical motives]]_.  This category has the advantage of being a [[semisimple category|semisimple]] [[abelian category]].  In fact, [[Uwe Jannsen]] proved that numerical equivalence is the only [[adequate equivalence relation]] that gives a semisimple abelian category of pure motives.

## Related concepts

* [[sheaf with transfer]]

* [[motive]]

* [[mixed motive]]

* [[Voevodsky motive]]

* [[noncommutative motive]]

## References

* [[Daniel Dugger]], *Navigating the Motivic World*.   ([pdf](http://www.stat.ucla.edu/~ywu/wbook.pdf))   A draft of a user-friendly introduction to motives, especially good if you're coming to this topic through algebraic topology.

* [[Uwe Jannsen]], Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107.3 (1992): 447-452.  ([pdf](https://epub.uni-regensburg.de/26642/1/jannsen10.pdf))

*  [[Minhyong Kim]], _Classical Motives: Motivic $L$-functions_ ([pdf](http://www.ucl.ac.uk/~ucahmki/ihes3.pdf))

* [[Bruno Kahn]], [pdf slides](http://www.aimath.org/WWN/motivesdessins/PaloAlto1.pdf) on pure motives

* [[Yuri Manin]], _Correspondences, motifs and monoidal transformations_ , Math. USSR Sb. 6 439, 1968([pdf](http://resources.agssp2012.torsor.org/documents/manin.pdf), [web](http://iopscience.iop.org/0025-5734/6/4/A01))

*  [[James Milne]], _Motives -- Grothendieck's Dream_ ([pdf](http://www.jmilne.org/math/xnotes/MOT.pdf))

* [[Tony Scholl]], _Classical motives_, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187 ([pdf](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/classical_motives.pdf))

* R. Sujatha, _Motives from a categorical point of view_,  Lecture notes (2008) ([pdf](http://www.math.tifr.res.in/~sujatha/ihes.pdf))
 {#Sujatha}

Section 8.2 of

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_  ([pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf))

category: algebraic geometry

[[!redirects pure motives]]
[[!redirects category of pure motives]]


pure set

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

# Pure sets
* table of contents
{: toc}

## Motivation

In [[material set theory]], there is an intuitive conception of what a [[set]] is, which may be stated informally as follows: a set is a collection of sets.  Actually, it is possible to have [[urelements]] in a material set theory (such as [[ZFA]]), although the most common axiom systems do not allow this; in any case we can say that a *pure* set is a collection of pure sets.

The primary motivation for [[structural set theory]] is that this conception of a set is not needed in ordinary [[mathematics]]; it is sufficient to characterise the [[category of sets]] (although a structural set theory can also be described in ways other than category-theoretic).  However, material set theory is itself part of mathematics, and we may want to describe the material notion of a pure set in structural terms.

There is also a very practical point to this exercise: the translation between material and structural set theories.  Any model of a material set theory is already a model of a corresponding structural set theory, but we go through the yoga below to construct a model of a material set theory out of a model of a structural set theory.  In particular, we must do this to prove that two set theories (one material and one structural) are equiconsistent.

See also at _[[material-structural adjunction]]_.

## Idea

Taking a base notion of [[set]] as granted, we wish to define a __pure set__ to be a set of pure sets.  On the face of it, this is a circular definition, but like many such definitions, it can be made precise in (at least) two ways: [[recursion|recursively]] and [[corecursion|corecursively]].

* The 'recursive' meaning is that all pure sets must be *constructed*, starting from nothing, as sets of other previously constructed pure sets.  This results in the **[[well-founded relation|well-founded]] pure sets**.  Thus, at first, the only well-founded set possible is the [[empty set]] $\empty = \{ \}$, the set of no pure sets.  Once you have that, you can form $\star = \{\empty\}$, then $\{\star\}$ and $\{\empty,\star\}$, and so on.  In this way we can obtain at least all [[hereditarily finite sets]]; if we use an [[axiom of infinity]] as well, we can jump to the [[countably infinite set]] of all hereditarily finite sets and continue from there.

* The 'corecursive' meaning is that any pure set can be *deconstructed* into a set of other pure sets, and every possible such deconstruction defines a unique pure set.  In this way we obtain not just well-founded pure sets but also **ill-founded pure sets**.  (The inclusion of well-founded sets among the ill-founded sets is an example of the [[red herring principle]].)  Possible examples of non-well-founded sets include a set $\bullet$ such that $\bullet = \{\bullet\}$ (a suggestive model for the [[point]]), or sets $A$ and $B$ such that $A = \{B\}$ and $B = \{\empty, A\}$.

In addition, the meaning of this definition changes according to whether our [[set theory|set-theoretic]] [[foundations|foundation]] is [[material set theory|material]] or [[structural set theory|structural]].

* If the foundation is 'material' or 'membership-based,' such as [[ZFC]], then the elements of a set can, in fact, *be* other sets.  Therefore, in this case we are defining (either recursively or corecursively) an *adjective* "pure" that can be applied to the noun "set": a set is pure if at no point in its construction or deconstruction into elements do we encounter anything that is not a pure set.

  In the most common material set theories, such as [[ZFC]], *all* sets are pure, since the only 'things' the theory deals with (hence the only things that can be elements of sets) are sets.  However, there are easy modifications of these theories that allow 'atoms' or [[urelement]]s that are not sets, and in this case the pure sets will be those that 'hereditarily' contain no atoms.  Many common material set theories (starting with von Neumann 1925, Zermelo 1930) also include an [[axiom of foundation]] asserting that all (pure) sets are well-founded; the dual [[axiom of anti-foundation]] (due to Aczel) allows and 'tames' the ill-founded sets.

* If the foundation is 'structural' or 'categorial', such as [[ETCS]] or [[SEAR]], then the elements of a set cannot *be* other sets.  Thus, in this case we are defining a single noun "pure set," with no *a priori* relation to the structural notion of "set" that occurs in the foundational theory.  A *pure set*, according to this definition, is a set *equipped with structure* assigning to each of its elements another pure set (interpreted either recursively or corecursively).

  From this point of view, the definition of pure set provides a construction of a model of material set theory within a model of structural set theory: a global relation of $\in$ can be defined between pure sets which will satisfy the axioms of some material set theory.  (Of course, the sets in a model of material set theory always model a structural set theory, so this is the 'difficult' direction of the equivalence between the two types of set theory.)  Whether we use the recursive or corecursive definition determines whether the resulting material set theory will satisfy the axiom of foundation or the axiom of anti-foundation.  Note that using these two choices, the composite operation "material $\to$ structural $\to$ material" reduces to the standard proofs of the relative consistency of the axioms of foundation and anti-foundation, since the passage from material to structural requires neither one.

## Formalisation in material set theory

In material set theory without urelements, every set is a pure set.  If there are urelements, so that not all sets are pure, then it is easy to define the class of pure sets as follows:

+-- {: .un_defn}
###### Definition
A set $x$ is **pure** if given any finite sequence $x_n \in x_{n-1} \in \dots \in x_2 \in x_1 \in x_0 = x$, all of the $x_i$ are sets.
=--

Whether this produces the well-founded pure sets or the ill-founded ones depends on whether the sets in the ambient set theory are well-founded or ill-founded.

If you are (unusually) working in an ill-founded material set theory and want only the well-founded pure sets, then you simply need to define the other adjective:

+-- {: .un_defn}
###### Definition
A class $C$ of sets is __$\in$-inductive__ if whenever all elements of some set $x$ are in $C$, then $x$ itself is in $C$.  A set $x$ is __well-founded__ if it belongs to every $\in$-inductive class $C$.
=--

(That *all* sets are well-founded is the [[axiom of foundation]].)  If $x$ is well-founded, then there can be no infinite sequence $\in \dots \in x_2 \in x_1 \in x_0 = x$, since the sets with no such sequence form an $\in$-inductive class; the converse (that a set is well-founded if it has no such infinite sequence) holds using [[excluded middle]].  Then a well-founded pure set is simply a set that is both well-founded and pure.

If you are (as usual) working in a well-founded material set theory and want to work with ill-founded pure sets, then you cannot do so directly; instead, follow the structural development in the next section.

## Formalization in structural set theory

In a structural set theory like [[ETCS]] or [[SEAR]], we can model a pure set by a graph describing its hereditary membership relation.

One way to state the basic theoretical idea is that the class of well-founded sets is the [[initial algebra]] of the covariant [[power set]] functor, while the class of ill-founded sets is the [[terminal coalgebra]] of the same functor.  Of course, neither of these algebras exists as a set, since this would violate [[Cantor's theorem]], but we can still describe what their elements would be like.  We can also define these algebras as [[discrete category|discrete]] [[large category|large categories]], or as proper classes in a structural set-class theory such as [[algebraic set theory]].

The discussion which follows is phrased informally, like most mathematics.  However, it is purely structural and can be interpreted in any structural set theory.  For instance, the definition below of a "graph" as a set with a binary relation should be formalized as a set $G$ together with an injection $R \hookrightarrow G \times G$.

### Membership graphs

To describe a pure set, we must give all the elements of that set, each of which is a pure set and thus must have its own elements, which are also pure sets, and so on.  A convenient way to "picture" a pure set is with a *graph*.

+-- {: .un_defn}
###### Definition
For the purposes of this page, a **graph** will mean a set $G$ of __nodes__ equipped with a binary [[relation]] $\to$ on the nodes.  A node $i$ is called a __child__ of a node $j$ if $i \to j$.
=--

The idea of a graph-picture of pure sets is that the *nodes* represent *pure sets*, and we have $i\to j$ precisely when $i\in j$.  Clearly the membership relations between any collection of pure sets should be representable by a graph.  To what extent the converse holds, i.e. which graphs can be interpreted as membership-relations between pure sets, depends on whether the pure sets in question are well-founded or ill-founded.  For purposes of drawing well-founded pure sets, only well-founded graphs should be used, whereas arbitrary graphs can be used to draw ill-founded sets.  (In fact, one form of the [[axiom of anti-foundation]] is that any graph is the picture of some pure set.)

+-- {: .un_defn}
###### Definition
A set $S$ of nodes in a graph is **$\to$-inductive** if whenever all children of some node $i$ are in $S$, then $i$ itself is in $S$.  A graph is **[[well-founded relation|well-founded]]** if the only $\to$-inductive set of nodes is the set of all nodes.
=--

Assuming the principle of [[excluded middle]], well-foundedness is equivalent to saying that the tree has no infinite paths.

If we want to use a graph to describe a *specific* pure set, then we should single out one particular node as representing *that* set.  Moreover, it makes sense to demand that the graph contain no "superfluous" data other than what is necessary to describe the hereditary membership structure of the particular set in question.

+-- {: .un_defn}
###### Definition
A graph is **pointed** if it is equipped with a specified node $\top$ called the **root**.  A pointed graph is **accessible** if for every node $x$, there exists a path $x = x_0 \to x_1 \to \dots \to x_n =\top$ to the root; this is equivalent to saying that $\top$ is a [[top element]] for the reflexive-[[transitive closure]] of $\to$.  An accessible pointed graph is abbreviated **APG**.
=--

An APG gives all the data necessary to characterize a pure set $X$, and the pure set in question is well-founded iff the APG is well-founded.  We can think of the root as representing $X$ itself, the root\'s children as the elements of $X$, the next level as those elements\' elements, and so on.

If we want to think of these elements as pure sets themselves, then we can "externalize" them into separate APGs.  Specifically, for any node $z$ in a graph $G$, the subgraph consisting of all those nodes which admit some path to $z$ is an APG, which describes the pure set represented by $z$; we write it as $G/z$ and call it the **full subgraph** of $G$ rooted at $z$.  If $G$ is an APG and $z$ is a child of the root, we say that $G/z$ is an **immediate subgraph** of $G$.  The immediate subgraphs of $G$ represent the pure sets that are the "elements" of the pure set represented by $G$.  Thus, the APG $G$ pictures the hierarchy of elements of $X$, where each immediate subgraph $G/z$ corresponds to its root $z$, and $\to$ is interpreted as membership.

### Examples of APGs

Note that the nodes in most of the pictures below have labels, but these are for illustration only; they are *not* part of the structure defining the tree.

First, $\empty = \{ \}$ consists of nothing but a root:
$$ \empty $$

The set $\star = \{\empty\}$ has the empty set attached to the root (which we draw at the top):
$$ \array { \star \\ \uparrow \\ \empty } $$
Notice that $\empty$ is an immediate subgraph, so we can write $\empty \in \star$.

Then the two most obvious ways of representing the ordinal $2$ as a pure set, namely $2_Z = \{\star\}$ and $2_N = \{\empty, \star\}$, can be drawn as follows:
$$ \array { 2_Z \\ \uparrow \\ \star \\ \uparrow \\ \empty } \quad \quad \quad \quad \array {
       &          & 2_N \\
       & \nearrow &     & \nwarrow \\
\empty &          &     &          & \star    \\
       &          &     &          & \uparrow \\
       &          &     &          & \empty
} $$
Looking at immediate subtrees, $\star \in 2_Z$, $\empty \in 2_N$, and $\star \in 2_N$, but $\empty \notin 2_Z$.

$2_N$ can also be represented as $\{\star, \empty\}$:
$$ \array {
         &          & 2_N \\
         & \nearrow &     & \nwarrow \\
\star    &          &     &          & \empty \\
\uparrow \\
\empty
} $$
Of course, as a graph this is isomorphic to $\{\empty, \star\}$, and hence should represent the same pure set.

These sets are all well-founded.  A non-well-founded set can be represented either by an infinite picture, or by a picture with loops.  Here is an infinite picture of $\bullet = \{\bullet\}$:
$$ \array { \bullet \\ \uparrow \\ \bullet \\ \uparrow \\ \bullet \\ \uparrow \\ \vdots } $$
and here is a finite picture:
$$ \array {
           & \bullet \\
/          &         & \nwarrow \\
\backslash &         & /        \\
} $$
In both cases, the unique immediate subgraphs verify that $\bullet \in \bullet$.

### Ambiguities

APGs are sufficient to describe all pure sets, but multiple APGs, even non-isomorphic ones, can represent the "same" pure set.  Firstly, we have so far not eliminated the possibility of duplicate branches.  For instance, the pure set $2_Z$ can also be represented as $\{\star, \star\}$:
$$ \array {
         &          & 2_Z \\
         & \nearrow &     & \nwarrow \\
\star    &          &     &          & \star    \\
\uparrow &          &     &          & \uparrow \\
\empty   &          &     &          & \empty
} $$
It is easy to eliminate APGs with such "obviously redundant" branches.

+-- {: .un_defn}
###### Definition
An APG $G$ is **rigid** if whenever $x$ and $y$ are two children of the same node $z$ such that $G/x \cong G/y$, then in fact $x=y$.
=--

Assuming [[excluded middle]], rigidity is equivalent to saying that any graph [[automorphism]] is the identity function.  Note also that if two rigid APGs are isomorphic, they must be uniquely isomorphic.  Thus, for rigid APGs, being isomorphic is a *property*, rather than *structure*.

Even among rigid APGs, however, there are are ambiguities.  For example, the pure set $2_N$ can also be drawn by the APG
$$ \array {
2_N \\
\uparrow & \nwarrow \\
|        &          & \star    \\
|        & \nearrow &  \\
\empty   &          &
} $$
The issue here is that the "same" pure set can occur in more than one place in the membership hierarchy of a pure set, and we have to choose whether or not to identify them in the graph.  There are three solutions to this problem:

1. Demand that duplicate occurrences of a pure set always be represented separately.
2. Demand that duplicate occurrences always be identified, wherever they occur.
3. Define a notion of "equivalence" between APGs which is looser than isomorphism.

These are of course related; the first two can be seen as specifying a particular isomorphism class within each of the equivalence classes defined by the third.  In particular, they all result in equivalent notions of pure set, when they all apply.  (However, in [[predicative mathematics|predicative]] set theories, it seems that only the first method is applicable.)  We will consider them all in turn.

### Membership trees

A natural way to ensure that the same node is never "shared" between two occurrences is to require that our APGs be *trees*.

+-- {: .un_defn}
###### Definition
An pointed graph is a **tree** if every node admits a *unique* path to the root.  Equivalently, $G$ is a tree if its root $\top$ is a [[terminal object]] of the [[quiver]] generated by the relation $\to$.
=--

A tree may or may not be well-founded.  Note, though, that in a tree, the [[opposite relation]] $\leftarrow$ is automatically well-founded: each node has a unique [[natural number]] *height*, namely the length of its unique path to the root, so well-foundedness of $\mathbb{N}$ implies well-foundedness of $\leftarrow$.

It now makes sense to define a *pure set* to be a *rigid accessible pointed tree*.  The idea is that two such trees represent the same pure set iff they are isomorphic.   We define "membership" $G\in H$ between such trees to mean that $G$ is isomorphic to some immediate subtree of $H$ (that is, a tree $H/z$ where $z$ is a child of the root of $H$).

For example, the tree representation of $2_N$ is the one we gave first:
$$\array {
       &          & 2_N \\
       & \nearrow &     & \nwarrow \\
\empty &          &     &          & \star    \\
       &          &     &          & \uparrow \\
       &          &     &          & \empty
}
$$
and the tree representation of $\bullet = \{\bullet\}$ is the infinite path:
$$ \array { \bullet \\ \uparrow \\ \bullet \\ \uparrow \\ \bullet \\ \uparrow \\ \vdots .} $$
Of course, a well-founded [[infinite set]] also has an infinite picture, but there are no infinite paths.  Here is $\omega_N$, the set of von Neumann [[natural numbers]]:
$$ \array {
        &          &          & \omega_N \\
        & \nearrow & \nearrow &          & \nwarrow & \cdots \\
\empty  & \star    &          &          &          & 2_N      & \cdots \\
        & \uparrow &          &          & \nearrow & \uparrow & \cdots \\
        & \empty   &          & \empty   &          & \star    & \cdots \\
        &          &          &          &          & \uparrow & \cdots \\
        &          &          &          &          & \empty   & \cdots \\
        &          &          &          &          &          & \vdots
} $$
So we have $\empty \in \omega_N$, $\star \in \omega_N$, $2_N \in \omega$, etc.

A tree models a _[[hereditarily finite set]]_ if every node has finitely many children; a tree models a well-founded hereditarily finite set if and only if it is finite and $\to$ is [[decidable subset|decidable]].  The relationship between these two facts and the no-infinite-path formulation of well-foundedness is a form of [[Konig's lemma|Kőnig's Lemma]].

Given these definitions, one can prove the various axioms of material set theory.  Specifically, if __[[ETCS]]__ holds for the structural sets (of nodes, etc) used in this definition, then $\mathbf{BZC}^-$ (which is __[[ZFC]]__ with only [[bounded separation]] and no axioms of [[axiom of replacement|replacement]] or [[axiom of foundation|foundation]]) holds for rigid trees, while $\mathbf{BZC}$ ($\mathbf{BZC}^-$ together with foundation) holds for well-founded rigid trees.  Similar results hold for [[constructive mathematics|constructive]] weakenings of ETCS and $\mathbf{BZC}^-$, since these definitions involve only function-sets and not power-sets.  Likewise, we can strengthen both sides with axioms of replacement, collection, or separation, or by adding [[Grothendieck universes]].

The argument for the [[axiom of extensionality]] is perhaps the most interesting; it goes as follows.  If each immediate subtree $G/x$ of $G$ is isomorphic to an immediate subtree $H/y$ of $H$ and conversely, then $y$ and $x$ are mutually uniquely determined by rigidity, so we have a bijection between the children of the roots of $G$ and of $H$.  Since the isomorphisms $G/x \cong H/y$ are also unique, again by rigidity, and each $G/x$ is disjoint from $G/x'$ for $x\neq x'$, by the tree property, we can piece together these isomorphisms to define an isomorphism $G\cong H$.  (In particular, the uniqueness of all these isomorphisms means that the [[axiom of choice]] is not needed.)

(Note, though, that this is only the "weak" version of extensionality.  This is sufficient for well-founded sets, but we may need to strengthen rigidity somehow to obtain stronger notions of [[extensional relation|extensionality]], which are generally more appropriate for ill-founded sets.  More precisely, the well-founded rigid trees model the usual axioms of extensionality and foundation, while arbitrary rigid trees model not Aczel's axiom of anti-foundation AFA, but its "Scott" variant, SAFA.)

### Extensional graphs

The "tree" representation can be thought of as an _unwrapped_ picture; we now consider a _wrapped_ picture in which there is no duplication at all.  The natural way to ensure this is to require our graphs to be [[extensional relation|extensional]].  For well-founded graphs, this is easy: a well-founded graph is *extensional* if for any nodes $x$ and $y$ such that for all $z$, we have $z\to x$ iff $z\to y$, then necessarily $x=y$.  For non-well-founded graphs, we need the notion of "strong extensionality" defined as follows.

+-- {: .un_defn}
###### Definition
If $G$ and $H$ are graphs, a **[[bisimulation]]** from $G$ to $H$ is a binary relation $\sim$ from $G$ to $H$ such that for any $x\in G$ and $y\in H$ with $x\sim y$, then:

* For any $a\to x$, there is some $b\in H$ with $b\to y$ and $a\sim b$, and
* For any $b\to y$, there is some $a\in G$ with $a\to x$ and $a\sim b$.

A graph $G$ is **extensional** if the largest bisimulation from $G$ to $G$ is the identity.
=--

The idea is that a bisimulation identifies pairs of nodes which "represent the same pure set."  One can show that a well-founded graph is extensional in this sense iff it is extensional in the weaker sense given above.  Note also that any extensional APG is necessarily rigid, since any isomorphism $G/x \cong G/y$ induces a bisimulation.

Thus, it also makes sense to define a *pure set* to be an *extensional APG*.  Two such APGs represent the same set iff they are isomorphic, and we define membership in the same way: $G\in H$ if $G\cong H/x$ for some child $x$ of the root of $H$.  Of course, one can again prove the axioms of material set theory from this definition.  If we restrict to well-founded extensional APGs, we get the axiom of foundation, while if we allow all extensional APGs, we get Aczel's axiom af anti-foundation (along with its strong version of the axiom of extensionality).  Note that this definition is not predicative, since the notion of extensionality involves [[quantification]] over bisimulations, which are relations, i.e. subsets.  Thus, it requires at least [[limited separation]].

For example, the extensional representation of $2_N$ is this one:
$$ \array {
2_N \\
\uparrow & \nwarrow \\
|        &          & \star    \\
|        & \nearrow &  \\
\empty   &          &
} $$
while the extensional representation of $\bullet = \{\bullet \}$ is the loop:
$$ \array {
           & \bullet \\
/          &         & \nwarrow \\
\backslash &         & /        \\
} $$
Finally, here is the set $\omega_N$ of von Neumann [[natural numbers]]:
$$ \array {
         &            & \omega_N \\
         & \nearrow   & \uparrow & \nwarrow & \cdots \\
|        &            & |        &          & 2_N      & \cdots \\
|        &            & |        & \nearrow & \uparrow & \cdots \\
|        &            & \star    &          & |        & \cdots \\
         & \backslash & \uparrow & /        & \cdots   \\
         &            & \empty
} $$
Again we have $\empty \in \omega_N$, $\star \in \omega_N$, $2_N \in \omega$, etc.

Now an extensional accessible graph models a [[hereditarily finite set]] iff the graph itself is finite and $\to$ is decidable.

### Equivalence

Finally, we consider the third solution to the ambiguity problem, and its relation to the first two: defining a looser notion of equivalence between APGs.  Recall the notion of bisimulation.  We say that two APGs $G$ and $H$ are **equivalent** if there is a bisimulation $\sim$ from $G$ to $H$ such that $\top\sim\top$.  Note that if $\sim$ is a bisimulation from $G$ to $H$ such that $x\sim y$, then the APGs $G/x$ and $H/y$ are equivalent; this justifies the interpretation of a bisimulation.

We can now define a *pure set* to be an APG, with *equality* represented by *equivalence* in the above sense.  Likewise, we define "membership" $G\in H$ to mean that $G$ is *equivalent* to some immediate subgraph of $H$.  One can again prove the various axioms of material set theory with these definitions (either using well-founded APGs or arbitrary ones).  As with extensional APGs, in this case we obtain the *strong* notion of [[extensional relation|extensionality]] (which is equivalent to the weak one for well-founded sets).  This definition is also not predicative.

The relationship of this approach to the previous ones is as follows.

1. Every APG is equivalent to an extensional one.  Namely, let $\approx$ be the maximal bisimulation on $G$ (the union of all bisimulations); it is an equivalence relation, and its quotient $G/\approx$ inherits the structure of an extensional APG which is equivalent to $G$.  This is the *extensional quotient*; see [[extensional relation]].

1. Two extensional APGs are equivalent iff they are isomorphic.  For if $\sim$ is a bisimulation from $G$ to $H$, then defining $x\approx y$ in $G$ if there exists a node $z$ in $H$ with $x\sim z$ and $y\sim z$ gives a bisimulation on $G$, which must be the identity, and similarly for $H$; hence $\sim$ must in fact be a bijection.  Thus, the notion of "pure set" obtained from "extensional APGs and isomorphism" is the same as that obtained from "arbitrary APGs and equivalence."

1. Every APG is equivalent to one that is a tree, its "unwrapping".  The nodes of the tree can be taken to be the finite paths $x_n \to \dots \to x_0 = \top$ ending at the root, with any "one-level extension" $x_{n+1} \to x_n \to \dots \to \top$ being a child of $x_n \to \dots\to \top$.  The bisimulation relates each path $x_n \to \dots\to \top$ to its initial node $x_n$.

1. Moreover, every APG is equivalent to a *rigid* accessible pointed tree.  The key here is to first take the extensional quotient, then perform the above "unwrapping" construction: the unwrapping of any extensional graph will always be rigid.

1. Two well-founded rigid accessible pointed trees are equivalent iff they are isomorphic.  For if $\sim$ is a bisimulation from $G$ to $H$, define a new relation $\simeq$ from $G$ to $H$ such that $x\simeq y$ iff there are paths from $x$ and $y$ to the corresponding roots which are pointwise related by $\sim$, i.e. we have $x = x_n \to x_{n-1} \to \cdots \to x_1 \to x_0 = \top$ and $y = y_n \to y_{n-1} \to \cdots \to y_1 \to y_0 = \top$, with $x_i \sim y_i$.  We can then prove by well-founded induction that whenever $x\simeq y$, then $\simeq$ is a bijection between the children of $x$ and of $y$.  For assume that all the children of $x$ and $y$ have this property.  Then if we have $x' \to x$ and $x''\to x$ with $x' \sim y$ and $x'' \sim y$, then by hypothesis $\sim$ is a bijection between $G/x'$ and $H/y$, and between $G/x''$ and $H/y$, hence we have $G/x' \cong G/x''$; thus by rigidity $x'=x''$.  Therefore, $\simeq$ is a bijection between the children of $x$ and the children of $y$.  The construction of $\simeq$ then ensures that for any child $x'\to x$, $\simeq$ relates each node in $X/x'$ to a node in at most one $Y/y'$; hence $\simeq$ is in fact a bijection from $X/x$ to $Y/y$.  Thus, by induction, $\simeq$ is a bijection.

In order to extend this last result to non-well-founded trees, we would need to use a stronger notion of rigidity.  (This makes sense since SAFA is incompatible with AFA.  We could expect to be able to use "Scott-extensionality" in place of extensionality to obtain a theory equivalent to that of rigid trees.)  For example, the following two ill-founded trees are equivalent but not isomorphic, yet both are rigid.

$$ \array{ \bullet \\ \uparrow \\ \bullet \\ \uparrow \\ \bullet \\ \vdots }
\quad\quad\quad\quad
\array{
       &          &          &          & \bullet \\
       &          &          & \nearrow &         & \nwarrow\\
       &          & \bullet  &          &         &          & \bullet\\
       & \nearrow &          &          &         & \nearrow &         & \nwarrow\\
\udots &          &          &          & \bullet &          &         &          & \bullet\\
       &          &          & \nearrow &         &          &         & \nearrow &         & \nwarrow\\
       &          & \udots   &          &         &          & \udots  &          &         &          & \ddots
}
$$

In conclusion, there are two extreme ways to represent a pure set by an isomorphism class of graphs, but any accessible pointed graph will represent a pure set one way or another.  For well-founded sets, the two extreme ways are equivalent, but for ill-founded sets the choice between them can make a difference (and in fact, there are a number of other possible variants, corresponding to the spectrum of anti-foundation axioms discussed by Aczel).

Note that the version on the left is the unwrapping of the loop; we would like a stronger version of rigidity that would rule out the version on the right.  As it is, the best that we have come up with is simply: isomorphic to the unwrapping of some extensional APG.  Technically this works, but it is not very satifsying.

## References

*  Peter Aczel (1988).  Non-well-founded sets.  CSLI 14; Stanford University.  [PDF](http://irafs.org/courses/materials/aczel_set_theory.pdf).

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

[[!redirects pure set theory]]


pure spinor

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

#Contents#
* table of contents
{:toc}

## Idea

A _pure spinor_ is a [[spinor]] with vanishing [spinor-to-vector pairing](spin+representation#SuperPoincareBrackets)

$$
  \overline{\psi} \Gamma^a \psi \;=\; 0
  \,.
$$

## Related concepts

* [[Berkovits superstring]]

## References

See also

* Wikipedia, _[Pure spinor](https://en.wikipedia.org/wiki/Pure_spinor)_

Discussion of [[superstring scattering amplitudes]] in terms of pure spinors:

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics_ ([arXiv:1812.10969](https://arxiv.org/abs/1812.10969))

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics_ ([arXiv:1812.10970](https://arxiv.org/abs/1812.10970))

* [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure_ ([arXiv:1812.10971](https://arxiv.org/abs/1812.10971))

Relation to [[picture changing operators]]:

* Andrei Mikhailov, Dennis Zavaleta, _Geometrical framework for picture changing operators in the pure spinor formalism_ ([arXiv:2003.13995](https://arxiv.org/abs/2003.13995))

On pure spinors in [[supergravity]]:

* [[Martin Cederwall]], *Pure spinors in classical and quantum supergravity*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.06141](https://arxiv.org/abs/2210.06141)]

Relating the [[D'Auria-Fré formulation of supergravity]] to pure spinor-formalism:

* [[Pietro Antonio Grassi]], *Remarks on the Integral Form of $D=11$ Supergravity* [[arXiv:2304.01743](https://arxiv.org/abs/2304.01743)]

[[!redirects pure spinors]]


pure state

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Algebraic Quantum Field Theory
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _pure state_ is a [[state on a star-algebra]] which is an extremal point in the [[convex set]] of all states.

In [[physics]], recall that a [[state]] of a [[physical system]] is (in the [[Bayesian interpretation of quantum mechanics|Bayesian interpretation]]) a specification of the information that one might have about the system (typically relative to some fixed background information).  States form (at least) a [[poset]] where $\psi \leq \phi$ means that $\phi$ contains all of the information of $\psi$ (and possibly more).  A __pure state__ is a [[maximal element]] of this poset: a state that specifies as much information as possible about the system.  A __mixed state__ is a state that is not pure.

## Definitions

Fairly generally, a physical system has a complex [[C*-algebra]] $\mathcal{A}$ of [[observables]] (or more generally a [[unital algebra|unital]] [[star algebra]]) and a __[[state on a star-algebra|state]]__ is a positive-semidefinite [[linear function]] $\rho\colon \mathcal{A} \to \mathbb{C}$ such that $\rho(1) = 1$.  We might write $\rho$ as a convex-[[linear combination]] of two other states:
\[ \rho = a \sigma + b \tau ,\label{combo}\]
where necessarily $0 \leq a, b \leq 1$ and $a + b = 1$.

The state $\rho$ is __pure__ if, whenever (eq:combo) holds for $\sigma \neq \tau$, then either $a = 0$ (hence $b = 1$ and $\rho = \tau$) or $a = 1$ (hence $b = 0$ and $\rho = \sigma$); conversely, $\rho$ is __mixed__ if we ever have (eq:combo) for $\sigma \neq \tau$ and $0 \lt a \lt 1$ (hence $0 \lt b \lt 1$ and $\rho \neq \sigma, \tau$).

Really, this definition makes sense as long as the states form a [[convex space]].

To define when one state gives at least as much information as another (the [[partial order]] from the Idea section), let $\rho \leq \sigma$ mean that the [[mutual information]] $I(\rho,\sigma)$ equals the [[entropy]] $H(\rho)$, or equivalently that the [[conditional entropy]] $H(\rho|\sigma)$ is zero.  (In the classical case, this partial order is attributed to [Shannon (1953)](#Shannon1953), which I have not read, by [Li & Chong (2011)](#LiChong2011), which I have only skimmed.)  The [[maximal elements]] under this partial order should be precisely the pure states, but the direct definition of pure states is much simpler.

+-- {: .query}
I need to check whether these are equivalent on any $C^*$-algebra.  —Toby
=--

## Special cases

If $A$ is the algebra of continuous complex-valued functions on some [[compactum]] $X$, then the __pure states__ on $A$ correspond precisely to the [[points]] in $X$; so pure states here are the states of [[classical mechanics]] (at least for a compact [[phase space]]).  The mixed states, however, correspond more generally to [[Radon measure|Radon]] [[probability measures]] on $X$, with the pure states as the [[Dirac delta measure]]s.

On the other hand, if $A$ is the algebra of all [[bounded operators]] on some [[Hilbert space]] $H$, then the pure states on $A$ correspond precisely to the rays in $H$, as is usual in [[quantum mechanics]].  The mixed states, however, correspond more generally to [[density matrices]] on $H$, with the pure states those matrices of the form ${|\psi\rangle}{\langle\psi|}$ for some unit vector ${|\psi\rangle}$.

## Classical versus quantum

In any case, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information.  In the classical case, we may say that a pure state is a state of *complete* information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of [[quantum physics]] that distinguishes it from [[classical physics]].

## Related concepts

* [[mixed quantum state]]

[[!include states and observables -- content]]

## References

For comprehensive references see those at

* [[quantum probability theory]]

Textbook accounts:

* {#Landsman17} [[Klaas Landsman]], Sections 1.3 and 2.3 of: _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

See also:

* Quantiki, _[Pure states](https://www.quantiki.org/wiki/pure-states)

Not really references on this subject, but ones referred to in the text:

* {#Shannon1953} [[Claude Shannon]], _The lattice theory of information_, IEEE Transactions on Information Theory 1, 105--107 (1953)

* {#LiChong2011} Hua Li and Edwin Chong, _On a connection between information and group lattices_, Entropy 13, 683--708
 (2011) ([mdpi:1099-4300/13/3/683](http://www.mdpi.com/1099-4300/13/3/683))

[[!redirects pure states]]

[[!redirects pure quantum state]]
[[!redirects pure quantum states]]