prequantum field theory > history

< [[prequantum field theory]]


prequantum geometry

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###Context###
#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#### Geometric quantization
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[[!include geometric quantization - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Traditional [[prequantum geometry]] is the [[differential geometry]] of [[smooth manifolds]] which are equipped with a _[[twisted cohomology|twist]]_   in the form of a [[circle group]]-[[principal bundle]] and a circle-[[principal connection]].
In the context of _[[geometric quantization]]_ of [[symplectic manifolds]] these arise as _[[prequantum bundles]]_. Equivalently, prequantum geometry is the _[[contact geometry]]_ of the  total spaces of these bundles, equipped with their [[Ehresmann connection]] [[differential 1-form]] and  thought of as _[[regular contact manifolds]]_.  Prequantum geometry notably studies the [[automorphisms]] of [[prequantum bundles]] covering [[diffeomorphisms]] of the base
-- the _[[prequantum operators]]_ or _[[contactomorphisms]]_ -- and the [[action]] of these on the space of [[sections]] of the [[associated bundle|associated]] [[line bundle]] -- the _[[prequantum states]]_. This is an intermediate step in the genuine _[[geometric quantization]]_ of the [[curvature]] [[differential 2-form]] of these bundles,
which is obtained by "dividing the above data in half" ([[polarization]]), important for instance in the the _[[orbit method]]_.

But prequantum geometry is of interest in its own right.
For instance the above automorphism group naturally provides the
[[Lie integration]] of the [[Poisson bracket]] [[Lie algebra]] of the underlying [[symplectic manifold]], together with the canonical
injection into the [[group of bisections]] of the [[Lie integration]] of the  _[[Atiyah Lie algebroid]]_ which is associated with the given circle bundle, all of which are fundamental objects of interest in the study of [[line bundles]] over [[manifolds]].

## Related concepts

* [[higher prequantum geometry]]

* [[prequantum field theory]]


prequantum line bundle

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###Context###
#### Geometric quantization
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[[!include geometric quantization - contents]]
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#### Bundles
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[[!include bundles - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

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

For $(X,\omega)$ a ([[pre-symplectic form|pre]]-)[[symplectic manifold]] such that $\omega$ is an [[integral form]], a **prequantum line bundle** is any [[line bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$ on $X$ such that

$$
  \omega = F_\nabla
$$

is the [[curvature]] 2-form of $\nabla$.

=--

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

Choosing a prequantum line bundle is the first step in the [[geometric quantization]] of $(X, \omega)$.

=--

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

In [[cohomology]], a choice of prequantum line bundle corresponds to a lift from [[curvature]] 2-forms to [[ordinary differential cohomology]] $H^2(X)_{diff}$ through the [[curvature]] projection

$$
  H^2(X)_{diff} \stackrel{F}{\to} \Omega^2_{int}(X)
  \,.
$$

=--

The above definition has an immediate generalization to [[n-plectic geometry]].

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

For $(X,\omega)$ an [[n-plectic manifold]] such that $\omega$ is an [[integral form]], a **prequantum circle n- bundle** is any [[circle n-bundle with connection]] $(P \to X, \nabla)$  such that

$$
  \omega = F_\nabla
$$

is the [[curvature]] $(n+1)$-form of $\nabla$.

=--

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

In [[cohomology]], a choice of prequantum circle $n$-bundle corresponds to a lift from [[curvature]] $(n+1)$-forms to [[ordinary differential cohomology]] $H^{n+1}(X)_{diff}$ through the [[curvature]] projection

$$
  H^{n+1}(X)_{diff} \stackrel{F}{\to} \Omega^{n+1}_{int}(X)
  \,.
$$

=--

## Related concepts

* [[prequantized Lagrangian correspondence]]

* [[prequantum circle n-bundle]]

* [[local prequantum field theory]]

* [[fiber bundles in physics]]

* [[circle bundle]], [[circle n-bundle with connection]]

[[!include extended prequantum field theory - table]]

* [[geometric quantization]], [[higher geometric quantization]]

## References

Lecture notes with more details are in the section _[Lagrangians and Action functionals](geometry+of+physics#LagrangiansAndActionFunctionals)_ of

* _[[geometry of physics]]_

Discussion of prequantized (and [[polarization|polarized]]) symplectic manifolds in the context of [[cobordism rings]] and [[Thom spectra]] is in

* {#Morava99} [[Jack Morava]], _Cobordism of symplectic manifolds and asymptotic expansions_, talk at the conference in honor of S.P. Novikov's 60th birthday ([arXiv:9908070](http://arxiv.org/abs/math/9908070))

[[!redirects prequantum line bundles]]

[[!redirects prequantum circle bundle]]
[[!redirects prequantum circle bundles]]

[[!redirects prequantum bundle]]
[[!redirects prequantum bundles]]

[[!redirects pre-quantum line bundle]]
[[!redirects pre-quantum line bundles]]

[[!redirects pre-quantum circle bundle]]
[[!redirects pre-quantum circle bundles]]

[[!redirects pre-quantum bundle]]
[[!redirects pre-quantum bundles]]

[[!redirects prequantization]]
[[!redirects prequantizations]]

[[!redirects pre-quantization]]
[[!redirects pre-quantizations]]

[[!redirects prequantum connection]]
[[!redirects prequantum connections]]


prequantum n-state

In [[higher geometric quantization]] for a given [[prequantum line n-bundle]] a _prequantum $n$-state_ is a [[section]] of this prequantum line $n$-bundle.

[[!redirects prequantum n-states]]

[[!redirects prequantum state]]
[[!redirects prequantum states]]


Presburger arithmetic

2 +2


presentable category > history

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


presentable module

#Contents#
* table of contents
{:toc}

## Idea

A [[module]] is _presentable_ if it is the [[cokernel]] of a [[homomorphism]] of [[free modules]]. If the free modules have [[finite set|finite]] [[rank]], one has a [[finitely presented module]].

## Related concepts

* [[quasicoherent sheaf]]

* [[generators and relations]]

[[!redirects presentable modules]]


presentable orbifold

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###Context###
#### Higher geometry
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[[!include higher geometry - contents]]
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#### Higher Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

An [[orbifold]] is called *presentable* if it is ([[Morita equivalence|Morita]]-)[[equivalence in an (infinity,1)-category|equivalently]] a [[global quotient orbifold]] of a [[smooth manifold]] by the [[group action|action]] of some  [[Lie group]] (sometimes assumed/required to be [[compact Lie group|compact]]).

For example, (very) [[good orbifolds]] are presentable, since ([[finite group|finite]]) [[discrete groups]] are examples of ([[compact Lie group|compact]]) [[Lie groups]].

More interestingly, every [[effective orbifold]] $\mathcal{X}$ is presentable, namely as the [[global quotient orbifold|quotient]] of its [[frame bundle]] by the [[general linear group]], or equivalently of its orthonormal frame bundle (with respect to some [[Riemannian structure]]) by the [[orthogonal group]]: $\mathcal{X} \,\simeq\, Fr_{orth}(\mathcal{X}) \sslash O(dim(\mathcal{X}))$ ([Satake 1956](#Satake56),  recalled in, e.g., [Henriques & Metzler 2004, Prop. 1.4](#HenriquesMetzler04)).

In fact, it is [[conjecture|conjectured]] that *every* orbifold is presentable as the [[global quotient orbifold|global quotient]] of a smooth manifold by some Lie group ([Adem, Leida & Ruan 2007, Conj. 1.55](#AdemLeidaRuan07)).
Some results in this direction are presented in [Henriques & Metzler 2004, Sec. 5](#HenriquesMetzler04)

## References

* {#Satake56} [[Ichiro Satake]], _On a generalisation of the notion of manifold_, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 ([doi:10.1073/pnas.42.6.359]( https://doi.org/10.1073/pnas.42.6.359))

* {#HenriquesMetzler04} [[André Henriques]], [[David Metzler]], *Presentations of Noneffective Orbifolds*, Trans. Amer. Math. Soc. 356 (2004), 2481-2499  ([arXiv:math/0302182](https://arxiv.org/abs/math/0302182), [doi:10.1090/S0002-9947-04-03379-3](https://doi.org/10.1090/S0002-9947-04-03379-3))

* {#AdemLeidaRuan07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

[[!redirects presentable orbifolds]]


presentation axiom

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### Context
#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
{: toc}

## Idea

In the [[foundations]] of [[mathematics]], it\'s interesting to consider the axiom that the [[Set|Category of Sets]] Has [[projective object|Enough Projectives]]; in short: **CoSHEP** (pronounced /ko:-shep/).  This is more commonly known as the **presentation axiom**: PAx.   It is a weak form of the [[axiom of choice]].

## Statement
 {#Statement}

In elementary terms, CoSHEP states

+-- {: .num_defn}
###### Axiom (CoSHEP)

For every [[set]] $A$, there exists a set $P$ and a [[surjection]] $P \to A$, such that every surjection $X \twoheadrightarrow P$ has a [[section]].
=--

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

The full axiom of choice states that every surjection $X \to A$ has a section; hence in the above $P$ may be chosen to be $A$ itself.
=--

This should be read in view of the definition of _[[projective objects]]_:

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

An [[object]] $P$ in a [[category]] $C$ is (externally) [[projective object|projective]] iff the [[hom-functor]] $C(P, -): C \to Set$
takes [[epimorphism|epis]] to epis. This is the same as saying: given an epi $p: B \to A$ and a map $f: P \to A$, there exists a lift $g: P \to B$ in the sense that $f = p \circ g$.
=--

Accordingly, in a [[topos]] the CoSHEP axiom says equivalently

+-- {: .num_defn}
###### Axiom (CoSHEP)

Every object has a [[projective presentation]]. Hence: There are [enough projectives](projective+object#EnoughProjectives).
=--

Borrowing from the philosophy of [[constructive mathematics|constructivism]], we may also call this a _complete presentation_.

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

The [[duality|dual]] axiom,  that $Set$ has [enough injectives](injective+object#EnoughInjectives) (that is, every set admits an injection into an injective set) is [[true]] in every [[topos]]: every [[power object]] is an [[injective object]], and every object $X$ embeds in its power object $P X$ via the [[singleton subset|singleton]] map $\{\cdot\}:X\hookrightarrow P X$.
=--

## Justification

Although perhaps not well known in the literature of [[constructive mathematics]], the CoSHEP axiom may be justified by the sort of reasoning usually accepted to justify the [[axiom of countable choice|axioms of countable choice]] and [[axiom of dependent choice|dependent choice]] (which it implies, by Proposition \ref{ImpliesDependentAndCountableChoice} below).

To be explicit, every set $A$ should have a 'completely presented' set of 'canonical' [[elements]], that is elements given directly as they are constructed without regard for the [[equality relation]] imposed upon them. For canonical elements, [[equality]] is identity, so the [[BHK interpretation]] of logic justifies the axiom of choice for a completely presented set. This set is $P$, and $A$ is obtained from it as a [[quotient set|quotient]] by the relation of equality on $A$. This argument can be made precise in many forms of [[type theory]] (including those of Martin-Löf and Thierry Coquand), which thus justify CoSHEP, much as they are widely known to justify dependent choice.

## Consequences

The existence of sufficiently many [[projective presentations]] plays a central role in [[homological algebra]] as a means to construct [[projective resolutions]] of objects. Tradtionally one often uses the [[axiom of choice]] to prove that [[categories of modules]] have [enough projectives](projective+object#EnoughProjectives), on the grounds that the [[free modules]] are projective.

But the weaker assumption of CoSHEP is already sufficient for this purpose: while not every free module will be projective, one can still use CoSHEP to find a [[projective presentation]] for every free module (and thus for every module). This is discussed in more detail [here](projective+object#EnoughWithCOSHEP).

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

The following three conditions on a [[W-pretopos]] with [enough projectives](projective+object#EnoughProjectives) are equivalent:

1. The axiom of [[dependent choice]] (DC),

2. The axiom of [[countable choice]] (CC),

3. Projectivity of the [[singleton]] (the [[terminal object]]) $1$.
=--

Note that we normally assume (3) for [[the category of sets]], which is true in any (constructively) [[well-pointed pretopos]] and true *[[internalization|internally]]* in any pretopos whatsoever, so one normally says that DC and CC simply follow from the existence of enough projectives (CoSHEP).  Equivalently, internal DC and internal CC follow from internal CoSHEP.

+-- {: .proof}
###### Proof
Condition 1 easily implies 2. Condition 2 says precisely that the [[natural numbers object]] $\mathbb{N}$ is externally projective, and since $1$ is a retract of $\mathbb{N}$, it is projective under condition 2, so 2 implies 3. It remains to show 3 implies 1.

Let $X$ be inhabited, so there exists an [[entire relation]] given by a jointly monic span

$$1 \stackrel{epi}{\leftarrow} U \stackrel{f}{\to} X,$$

and similarly let

$$X \stackrel{epi \pi_1}{\leftarrow} R \stackrel{\pi_2}{\to} X$$

be an [[entire relation|entire binary relation]]. Let $p: P \to X$ be a projective cover. Since $1$ is assumed projective, the cover $U \to 1$ admits a section $\sigma: 1 \to U$, and the element $f \sigma: 1 \to X$ lifts through $p$ to an element $x_0: 1 \to P$. Next, in the diagram below, $p$ lifts through the epi $\pi_1$ to a map $q: P \to R$, and then $\pi_2 q$ lifts through $p$ to a map $\phi$ (since $P$ is projective):

$$\array{
 & & P  & \stackrel{\phi}{\to} & P \\
 & \swarrow p & \downarrow q & & \downarrow p \\
X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X
}$$

By the universal property of $\mathbb{N}$ (see [[recursion]]), there exists a unique map $h: \mathbb{N} \to P$ rendering commutative the diagram

$$\array{
1 & \stackrel{0}{\to} & \mathbb{N} & \stackrel{s}{\to} & \mathbb{N} \\
id \downarrow & & \downarrow h & & \downarrow h \\
1 & \underset{x_0}{\to} & P & \underset{\phi}{\to} & P \\
 & \swarrow p & \downarrow q & & \downarrow p \\
X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X
}$$

Clearly $\langle p h, p h s \rangle : \mathbb{N} \to X \times X$ factors through $\langle \pi_1, \pi_2 \rangle : R \to X \times X$, i.e., $\forall_{n: \mathbb{N}} (p h(n), p h(n+1)) \in R$, thus proving that dependent choice holds under CoSHEP.
=--

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

A topos in which CoSHEP holds but $1$ is not projective is $Set^C$, where $C$ is the category with three objects and exactly two non-identity arrows $a \to b \leftarrow c$. For if $U: C \to Set$ is a functor with $U(a) = \{a_0\}$, $U(b) = \{b_0, b_1\}$, and $U(c) = \{c_0\}$, with $U(a \to b)(a_0) = b_0$ and $U(c \to b)(c_0) = b_1$, then the map $U \to 1$ is epi but has no section, so $1$ is not projective. On the other hand, as noted below, every presheaf topos satisfies CoSHEP, assuming that $Set$ itself does.
=--

CoSHEP also implies several weaker forms of choice, such as the [[axiom of multiple choice]] and [[WISC]].  In weakly [[predicative mathematics]], it can be combined with the existence of [[function sets]] to show the [[subset collection]] axiom.

## In a topos
 {#topos}

When working in the [[internal logic]] of a [[topos]], the "internal" meaning of CoSHEP is "every object is covered by an [[internally projective object]]."  (Compare with the internal axiom of choice: every object is internally projective.)  As regards foundational axioms for toposes (in the sort of sense that the axiom of choice is regarded as "foundational"), the internal version of the presentation axiom should be taken as the default version.

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

Suppose that $1$ is (externally) projective in $E$. Then $E$ satisfies PAx whenever it satisfies internal PAx.
=--

Internal PAx does not follows from external PAx; see Counterexample 5.3. However, if _every_ object is projective (AC), then every object is internally projective (IAC).

A stronger version of PAx may be worth considering. Say that an object is **stably projective** if its [[pullback]] to any [[slice category]] is projective. Then stably projective objects are internally projective (proof?). Similarly, if we say that a topos $E$ satisfies stable PAx if every object is covered by a stably projective object, then a topos satisfies internal PAx if it satisfies stable PAx.

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

Every [[presheaf topos]] $Set^{C^{op}}$ has enough projectives, since any coproduct of representables is projective. If in addition $C$ has binary products, then by [this result](/nlab/show/internally+projective+object#presheaf), $Set^{C^{op}}$ validates internal PAx.
=--

+-- {: .num_example #counter}
###### Counterexample

However, not every presheaf topos validates internal PAx, even though every presheaf topos validates external PAx. As an example, let $C$ be the category where $Ob(C)$ is the disjoint sum $\mathbb{N} \cup \{a, b\}$, and preordered by taking the reflexive transitive closure of relations $n \leq n+1$, $n \leq a$, $n \leq b$. The claim is that neither $C(-, a)$, nor any presheaf $P$ that maps epimorphically onto $C(-, a)$, can be internally projective. Indeed, consider the presheaf $F$ defined by $F(a) = F(b) = \emptyset$ and $F(n) = [n,\infty)$, with $F(n+1 \to n)$ the evident inclusion. Let $G$ be the [[subterminal object|support]] of $F$, so that we have an epi $e: F \to G$.

The objects $C(-, a), C(-, b)$, and $G$ are [[subterminal object|subterminal]] and $G \cong C(-, a) \cap C(-, b) \cong C(-, a) \times C(-, b)$. The set $F^{C(-, a)}(b)$ is empty because there is no $C(-, a) \times C(-, b) \to F$ (it would give a section $G \to F$ of $e: F \to G$, but none exists), whereas $G^{C(-, a)}(b)$ is inhabited by $C(-, a) \times C(-, b) \cong G$. For any $P$ covering $C(-, a)$, the set $F^P(b)$ is empty (because any section $s: C(-, a) \to P$ of $P \to C(-, a)$ induces a function $F^P(b) \to F^{C(-, a)}(b) = 0$), and the set $G^P(b)$ is inhabited (the map $P \to C(-, a)$ induces a map $1 \cong G^{C(-, a)}(b) \to G^P(b)$). Thus the map $e^P \colon F^P \to G^P$ cannot be epic.
=--

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

Any topos that violates countable choice, of which there are plenty, must also violate internal PAx.
=--

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

An interesting example of a topos that has enough projectives and satisfies internal CoSHEP (at least, assuming the axiom of choice in [[Set]]), although it violates the full (internal) axiom of choice, is the [[effective topos]], and more generally any [[realizability topos]]. The reason for this is quite similar to the intuitive justification for CoSHEP given above. Technically, it results from the fact that realizability toposes are [[exact completions]]; an explanation is given in [this remark](/nlab/show/realizability+topos#pax).
=--

For a Grothendieck topos example, the sheaves on the interval $[0,1]$ with its usual topology give a topos in which the internal axiom of countable choice fails, hence internal PAx must also fail.

## In categories which are not topoi

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

According to Peter Scholze in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057798), an example of a category that satisfies external CoSHEP is the category of [[condensed sets]], assuming that [[Set]] satisfies the axiom of choice. The category of condensed sets do not form a topos, only an [[infinitary pretopos]].

However, internal CoSHEP fails in condensed sets.
=--

## Further properties

Since [[Set]] is (essentially regardless of foundations) an [[exact category]], if it has [enough projectives](projective+object#EnoughProjectives) then it must be the _free_ [[exact category]] $PSet_{ex/lex}$ generated by its [[subcategory]] $PSet$ of [[projective objects]].  By the construction of the [ex/lex completion](regular+and+exact+completions#TheExLexCompletion) $PSet_{ex/lex}$, it follows that every set is the quotient of some "pseudo-equivalence relation" in $PSet$; i.e., the result of imposing an equality relation on some completely presented set.  See [[SEAR plus epsilon|SEAR+ε]] for an application of this idea.

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

CoSHEP as a choice principle added to [[ZF]] implies a [[proper class]] of [[regular cardinals]].
=--

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

Since CoSHEP implies [[WISC]], and WISC has this implication (a result of [[Benno van den Berg|van den Berg]]).
=--

## In higher category theory

In [[higher category theory]], there are different versions of CoSHEP:

* For every [[0-groupoid]] $A$, there exists 0-groupoid $P$ and [[effective epimorphism]] $P \to A$, such that for all 0-groupoids $X$ there exists effective epimorphism $X \to P$ with [[section]].

* For every [[infinity-groupoid]] $A$, there exists 0-groupoid $P$ and [[effective epimorphism]] $P \to A$, such that for all 0-groupoids $X$ there exists effective epimorphism $X \to P$ with [[section]].

The difference between these versions of CoSHEP is that [[sets cover]].

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

  * [[internally projective object]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

When [[Peter Aczel]] was developing $CZF$ (a constructive predicative version of [[ZFC]]), he considered this axiom, under the name of the _presentation axiom_, but ultimately rejected it.

* [[Peter Aczel]], _The type theoretic interpretation of constructive set theory_. Logic Colloquium '77 (Proc. Conf., Wroclaw, 1977), pp. 55--66, Stud. Logic Foundations Math., 96, North-Holland, Amsterdam-New York, 1978. doi:[10.1016/S0049-237X(08)71989-X](https://doi.org/10.1016/S0049-237X%2808%2971989-X)

The presentation axiom was, however, adopted by [[Erik Palmgren]] in $CETCS$ (a constructive predicative version of [[ETCS]]):

*  [[Erik Palmgren]],  _Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets_, Annals of Pure and Applied Logic
 **163** Issue 10 (2012) 1384–1399, doi:[10.1016/j.apal.2012.01.011](https://doi.org/10.1016/j.apal.2012.01.011) arXiv:[1201.6272](https://arxiv.org/abs/1201.6272),  [author pdf](http://www.math.uu.se/~palmgren/cetcs.pdf).

Its relationship to some other weak axioms of choice is studied in

* [[Michael Rathjen]], _Choice principles in constructive and classical set theories_, In: Z. Chatzidakis, P. Koepke, & W. Pohlers (Eds.), Logic Colloquium '02 (Lecture Notes in Logic, pp. 299-326) (2006). Cambridge: Cambridge University Press. doi:[10.1017/9781316755723.014](https://doi.org/10.1017/9781316755723.014). [author pdf](http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf).

category: foundational axiom

[[!redirects presentation axiom]]
[[!redirects PAX]]
[[!redirects PAx]]
[[!redirects pax]]
[[!redirects COSHEP]]
[[!redirects CoSHEP]]
[[!redirects coshep]]
[[!redirects completely presented set]]
[[!redirects completely presented sets]]


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### Context
#### Category theory
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[[!include category theory - contents]]
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=--
=--

# Contents
* the following line creates the automatic table of contents
{: toc}

## Idea

Similarly to the presentation of [[groups]] by [[generators and relations]], a
category may be presented by a set of generating arrows subject to
certain relations.

## Definition

Let $G$ be a [[quiver|directed graph]], and let $R$ be a function that assigns to each pair $a,b$ of objects of the [[free category]] $F(G)$ a [[binary
relation]] $R_{a,b}$ on the [[hom-set]] $F(G)(a,b)$.  The
**category with generators $G$ and relations $R$** is the [[quotient category]] (as defined in [Mac Lane](#MacLane) or [Mitchell](#Mitchel), for example -- this is _not_ the nLab definition due to issues of [[evil]]) $F(G)/R$. For a category $C$, an [[isomorphism of categories|isomorphism]] $C\to F(G)/R$ is called a **presentation** of $C$.

## Properties

Writing $can\colon F(G)\to F(G)/R$ for the canonical functor, it follows
from the universal property of the quotient category that for any
functor $S\colon F(G)\to D$ that respects the relation $R$ ($f
R_{a,b}g$ implies $S(f)=S(g)$), there exists a unique functor $S'\colon
F(G)/R\to D$ with $S = S'\circ can$.

## Examples

1. Every category $C$ has a presentation by generators and relations:
Take $G$ as the underlying graph of $C$, and for objects $a$, $b$,
let $R_{a,b}$ be the relation on $F(G)(a,b)$ consisting of all pairs
of paths from $a$ to $b$ in $G$ whose arrows have the same composition
in $C$. However, there are sometimes more economical presentations for a
category, as the following example shows.

2. The augmented simplex category $\Delta_a$ is generated by the face
maps and the degeneracy maps, subject to the simplicial relations (see
[[simplex category]] for details).  The existence of a functor from the
quotient category to $\Delta_a$ follows from the fact that the arrows of
$\Delta_a$ do satisfy the simplicial relations, and the fact that this
functor is an isomorphism may be verified using the unique
decomposition of an arrow of $\Delta_a$ as the composition of
degeneracies of decreasing index followed by the composition of face
maps of increasing index (see the lemma on p. 177 of [Mac Lane](#MacLane)). Similarly, the subcategory $(\Delta_a)_{inj}$ consisting of all monics (injective monotone functions in our case) is generated by the face maps subject to the single simplicial relation involving only face maps.

##2-cells

A useful way to think of the relations is as being 2-cells between parallel pairs of arrows, thus if $a, b$ are objects, and $(u,v)\in R_{a,b}$, we think of $(u,v)$ as a 2-cell (initially _from $u$ to $v$_).  In this way, one can encode [[rewriting]] systems of a certain kind in terms of the embryonic data for a 2-category. This is discussed more in the entry on [[computad]], which are also called [[polygraphs]].

## References

* [[Saunders Mac Lane]],   [[Categories Work|Categories for the Working Mathematician]], pp. 51-52
{#MacLane}

* [[Barry Mitchell]], _Introduction to Category Theory and
Homological Algebra_, in P. Salmon (Ed.), _Categories and Commutative
Algebra_.  Springer, 2010. pp. 108-112.{#Mitchel}

* [[Barry Mitchell]], _Rings with several objects_, Advances in Mathematics 8 (1972), 1–161.

[[!redirects presentation of a category by generators and relations]]
[[!redirects presentations of a category by generators and relations]]
[[!redirects presentations of categories by generators and relations]]


presentations of (infinity,1)-sheaf (infinity,1)-toposes

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

#Contents#
* tic
{:toc}

# Idea #

An [[(∞,1)-sheaf]] -- often called a [[∞-stack]] -- is the
[[(infinity,1)-category|(∞,1)-categorical]] analog of a [[sheaf]].
Just like a [[category of sheaves]] is a [[topos]],
an [[(∞,1)-category of (∞,1)-sheaves]] is an
[[(∞,1)-topos]].

There is good [[motivation for sheaves, cohomology and higher stacks]].

Here we recall basic definitions and then concentrate on
[[model category|1-categorical models]] that
[[presentable (infinity,1)-category|present]] [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-categories of ∞-stacks]].

What we describe is effectively the old theory of the
[[model structure on simplicial presheaves]] seen in the new light of
[[Higher Topos Theory]].

# Plan #

We proceed as follow.

* As a preparation,

  * we recall 1-categorical [[presheaf|presheaves]]

  * and how a [[category of sheaves]] may equivalently be thought of as

    * a category [[geometric embedding|geometrically embedded]] into that of presheaves

    * or equivalently as the [[localization]] of presheaves at [[local isomorphisms]].

* Then in the main part, after recalling

  * the definition of [[(∞,1)-category]]

    * and the [[presentable (infinity,1)-category|presentation]] of
      $(\infty,1)$-categories by [[model category|1-categorical models]]

  we give

  * the general definition of [[(∞,1)-presheaves]];

    * and their [[presentable (infinity,1)-category|presentation]] by the [[global model structure on simplicial presheaves]]

  and finally

  * the definition of the [[(∞,1)-category of (∞,1)-sheaves]] by [[reflective (infinity,1)-subcategory|(∞,1)-geometric embedding]];
  * and the [[model category|1-categorical model]] of this by [[Bousfield localization]] to the [[local model structure on simplicial presheaves]].

# Sheaf toposes #

It is helpful to briefly recall the story that we want to tell in the [[category theory]] context, because in the full [[higher category theory]] context it will be **literally the same** with all notions such as [[adjoint functor]], [[exact functor]] etc suitably regarded in the context of [[(∞,1)-functors]].

## Presheaves ##

Consider a [[category]] $C$ that we want to think as a category
of "test spaces". Classical choices would be $C = $ [[Top]], the
category of [[topological spaces]], $C = $ [[Diff]], the category
of smooth manifolds or $C = Op(X)$, the [[category of open subsets]]
of some [[topological space]] $X$.

Let [[Set]] be the category of sets. We write

$$
  PSh(C) := [C^{op}, Set] := Func(C^{op}, Set)
$$

for the category of [[presheaf|presheaves]] on $C$.
This is like a category of [[space and quantity|very general spaces]] modeled on $C$
as described at [[motivation for sheaves, cohomology and higher stacks]].

## Sheaves ##

In fact, this is a bit too general for most purposes:
the objects of $PSh(C)$ may be very non-local in that they don't
respect the way test objects in $C$ are supposed to glue together.
The full subcategory on those presheaves that do respect some
kind of gluing of test objects is the [[category of sheaves]].

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

A [[category of sheaves]] on $C$ is a category $Sh(C)$
equipped with a [[geometric embedding]] into $PSh(C)$

$$
  Sh(C) \stackrel{\leftarrow}{\to} PSh(C)
  \,.
$$

=--

Recall that this means that

* $Sh(C) \hookrightarrow PSh(C)$ is a [[full and faithful functor]];

* $\bar {(\cdot)} : PSh(C) \to Sh(C)$ is a [[exact functor|left exact]] [[left adjoint]].

in other words that

* $Sh(C) \hookrightarrow PSh(C)$ is the inclusion of a [[reflective subcategory]];

* with the special property that the [[left adjoint]] to the inclusion is [[exact functor|left exact]] (i.e. preserves finite [[limits]]).

In view of our models for $\infty$-sheaves it is of importance that
this implies an equivalence characterization

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

The category $Sh(C)$ is equivalent to the full [[subcategory]] of
$S$-[[local object|local]] presheaves, where $S$ is the set of [[local isomorphisms]].

=--

Another useful kind of [[geometric embeddings]] is that of the point:

let
${*}$ be the category with a single morphism (the identity on a single object).
Then $PSh({*}) \simeq Sh({*}) \simeq Set $. Geometric embeddings

$$
  x : Sh({*}) \stackrel{\leftarrow}{\to} Sh(C)
$$

are called [[point of a topos|points]] of $Sh(C)$. We say that
$Sh(C)$ has _enough points_ if isomorphisms of sheaves can be tested on
points

$$
  (f : A \stackrel{\simeq}{\to} B)\in Sh(C)
  \;\;
  \Leftrightarrow
  \;\;
  \forall x : (x^* f : x^* A \stackrel{\simeq}{\to} x^* B)
  \,.
$$

This is the situation we shall concentrate on here.

* The topos $Sh(Diff)$ has enough points, one for every $n \in \mathbb{N}$.

* The topos $Sh(Op(X))$ has enough points: one for every ordinary point of $X$.

If $Sh(C)$ has enough points, we may characterize sheaves in yet another way, which is the one that directly suggests the  [[local model structure on simplicial presheaves]] discussed below:

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

Let $S \subset Mor(PSh(C))$ be the set of _[[stalk]]wise isomorphisms_, i.e. those morphisms $f : A \to B$ of presheaves such that for all [[point of a topos|points]]
$x$ the morphism $x^* f : x^* A \to x^* B$ is
an [[isomorphism]] (of [[sets]]).

If $Sh(C)$ has [[point of a topos|enough points]], then $Sh(C)$ is equivalent to the full [[subcategory]] of $S$-local presheaves.

=--

The [[local model structure on simplicial presheaves]] that we are going to describe is obtained from this description of sheaves by

* replacing [[sets]] by [[simplicial set]]

* replacing [[stalk]]wise [[isomorphism]] of sets by [[stalk]]wise [[model structure on simplicial sets|weak homotopy equivalences]] of simplicial sets.

So the model structures we shall encounter are plausible guesses. What is less trivial is that this plausible structure indeed [[presentable (infinity,1)-category|presents]] the fully general notion of [[(∞,1)-sheaf]]/[[∞-stack]].

This fully general notion we introduce now.

# $(\infty,1)$-categories and their presentation #

An ordinary [[locally small category]] is a
[[enriched category|category enriched]] over the category [[Set]]
of [[sets]].

An [[(∞,0)-category]] is an [[∞-groupoid]]
which we think of as modeled by a [[simplicial set]] that
is a [[Kan complex]].

Recall that there is a notion of
[[nerve and realization]]

$$
  N : SSet\text{-}Cat \stackrel{\leftarrow}{\to} SSet : |-|
$$

for [[SSet]]-[[enriched category|enriched categories]]
induced by a [[simplicial object|cosimplicial]] [[simplicially enriched category]]

$$
 \Delta_{SSet-Cat} : \Delta \to SSet-Cat
$$

where the [[nerve]] operation $N$ is called the [[homotopy coherent nerve]]
of [[simplicially enriched category|simplicially enriched categories]].

+-- {: .un_defn}
###### Definition ($(\infty,1)$-category)

An [[(∞,1)-category]] is a
[[enriched category|category enriched]] over $\infty$-groupoids,
i.e. an [[SSet]]-[[enriched category]] all whose [[hom-objects]] happen
to be [[Kan complexes]].

Given two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$
the [[(∞,1)-functor]] $(\infty,1)$-category is

$$
  Func(\mathbf{C}, \mathbf{D}) :=
  |SSet(N(\mathbf{C}), N(\mathbf{D}))|
  \,.
$$

This is indeed itself an $(\infty,1)$-category ([[Higher Topos Theory|HTT, prop 1.2.7.3]]).

The [[(∞,1)-category of (∞,1)-categories]]
$(\infty,1)Cat$ is that whose

* objects are $(\infty,1)$-categories;

* for $\mathbf{C}$ and $\mathbf{D}$ two $(\infty,1)$-categories the $\infty$-groupoid $(\infty,1)Cat(\mathbf{C}, \mathbf{D})$  is the maximal [[Kan complex]] inside the [[simplicial set]] of maps between the [[homotopy coherent nerves]]

  $$
    (\infty,1)Cat(\mathbf{C}, \mathbf{D})
    :=
    Core( SSet(N(\mathbf{C}), N(\mathbf{D})) )
    \,.
  $$

=--

**Examples**

* Using the [[monoidal category|monoidal]] embedding $const : Set \hookrightarrow \infty Grpdf \subset SSet$
  every ordinary category is an $(\infty,1)$-category.

* The $(\infty,1)$-category $\infty Grpd$ ([[∞Grpd]]) is the full [[SSet]-subcategory of [[SSet]]
  on [[Kan complexes]].

+-- {: .un_defn}
###### Definition (homotopy category)

The [[simplicial homotopy groups|simplicial connected components]] functor

$$
  \pi_0 : SSet \to Set
$$

is strong monoidal and hence induces a functor

$$
  H : (\infty,1)Cat \to Cat
  \,.
$$

The image $H(\mathbf{C})$ of an $(\infty,1)$-category $\mathbf{C}$
with $H(\mathbf{C})(x,y) = \pi_0(\mathbf{C}(x,y))$
is the [[homotopy category of an (∞,1)-category]].

Two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$ are
**equivalent** if they are isomorphic in $H((\infty,1)Cat)$

$$
  (f : \mathbf{C} \to \mathbf{D} \;is equivalence)
  \;\;
  \Leftrightarrow
  \;\;
  (H_{(\infty,1)Cat}(f) : \mathbf{C} \to \mathbf{D}
  \;
  is isomorphism)
  \,.
$$

=--

## Presentations ##

It is often convenient to [[presentable (infinity,1)-category|present]]
$(\infty,1)$-categories by 1-categorical [[model category|models]].

+-- {: .un_defn}
###### Definition ($(\infty,1)$-category presented by a model category)

For $\mathbf{A}$ a [[combinatorial simplicial model category]],
the $(\infty,1)$-category [[presentable (infinity,1)-category|presented]]
by it is the full
subcategory $\mathbf{A}^\circ \subset \mathbf{A}$ on objects that
are both cofibrant and fibrant.

=--

**Remark** The axioms of a simplicial model category
ensure that the [[hom-object|hom-simplicial sets]] of
$\mathbf{A}^\circ$ are indeed [[Kan complexes]].
(for instance [[Higher Topos Theory|HTT, remark 3.1.8]]).

+-- {: .un_prop }
###### Proposition ([[Higher Topos Theory|HTT, remark A.3.7.7]])

Let $\mathbf{A}$ and $\mathbf{B}$ be
[[combinatorial simplicial model category|combinatorial simplicial model categories]].
Then the corresponding $(\infty,1)$-categories $\mathbf{A}^\circ$ and
$\mathbf{B}^\circ$ are equivalent precisely if there is a sequence of
[[SSet]]-[[enriched functor|enriched]] [[Quillen equivalences]]
$$
  \mathbf{A}
  \stackrel{\leftarrow}{\to}
  \stackrel{\to}{\leftarrow}
  \stackrel{\leftarrow}{\to}
  \cdots
  \mathbf{B}
  \,.
$$

=--

# $(\infty,1)$-Sheaf $(\infty,1)$-toposes #

There is now an obvious definition of $(\infty,1)$-categories of $(\infty,1)$-presheaves and
of $(\infty,1)$-sheaves by interpreting the 1-categorical story in the $(\infty,1)$-categorical context.

## $(\infty,1)$-Presheaves ##

Now we generalize the above from sheaves to
[[(∞,1)-sheaves]] also known as
[[∞-stacks]].

+-- {: .un_defn}
###### Definition ($(\infty,1)$-presheaves)

The $(\infty,1)$-category of [[(∞,1)-presheaves]]
on $C$ is

$$
  PSh_\infty(C) := [C^{op}, \infty Grpd] =
  Func( C^{op}, \infty Grpd )
  \,.
$$

=--

+-- {: .un_prop}
###### Proposition (models for $(\infty,1)$-presheaves) ([[Higher Topos Theory|HTT, prop. 4.2.4.4]],
see also
[[Higher Topos Theory|HTT, prop. 5.1.1.1]])

The $(\infty,1)$-category
presented by the
[[global model structure on simplicial presheaves]]
$SPSh(C)_{proj}$
on $C$
(either the projective or the injective one)
is equivalent to that of $(\infty,1)$-presheaves on $C$:
$$
 (SPSh(C)_{proj})^{\circ}
 \simeq
 (SPSh(C)_{inj})^{\circ}
 \simeq
 PSh_\infty(C)
 \,.
$$

=--

## $(\infty,1)$-sheaves ##

There are $(\infty,1)$-category analogs of all the familiar
notions from [[category theory]], in particular

* [[adjoint functor]]

* [[exact functor]] (preserving finite [[limits]]).

Using this we obtain a definition of [[geometric embedding]]
of $(\infty,1)$-toposes , i.e. left exaxt
[[reflective (∞,1)-subcategories]]
by literally copying the 1-categorical definition.

+-- {: .un_defn}
###### Definition ($(\infty,1)$-sheaves) ([[Higher Topos Theory|HTT, def. 6.1.0.4]])

An [[(∞,1)-category of (∞,1)-sheaves]] is a [[geometric embedding]]
into an [[(∞,1)-category of (∞,1)-presheaves]]

$$
  Sh_\infty(C) \stackrel{\leftarrow}{\to} PSh_\infty(C)
  \,.
$$

=--

+-- {: .num_lemma }
###### Proposition (models for reflective $(\infty,1)$-subcategories)

Let
the [[combinatorial simplicial model category]] $\mathbf{B}$
be a left [[Bousfield localization]] of
the
[[combinatorial simplicial model category]]
$\mathbf{A}$
then

$$
  \mathbf{B}^\circ \stackrel{\leftarrow}{\to}
  \mathbf{A}^\circ
$$

is the inclusion of a [[reflective (∞,1)-subcategory]].

=--

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

By [[Higher Topos Theory|HTT, prop A.3.7.4]] every
combinatorial simplicial left  [[Bousfield localization]]
is given by a set $S$ of cofibrations such that

* the fibrant objects of $\mathbf{B}$ are
  precisely the fibrant objects in $\mathbf{A}$
  that are $S$-[[local object]];

* the weak equivalences of $\mathbf{B}$
  are the $S$-local morphisms in $\mathbf{A}$.

Accordingly $\mathbf{B}^\circ$ is the full $\infty Grpd$-enriched
subcategory of $\mathbf{A}^\circ$ on $S$-[[local objects]].
(see also [[Higher Topos Theory|HTT, prop 6.5.2.14]]).

By [[Higher Topos Theory|HTT, prop. 5.5.4.15]] this means that
$\mathbf{B}$ is a [[reflective (∞,1)-subcategory]] of $\mathbf{A}$.

=--

**Remark**
Notice that this does not yet say that the localization is
_left exact_ .

But this makes at least plausible that
the [[local model structure on simplicial presheaves]] is a
[[presentable (infinity,1)-category|presentation]] for an
[[(∞,1)-category of (∞,1)-sheaves]].

That this is indeed the case is

+-- {: .num_lemma #SizeOfLeftCoset}
###### Proposition (model for hypercomplete $(\infty,1)$-sheaves) ([[Higher Topos Theory|HTT, prop. 6.5.2.14]])

The [[local model structure on simplicial presheaves]]
$SSh(C)^{l loc}_{proj}$ presents the [[hypercompletion|hypercompleted version]]
of the [[(∞,1)-category of (∞,1)-sheaves]]
$Sh^{hc}(C)$ on $C$.

$$
  (SPSh(C)_{proj}^{loc})^\circ
  \simeq
  Sh^{hc}(C)
  \,.
$$

=--

**Remark** See the discussion at [[?ech cohomology]] for
the  role of [[hypercompletion]].

# Applications #

## Abelian sheaf cohomology as special case of $\infty$-stackification ##

The [[nerve]] operation of the [[Dold-Kan correspondence]]

$$
  N : Ch_+ \to SimpAb \subset \infty Grpd
$$

embeds [[sheaf|sheaves]] with values in non-negatively graded [[chain complex]]es
of abelian groups
into simplicial sheaves as those simplicial sheaves with values in
[[Kan complex]]es that carry a struict abelian group structure.
This way [[homological algebra]] and [[abelian sheaf cohomology]]
are realized as special cases of models for $\infty$-stacks:
a complex of abelian sheaves presents a stably abelian $\infty$-stack.

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

Under the [[Dold-Kan correspondence]] [[abelian sheaf cohomology]]
identifies with the [[hom-set]] of the
[[homotopy category of an (infinity,1)-category|homotopy category]] corresponding
[[infinity-stack]] [[(infinity,1)-topos]].

More precisely, let

* the underlying [[site]] be the [[category of open subsets]]  $C = Op(X)$
  of a [[topological space]] $X$,

* let $A \in Sh(X)$ be a sheaf with values in abelian groups on $X$;

* let $\mathbf{B}^n A \in Sh(X,SSet)$ be the image of the
  complex of sheaves $A[-n]$ concentrated in degree $n$ under the
  [[Dold-Kan correspondence|Dold-Kan]] [[nerve]];

* write $X \in Sh(X)$ for the [[terminal object]] sheaf in $Sh(X)$ (the sheaf constant
  on the singleton set).

Then degree $n$ [[abelian sheaf cohomology]] of $X$ with coefficients in $A$ is
homotopy classes of maps from $X$ to $\mathbf{B}^n A$:

$$
  H^n(X,A) \simeq Ho_{SSh(X)}(X, \mathbf{B}^n A)
  \,.
$$
=--

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

The original proof was given in [[BrownAHT]] in terms of the [[category of fibrant objects]] structure on locally Kan simplicial sheaves.

The analogous arguments in terms of the full injective model structure were  given by Jardine. See section 6 of his [lecture notes](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf).
=--

[[!redirects presentations of (∞,1)-sheaf (∞,1)-toposes]]

[[!redirects models for infinity-stack (infinity,1)-topoi]]
[[!redirects models for ∞-stack (∞,1)-toposes]]
[[!redirects models for ∞-stack (∞,1)-topoi]]
[[!redirects models for infinity-stack (infinity,1)-toposes]]


presented stack

#Definition#

A [[stack]] $X$ is _presented_ by a [[groupoid]] $G$, if it is the stack (then [[differentiable stack|usually]] regarded as a stack on the [[site]] [[Top]] or [[Diff]]) which assigns to each test domain $U$ the category of $G$-principal bundles over $U$

$$
  X := G-Bund(-) : U \mapsto \{G-principal bundles on U\}
  \,.
$$

#Formulation in terms of nonabelian cohomology#

This can be reformulated as follows: for $X$ a [[manifold]] let
$hom(X,G)$ denote the [[internal hom]] of [[groupoid]]s (or of categories with topological or smooth structure), with $X$ regarded as the [[discrete category|discrete]] groupoid over $X$. We can regard this as the groupoid of _trivial_ $G$-principal bundles over $X$:

This is contravariantly functorial in $X$ and indeed yields a groupoid-valued presheaf

$$
  G-TrivBund : X  \mapsto hom(X,G)
$$

the presheaf of trivial $G$-principal bundles.

So the stack presented by $G$ is the [[stackification]] of this groupoid-valued presheaf.

In yet other words this means nothing but that the stack presented by $C$ is the [[nonabelian cohomology]] $H(-,G)$ with coefficients in $G$:

$$
  G-Bund(-) := H(-,G)
  \,.
$$

#Further resources#

There is discussion of this and related aspects in [[Schreiber:Differential Nonabelian Cohomology]] in the [private part](http://ncatlab.org/schreiber/show/HomePage) of the $n$Lab.


preserved limit

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

# Preservation of limits
* table of contents
{: toc}

## Idea

If $J\colon I \to C$ is a [[diagram]] and $x$ is its [[limit]] in $C$, then we may naïvely say that this limit is _preserved_ by a [[functor]] $F\colon C \to D$ if $F(x)$ is the limit of the [[composite]] diagram $I \overset{J}\to C \overset{F}\to D$.  However, it is not enough to state this at the level of objects; we also need to impose some coherence conditions, preserving the entire universal [[cone]].  Furthermore, we can use a trick involving the [[Yoneda embedding]] to get a meaningful condition even if $J$ has no limit in $C$ at all.

## Definitions

Let $J\colon I \to C$ be a [[diagram]] and let $F\colon C \to D$ be a [[functor]].

Recall (see [[limit]]) that a [[cone]] over $J$ in $C$ may be defined as an [[object]] $x$ of $C$ together with a [[natural transformation]] to $J$ from the composite $\const^I_x\colon I \overset{!}\to \mathbf{1} \overset{\{x\}}\to C$, where $\mathbf{1}$ is the [[terminal category]].  Then a [[terminal object]] in the category of these cones (if it exists) is a [[limit]] of $J$ in $C$.  Thus, a limit consists of an object $x$ and a natural transformation $\eta\colon \const^I_x \to J$.

The functor $F$ __preserves__ the limit $(x,\eta)$ if $(F(x),F\cdot\eta)$ is a limit of the functor $F \circ J$ in $D$.  (Here, $F\cdot\eta\colon \const^I_{F(x)} \to F \circ J$ is a [[whiskering]].)

Dually, $F$ __preserves__ a [[colimit]] of $J$ if $F^\op\colon C^\op \to D^\op$ preserves it as a limit of $J^\op\colon I^\op \to C^\op$.

For instance:

* Let $I$ be the [[empty category]], so that a limit of the unique functor $J\colon I \to C$ is a [[terminal object]] $1$.  Then $F$ preserves this terminal object if and only if $F(1)$ is a terminal object of $D$.

* Let $I$ be the [[discrete category]] $\mathbf{2}$, so that $J$ picks out two objects $a$ and $b$ of $C$ and the limit of $J$ is a [[product]] $a \times b$ of $a$ and $b$.  Note that this product comes equipped with product projections $\pi\colon a \times b \to a$ and $\rho\colon a \times b \to b$.  Then $F$ preserves this product if and only if $F(a \times b)$ is a product of $F(a)$ and $F(b)$ and furthermore the product projections are $F(\pi)$ and $F(\rho)$.

If $F$ preserves all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ preserves that sort of limit (e.g. $F$ preserves [[products]], $F$ preserves [[equalizers]], etc.).

A functor that preserves all small limits in $C$ that exist is called a __[[continuous functor]]__.  Usually this term is only used when $C$ has all small limits, i.e. is a [[complete category]].

## Examples
  {#Examples}

* [[limits preserve limits]]

* [[adjoints preserve (co-)limits]]

* [[hom-functor preserves limits]]

## Preservation of weighted limits

Analogously, an [[enriched functor]] between [[enriched categories]] may preserve [[weighted limits]].  Are there any tricky points that we should mention?

## Preservation of limits that don\'t exist

Sometimes we want to say that a functor $F\colon C \to D$ preserves a limit that does not actually exist in $C$.  For instance, a __finitely continuous functor__ is usually defined as one that preserves all finite limits.  If $C$ is a [[finitely complete category]], then this is fine; such a functor is called __[[left exact functor|left exact]]__.  But what if $C$ does not have all finite limits?

If $C$ and $D$ are [[locally small category|locally small]], then we can use the [[Yoneda lemma]] to turn the question into one involving categories that *do* have the required limits (and in fact have all limits), the [[presheaf categories]] $[C^op,Set]$ and $[D^op,Set]$.  (For colimits, use $[C,Set]$ and $[D,Set]$; for $V$-enriched categories, use $[C^op,V]$ and $[D^op,V]$, which will work if $V$ is complete.)

The left [[Kan extension]] of the composite $C \overset{F}\to D \overset{Yon}\hookrightarrow [D^\op,Set]$ along the [[Yoneda embedding]] $C \overset{Yon}\hookrightarrow [C^\op,Set]$ (which always exists) is a functor from $[C^op,Set]$ to $[D^op,Set]$, which may be written as $- \otimes F$ (alluding to the [[bimodule]] nature of [[profunctors]]).  A diagram $J\colon I \to C$ becomes a diagram $I \overset{J}\to C \overset{Yon}\hookrightarrow [C^op,Set]$ in $[C^op,Set]$, where it has a limit.  If $- \otimes F$ preserves this limit, then we say that $F$ __preserves__ the hypothetical limit of $J$.

Since the Yoneda embedding preserves and [[reflected limit|reflects]] all limits, if $J$ has a limit in $C$, then this condition is equivalent to the condition that $F$ preserve it in the ordinary sense, but in general it is stronger than requiring that $F$ preserve the limit only if it exists in $C$.

Finishing the motivating example, a __[[flat functor]]__ may be defined as one that preserves all finite limits, whether or not they exist.

## Related concepts

* ([[cocontinuous functor|co]])[[continuous functor]]

* [[reflected limit]]

* [[created limit]]

* [[lifted limit]]

## References

* [[Saunders MacLane]], §V.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], §2.4 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* [[Emily Riehl]], §3.3 in: *[[Category Theory in Context]]*, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf), [book website](http://www.math.jhu.edu/~eriehl/context/)]

[[!redirects preserved limit]]
[[!redirects preserved limits]]
[[!redirects preserves limits]]
[[!redirects preservation of limits]]
[[!redirects preserved colimit]]
[[!redirects preserved colimits]]
[[!redirects preserves colimits]]
[[!redirects preservation of colimits]]

[[!redirects preserves]]
[[!redirects preserved]]

[[!redirects preserves colimits]]
[[!redirects preserve colimits]]
[[!redirects preserving colimits]]


preset

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

# Presets
* table of contents
{: toc}

## Idea

A _preset_ is a [[set]] without an [[equality relation]].  Conversely, a set may be defined as a preset $X$ equipped with an equality relation, an [[equivalence relation|equivalence]] _prerelation_ on $X$ which satisfies the [[principle of substitution]] and the [[identity of indiscernibles]].

In his seminal work _The Foundations of Constructive Analysis_ (1967), [[Errett Bishop]] explained what you must do to define a _set_ (see also [[Bishop set]]) in three steps:

1. You must state what one must do to construct an element of the set;
2. Given two elements constructed as in (1), you must state what one must do to prove that the elements are equal;
3. You must prove that the relation defined in (2) is reflexive, symmetric, and transitive (which can be phrased in similar 'what one must do' terms, but that\'s kind of wordy).

If you only do (1), then you don\'t have a set, according to Bishop; you only have a __preset__.

## Prefunctions and prerelations

As [[functions]] go between sets, so __prefunctions__ go between presets. (Bishop used the term 'operation' instead of 'prefunction', but 'operation' has many other meanings.) Even if $X$ and $Y$ are sets, a prefunction from $X$ to $Y$ is not the same as a function from $X$ to $Y$, because a prefunction need not preserve equality; that is, we may have $a = b$ without $f(a) = f(b)$.  Conversely, we may define a function as a prefunction (between sets) that preserves equality; such a prefunction is said to be __extensional__.

For example, consider the identity prefunction on the underlying preset of both $Q^+$ and $Z^+ \times Z^+$, as defined above.  From $Z^+ \times Z^+$ to $Q^+$, this is a function, since $a/b = c/d$ if $(a,b) = (c,d)$.  But from $Q^+$ to $Z^+ \times Z^+$, it is not a function, since (for example) $2/4 = 3/6$ but $(2,4) \neq (3,6)$.  A related example is the operation of taking the numerator of a (positive) rational number; from $Q^+$ to $Z^+$, we may view this as a prefunction but not as a function, although it is a function on $Z^+ \times Z^+$.

In general, the prefunctions from $X$ to $Y$ form a preset, since there is no way to compare them for equality.  (Of course, it is still [[predicative mathematics|impredicative]], at least in the classical sense, to form this preset.)  However, if $Y$ is a set, then these prefunctions do form a set, with $f = g$ defined to mean that $f(a) = g(a)$ for every $a$ in $X$.  If $X$ is also a set, then the [[function set]] from $X$ to $Y$ is a [[subset]] of this set of prefunctions.

Composition of prefunctions is also possible, but likewise does not preserve equality.

A (say binary) __prerelation__ between $X$ and $Y$ may be thought of as a prefunction from $X \times Y$ to [[truth values]].  Even if one is too predicative to allow a (pre)set of truth values, still one may have a notion of prerelation, by fiat if nothing else.  Note that one *can* compare prerelations for equality; $R = S$ means that $a \sim_R b$ if and only if $a \sim_S b$.  (In other words, a preset of truth values becomes a set under the biconditional, so we can compare functions to it.) In particular, the [[principle of substitution]] and [[identity of indiscernibles]] together imply that any two equality relations on a preset are equal to each other, so equality is unique if it exists.

We define a [[relation]] between sets to be a prerelation that respects equality; such a prefunction is said to be __extensional__.

Many properties of relations can also be predicated of prerelations, but not all. In particular, prerelations may be [[reflexive relation|reflexive]], [[symmetric relation|symmetric]], and [[transitive relation|transitive]], so we have a notion of equivalence prerelation, which completes the definition of sets in terms of presets. A prerelation may also be [[entire relation|entire]], but it makes no sense to ask if it is [[functional relation|functional]] unless $Y$ is a set.  In that case, there is a correspondence between prefunctions and functional entire prerelations as usual.  In general, however, there is no way to define the prerelation corresponding to a given prefunction (which would be a sort of pre-[[graph of a function|graph]]).  In other words, the idea that functions are certain relations (namely the functional entire ones) does not extended to prefunctions and prerelations unless $Y$ is a set.

In general, prerelations are

## Presets, types, sets, and setoids

Presets do not have equality. However, the types in many type theories come equipped with some kind of [[identity type]], where the only difference from equality is that they are not predicates. Thus, strictly speaking, the types in the type theories are not presets.

The sets defined by Bishop do not have [[quotient sets]], so there is still a distinction between Bishop sets and the quotient set of an object with an equivalence relation: this is the difference between a [[set]] and a [[setoid]]. Some people call a set with no quotient sets a "preset", however, this use of "preset" is at odds with preset as something that does not have equality.

While there is at most one [[equality]] on a preset due to the [[principle of substitution]] and the [[identity of indiscernibles]], a given preset may have many different equivalence relations, giving rise to setoids. For example, the setoid $Q^+$ of positive rational numbers may be defined using the same underlying set as the setoid $Z^+ \times Z^+$ of ordered pairs of positive integers, but the equivalence relation is different; two pairs $(a,b)$ and $(c,d)$ of positive integers are equivalent iff $a = c$ and $b = d$, while two positive rational numbers $a/b$ and $c/d$ are equivalent iff $a d = b c$. (Of course, these definitions require that one already has the set $Z^+$ of positive integers with its equality relation, and the operation of multiplication on it. Additionally, the [[cartesian product]] of two [[sets]] is itself a [[set]].)

## Formalisation

### In category theory

One may hope to formalise the category of presets and prefunctions between presets, but unfortunately, presets and prefunctions do not form a category. Instead, they only form a [[magmoid]].

### In type theory

It is possible to develop type-theoretic foundations in which presets are *not* equipped with identity relations (only metamathematical identity or interconvertibilty *judgements*); see [[tobybartels:preset]] for some discussion. Unlike the case for types with identity relations, the [[presentation axiom]] is not provable in the base theory, although it is provable in the impredicative version (where identity relations can be defined, following [[Gottfried Leibniz|Leibniz]]\'s definition of equality).

### In logic

If an untyped [[first-order logic]] does not have equality of propositions or equality of predicates, then the [[domain of discourse]] is a preset instead of a set. This remains true for typed first-order logic with multiple types, which are presets if each type does not have local equality.

The sorts in [[Michael Makkai]]\'s [[FOLDS]] are presets.  FOLDS is very different from the other foundations considered above, since it is based strictly on prerelations and has no notion of prefunction.  As far as I can tell, it therefore does not prove the [[presentation axiom]].

## Applications

To make the [[principle of equivalence]] hold automatically, a [[category]] should have only a preset of [[objects]] and only its [[hom-sets]] as sets.  Then a category whose set of objects *is* a set may be called a [[strict category]], which is really a special case of a [[strict ∞-category]].  Alternatively, one may keep sets as sets but adopt pre[[class]]es; then a [[small category]] is strict but a large category is not.

## Related concepts

* [[setoid]]

* [[type theory]]

* [[Bishop set]], [[predicative topos]]

* [[prefunction]]

[[!redirects preset]]
[[!redirects presets]]
[[!redirects pre-set]]
[[!redirects pre-sets]]

[[!redirects prerelation]]
[[!redirects prerelations]]
[[!redirects pre-relation]]
[[!redirects pre-relations]]

[[!redirects preset theory]]

[[!include types and logic - table]]


presetoid

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

# Contents
* table of contents
{: toc}

## Idea

In set theory, a [[setoid]] is usually described as a *set* with a [[pseudo-equivalence relation]]. However, in [[homotopy type theory]] or more generally in [[intensional type theory]], there are multiple different ways to define the concept of a "[[collection]] with a pseudo-equivalence relation". Presetoids arise from the most general possible notion of "[[collection]] with a pseudo-equivalence relation" in homotopy type theory , in the same way that the notion of [[precategory]] and [[pregroupoid]] arise when defining categories and groupoids.

## Definition

A **pseudo-equivalence relation** on a type $A$ is a family of sets $Hom(a, b)$ of **morphisms** for each element $a:A$ and $b:A$, which comes with the following additional structure

* for each element $a:A$ a family of morphisms
$$\mathrm{id}(a):Hom(a, a)$$

* for each object $a:A$ and $b:A$ a family of functions
$$\mathrm{dagger}(a, b):Hom(a, b) \to Hom(b, a)$$

* for each object $a:A$, $b:A$, and $c:A$, a family of functions
$$\mathrm{comp}(a, b, c):(Hom(a, b) \times Hom(b, c)) \to Hom(a, c)$$

A **presetoid** $A$ consists of a type $Ob(A)$ of **objects** and a pseudo-equivalence relation $(\mathrm{Hom}, \mathrm{id}, \mathrm{dagger}, \mathrm{comp})$.

## Dagger functors

The structure-preserving maps between two presetoids are called **dagger-functors**. A dagger-functor $F:A \to B$ between two presetoids $A$ and $B$ consists of

* a function $f_{Ob}:\mathrm{Ob}(A) \to \mathrm{Ob}(B)$

* a family of functions $f_{hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b))$

* and [[identifications]] indicating that the identity morphism, dagger, and the composition operation are preserved:

$$\mathrm{idpreserve}(a):f_{Hom}(a, a)(\mathrm{id}_A(a)) = \mathrm{id}_B(f_{Ob}(a))$$
$$\mathrm{daggerpreserve}(a, b, f):f_{Hom}(a, b)(\mathrm{dagger}_A(a, b)(f)) = \mathrm{dagger}_B(f_{Ob}(a), f_{Ob}(b))(f_{Hom}(a, b)(f))$$
$$\mathrm{comppreserve}(a, b, c, f, g):f_E(a, c)(\mathrm{comp}_A(f,g)) = \mathrm{comp}_B(f_{Hom}(a, b)(f), f_{Hom}(b, c)(g)$$

An dagger functor between two presetoids is **faithful** if every morphism function $f_{Hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b))$ is an [[injection]]. A dagger functor is **embedding-on-objects** if the object function $f_V:V_A \to V_B$ is an [[embedding]], and **equivalent-on-objects** if the object function is a [[equivalence in homotopy type theory|equivalence of types]].

## Core of a presetoid

A subpresetoid $G$ of a presetoid $A$ is a presetoid $G$ with an faithful embedding-on-objects dagger functor $f:G \to A$. A subpresetoid $G$ of $A$ is a subpregroupoid if the pseudo-equivalence relation of $G$ additionally come with:

* for every object $a:Ob(G)$, $b:Ob(G)$, $c:Ob(G)$, and $d:Ob(G)$ and morphism $f:Hom_G(a, b)$, $g:Hom_G(b, c)$, and $h:Hom_G(c, d)$, an identification
$$\mathrm{assoc}(a, b, c, d, f
g, h):\mathrm{comp}(a, b, d)(f, \mathrm{comp}(b, c, d)(g, h)) = \mathrm{comp}(a, c, d)(\mathrm{comp}(a, b, c)(f, g), h)$$

* for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification
$$\mathrm{runital}(a, b, f):\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f$$

* for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification
$$\mathrm{lunital}(a, b, f):\mathrm{comp}(a, a, b)(\mathrm{id}(a), f) = f$$

* for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification
$$\mathrm{sec}(a, b, f):\mathrm{comp}(a, b, a)(f, \mathrm{dagger}(a, b)(f)) = \mathrm{id}(a)$$

* for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification
$$\mathrm{ret}(a, b, f):\mathrm{comp}(b, a, b)(\mathrm{dagger}(a, b)(f), f) = \mathrm{id}(b)$$

A subpregroupoid $G$ of a presetoid $A$ is a maximal subpregroupoid if the dagger functor $f:G \to A$ is equivalent-on-objects, and additionally if, for every other subgroupoid $H$ of $A$ with a faithful embedding-on-objects dagger functor $g:H \to A$, the type of faithful embedding-on-objects dagger functors $h:H \to G$ with an identification $\mathrm{up}(f, g, h):h \circ f = g$ is [[contractible type|contractible]].

The **core [[pregroupoid]]** $Core(A)$ of a setoid $A$ is the maximal subpregroupoid of $A$. Since the core is a pregroupoid, every morphism in the core is an isomorphism, and the type of morphisms between two objects $a$ and $b$ in the core of $A$ is denoted as $a \cong_A b$.

## Univalent setoids

There are two ways to show that two objects of a presetoid are identified with each other: by way of the [[identity type]], and by way of the type of [[isomorphisms]] in the [[core]] [[pregroupoid]] of a presetoid. Univalent setoids are precisely the presetoids which the two notions of identifying objects with each other coincide.

For each object $a:Ob(A)$ and $b:Ob(A)$ of a presetoid $A$, there is a function
$$\mathrm{idtoiso}(a, b):(a =_A b) \to (a \cong_A b)$$
from the [[identity type]] to the type of isomorphisms in the [[core]] [[pregroupoid]] $\mathrm{Core}(A)$ of $A$. A presetoid is a **univalent setoid** if for each object $a:Ob(A)$ and $b:Ob(A)$ the function
$$\mathrm{idtoiso}(a, b):(a =_A b) \simeq (a \cong_A b)$$
is an equivalence of types.

Since every type of morphisms in the core of the presetoid is a [[h-set]], this means that every univalent setoid is an [[h-groupoid]].

## See also

* [[setoid]]
* [[univalent setoid]]
* [[pregroupoid]]


presheaf

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###Context###
#### Category Theory
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[[!include category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a small category $C$ of "primitive objects", we can think of a functor $F:\: C^{op} \to Set$ as being a more complex object built out of primitive objects:

 * Given a primitive object $X$ in $C$, we interpret $F(X)$ as a set representing the ways $X$ occurs inside $F$

 * Given a morphism $f:\: X \to Y$ in $C$, we interpret $F(f):\: F(Y) \to F(X)$ as the function mapping each occurrence $y$ of $Y$ in $F$ to the corresponding suboccurrence $x$ (included in $y$ through $f$) of $X$ in $F$

Such functors are called presheaves.

## Definition

A **presheaf** on a [[small category]] $C$ is a [[functor]]

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

from the [[opposite category]] $C^{op}$ of $C$ to the category [[Set]] of [[set]]s. Equivalently this may be thought of as a [[contravariant functor]] $F:\: C \to Set$.

More generally, given any category $S$, an **$S$-valued presheaf** on $C$ is a functor

$$
  F:\: C^{op} \to S.
$$

While, hence, presheaves are just [[functors]] (on [[small categories]]), one says "presheaf" to indicate a specific perspective or interest, namely interest in the _[[sheafification]]_ of the functor/presheaf, or at least interest in the [[functor category]] as a [[topos]] (the [[presheaf topos]]). Hence "presheaf" is a [[concept with an attitude]].

Historically, the initial applications of presheaves and sheaves involved cases like $S = $ [[CRing]] (the category of [[commutative ring|commutative rings]]), $S = $[[Ab]] ([[abelian groups]]), $S = $ [[Mod|RMod]] ([[modules]]), etc. Later, especially with the development of [[topos theory]], the primary importance of the [[sheaf topos|category of set-valued (pre)sheaves]] as a [[topos]] was recognized; these other cases could be considered algebraic objects which live in the topos. This article and the one on [[sheaf topos]] recognize these later developments by making the set-valued case the default (in other words, presheaf or sheaf without further qualification is understood to refer to the set-valued case).

The **[[category of presheaves]]** on $C$, usually denoted $Set^{C^{op}}$ or $[C^{op},Set]$, but often abbreviated as $\widehat{C}$, has:

* functors $F:\: C^{op} \to Set$ as objects;

* [[natural transformation|natural transformations]] between such functors as morphisms.

As such, it is an example of a [[functor category]].

## Remarks

* Speaking of functors as presheaves indicates operations that one wants to do apply to these functors, or certain properties that one wants to check.

  * when $S = Set$, and especially one is interested in the [[Yoneda embedding]] of a category $C$ into its presheaf category $[C^{op}, Set]$ for purposes of studying, for instance, [[limit]]s, [[colimit]]s, [[ind-object]]s, and [[pro-object]]s of $C$;

  * or when there is the structure of a [[site]] on $C$, such that it makes sense to ask if a given presheaf is actually a [[sheaf]].

* One generally useful way to think of presheaves is in the sense of [[space and quantity]].

* In the case where $S = Set$ and $C$ is [[small category|small]], an important general principle is that the presheaf category $[C^{op},Set]$ is the [[free cocompletion]] of $C$; see [[Yoneda extension]].  Intuitively, it is formed by taking $C$ and 'freely throwing in small colimits'.  The category $C$ is contained in $[C^{op},Set]$ via the [[Yoneda embedding]]
$$ Y:\: C \to [C^{op},Set]$$
The Yoneda embedding sends each object $c \in C$ to the presheaf
$$ F(-) = hom(-, c) $$
Presheaves of this form, or isomorphic to those of this form, are called [[representable functors|representable]]; among their properties, representable presheaves always turn colimits into limits, in the sense that a representable functor from $C^{op}$ to $Set$ turns colimits in $C$ (i.e., limits in $C^{op}$) into limits in $Set$ (i.e., colimits in $Set^{op}$). In general, such continuity is a necessary but not sufficient criterion for representability; however, nicely enough, it _is_ sufficient when $C$ itself is a presheaf category. To see this, suppose $K$ is such a presheaf on $C = [D^{op}, Set]$, and let $G = K Y$, a presheaf on $D$. By the [[Yoneda lemma]], we have a natural isomorphism between $[D^{op}, Set](Y(-), G)$ and $K Y(-)$. But by the free cocompletion property of the Yoneda embedding, a colimit-preserving functor on presheaves is entirely determined by its precomposition with $Y$; accordingly, our isomorphism must extend to an identification of $[C^{op}, Set](-, G)$ with $K(-)$, thus establishing the representability of $K$.

## Properties

### Limits and colimits

Any [[category of presheaves]] is [[complete category|complete]] and [[cocomplete category|cocomplete]], with both [[limit|limits]] and [[colimit|colimits]] being computed _pointwise_.  That is, to compute the limit or colimit of a diagram $F:\: D \to Set^{C^op}$, we think of it as a functor $F:\: D \times C^{op} \to Set$ and take the limit or colimit in the $D$ variable.

### Presheaves are colimits of representables
 {#PresheavesAreColimitsOfRepresentables}

\begin{proposition}\label{EveryPresheafIsColimitOfRepresentables}
Every presheaf is a [[colimit]] of [[representable functor|representable presheaves]].
\end{proposition}

An elegant way to express this colimit for a presheaf $F:\: C^{op} \to Set$ is in terms of the [[coend]] identity

$$
  F(-) = \int^{c \in C} F(c) \times hom_C(-,c)
  \,,
$$

which follows by [[Yoneda reduction]]. See also at *[[co-Yoneda lemma]]*.

More concretely: let $Y:\: C \to [C^{op}, Set]$ denote the [[Yoneda embedding]] and let $C_F \coloneqq Y/F$ be the corresponding [[comma category]], the [[category of elements]] of $F$:

$$
  C_F \coloneqq
  \left\lbrace
     \array{
        Y(V) &&\stackrel{Y(g)}{\to}&& Y(V')
        \\
        & {}_f\searrow && \swarrow_{f'}
        \\
        && F
     }
  \right\rbrace
$$

and let $p:\: C_F \to C$ the canonical forgetful functor. Then the colimit over representables expression $F$ is

$$
   F \simeq colim_{(Y(V) \to F) \in C_F} (Y\circ p)
  \,.
$$

This is often written with some convenient abuse of notation as

$$
  F \simeq colim_{V \to F} V
  \,.
$$

Notice that these formulas can also be understood as those for the left [[Kan extension]] (see there) of $F$ along the identity functor.

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

Notice that for every $B \in [C^{op}, Set]$
and using the property of the [[hom-functor]] we have

$$
 \begin{aligned}
  Hom_{[C^{op}, Set]}(colim_{(Y(V) \to F) \in C_F} (Y\circ p),B)
  &\simeq
  lim_{(Y(V) \to F) \in C_F} Hom_{[C^{op}, Set]}(Y(V),B)
  \\
  & \simeq
  lim_{(Y(V) \to F) \in C_F} B(V)
 \end{aligned}
$$
by the [[Yoneda lemma]].

By the definition of [[limit]] we have that

$$\cdots=Hom_{[C_F^{op}, Set]}(pt,B),$$

so for each natural transformation $\alpha \in Hom_{[C_F^{op}, Set]}(pt,B)$ and each object $h:\: Y(V)\to F\in C_F$, $\alpha_h$ is a map $\{*\}\to B(V)$, that is, it is an element of $B(V)$. However, by Yoneda, we know that each object $h:\: Y(V)\to F\in C_F$ specifies a unique element $h\in F(V)$. Then rephrasing this, $\alpha$ specifies a [[function]] $F(V)\to B(V)$.  The naturality of this assignment is guaranteed by the naturality of the map $\alpha$.  Then $\alpha$ induces a natural transformation $k^\alpha:\: F\to B$. It's easy to check that $k$ defines an isomorphism:

$$
  Hom_{[C_F^{op}, Set]}(pt,B) \simeq Hom_{[C^{op}, Set]}(F,B)
  \,.
$$
Since this holds for all $B$, the claim follows, again using the [[Yoneda lemma|Yoneda lemma]].

=--

## Special cases

* [[representable functor|representable presheaf]]

* [[concrete presheaf]]

## Examples

Examples for presheaves are abundant. Here is a non-representative selection of some examples.

* For $C$ a [[locally small]] category, every object $c \in C$ gives rise to the [[representable functor|representable presheaf]] $Hom_C(-, c):\: C^{op} \to Set$.

* More generally, for $i:\: C \hookrightarrow D$ a [[subcategory]] of a locally small category $D$, every object $d \in D$ gives rise to the presheaf

  $$
    Hom_D(i(-), d):\: C^{op} \to Set
    \,.
  $$

  Let's spell this out in more detail: given a morphism $\phi:\: V \to U$
  in $C$, we can take any morphism $f:\: i(U) \to X$ in $Hom_{D}(U,X)$
  and turn it into a morphism $V \stackrel{\phi}{\to} U \stackrel{f}{\to} X$
  in $Hom_{D}(i(V),X)$. This determines a map of set

  $$
    f^*:\: Hom_{D}(i(U),X) \to Hom_{D}(i(V),X)
    \,.
  $$

  So we have a functorial assignment of the form

  $$
     \array{
        W && \mapsto && Hom_{Diff}(i(W),X)
         \\
        \downarrow^g &&&& \uparrow^{g^*}
        \\
         V && \mapsto && Hom_{Diff}(i(V),X)
         \\
        \downarrow^f &&&& \uparrow^{f^*}
        \\
        U && \mapsto && Hom_{Diff}(i(U),X)
     }
     \,.
  $$

  Of course $i$ here could be any functor whatsoever.
  Asking if such a presheaf
  is [[representable functor|representable]] is asking
  for a right [[adjoint functor]] of $i$.

* A [[simplicial set]] is a presheaf on the [[simplex category]]

  A [[globular set]] is a presheaf on the [[globe category]].

  A [[cubical set]] is a presheaf on the [[cube category]].

* A [[diffeological space]] is a [[concrete presheaf]] on [[CartSp]].

* An important class of presheaves is those on a
  [[category of open subsets]] $Op(X)$ of a [[topological space]]
  or [[smooth manifold]] $X$.

*  Traditional standard examples include: the presheaf of [[smooth function]]s
  on $X$, that assigns to each $U \subset X$ the set
  $C^\infty(U,\mathbb{R})$ of smooth functions and to each inclusion
  $V \subset U$ the corresponding restriction operation of functions. This is further a sheaf.

*  Traditional standard example which is a presheaf but not a sheaf: the presheaf of exact forms on $X$, that assigns to $U \subset X$ the set
  $\Omega^\bullet_{exact}(U)$ of exact forms on $U$ and to each inclusion
  $V \subset U$ the corresponding restriction operation of functions. Here, and like above, the site is made up by open sets in $X$ with inclusions as morphisms.

... etc. pp.

## Related concepts

* [[(0,1)-presheaf]]

* **presheaf**

  [[constant presheaf]]

* [[(2,1)-presheaf]]

  [[presheaf of groupoids]]

* [[(∞,1)-presheaf]]

  [[simplicial presheaf]]

* [[(∞,n)-presheaf]]

* [[Yoneda lemma]], [[Yoneda extension]]

[[!redirects presheaves]]
[[!redirects presheaf category]]
[[!redirects presheaf categories]]


presheaf of groupoids

[[!redirects presheaves of groupoids]]

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#### Category theory
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#### Enriched category theory
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#### $(\infty,1)$-Category theory
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#### Higher category theory
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## Idea

The idea of _presheaf of groupoids_ may refer to one of the following distinct but closely related concepts:

1. a [[functor]] from some [[opposite category]] to the [[1-category]] [[Grpd]] of all [[small groupoids]];

1. a [[Grpd]]-[[enriched functor]] from the opposite of a [[Grpd]]-[[enriched category]] to [[Grpd]] canonically regarded as enriched over itself, all with respect to its [[cartesian closed category]]-structure;

1. a [[pseudofunctor]] from the opposite of a plain category to the [[2-category]] (in fact: [[(2,1)-category]]) [[Grpd]] of all small groupoids;

1. a [[2-functor]] from the [[opposite 2-category]] of some [[2-category]] to the [[2-category]] [[Grpd]] of all small groupoids.

In the last two cases one also speaks of _[[pre-stacks]] of groupoids_ or _[[(2,1)-presheaves]]_.

The first two of these carry [[structures]] of [[homotopical categories]], even of  [[model categories]], that make them [[presentable (infinity,1)-category]]-presentations of the latter two.

Under the [[simplicial nerve]]-constructions, presheaves of groupoids in the sense of the first two items form a [[full subcategory]] of [[simplicial presheaves]], and their [[model category]]-structures enhance to the corresponding [[model structures on simplicial presheaves]], which [[presentable (infinity,1)-category|present]] _[[(∞,1)-presheaves]] of [[∞-groupoids]]_, hence _pre-[[∞-stacks]]_.

## Examples
 {#Examples}

* For $\mathcal{C}$ an ordinary [[site]], each [[covering]] family $\{U_i \overset{\iota_i}{\to} X\}$ induces a _[[Cech groupoid|Čech groupoid]]_, which is really a presheaf of groupoids

$$
  C(\{U_i\}) \in [\mathcal{C}^{op}, Grpd]
  \,.
$$

## Related concepts

* [[model structure on simplicial groupoids]]

## References

A [[model category]]-[[structure]] on categories of presheaves of groupoids, modeling [[(2,1)-categories]] of [[stacks]] the way that the [[model structure on simplicial presheaves]] model [[(∞,1)-categories]] of [[∞-stacks]] is discussed in

* {#Hollander01} [[Sharon Hollander]], _A homotopy theory for stacks_, Israel Journal of Mathematics, 163(1), 93-124 ([arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247))

[[!redirects presheaf of groupoids]]


presheaf on open subsets

The notion of [[presheaf]] today is technically synonymous with ([[contravariant functor|contravariant]]) [[functor]] $F : S^{op} \to A$ -- albeit a functor is called a [[presheaf]] only if it is being used in a certain way, notably when its domain has the structure of a [[site]] and there is the potential to consider the [[sheafification]] of the functor to a [[sheaf]].

Historically this formalization was abstracted from the special case where the domain category $S = O(X)$ is the [[category of open subsets]] of a [[topological space]].

(...say more...)


presheaf with transfer

## References

* [[Vladimir Voevodsky]], _Cohomological Theory of Presheaves with Transfers_, in V. Voevodsky, A. Suslin and E. Friedlander, eds., _Cycles, Transfers and Motivic Homology Theories_ , Annals of
Math. Studies, Princeton University Press, 2000. ([web](http://math.uiuc.edu/K-theory/0368/))


pressure

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#### Physics
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## Related concepts

* [[isobar]]

* [[thermodynamics]]

* [[mass]], [[volume]]

* [[stress-energy tensor]]

## References

See also

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


prestable (∞,1)-category

[[!redirects prestable ∞-category]]
# Prestable ∞-category
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## Idea

A _prestable ∞-category_ axiomatizes the properties
of the connective part of a [[t-structure]] on a [[stable ∞-category]].

## Definition

A _prestable ∞-category_ is a [[pointed]] [[finitely cocomplete]] [[∞-category]] $C$ with a fully faithful [[suspension functor]] such that the [[base change]] of any morphism $Y\to\Sigma Z$ along a map $0\to\Sigma Z$ exists and the resulting pullback square is also a pushout square.

The fully faithfulness condition can be rephrased
by saying that the functor $C\to SW(C)$ is fully faithful,
and then the base change condition can be reformulated
by saying that the image of $C\to SW(C)$ is closed under extensions.
Here $SW(C)$ is the [[Spanier-Whitehead category]] of $C$.

## Properties

An ∞-category is prestable if and only if it is a full subcategory of a stable ∞-category closed under finite colimits and extensions.

## Examples

Any stable ∞-category is prestable.

If $C$ is a stable ∞-category with a [[t-structure]] $(C_{\ge0},C_{\le0})$, then $C_{\ge0}$ is prestable.
Any finitely complete prestable ∞-category arises in such a fashion, and there are two canonical choices for $C$:
the [[Spanier-Whitehead category]] and the category of [[spectrum objects]].
In fact, any other choice can be squeezed in between these two.

## Grothendieck prestable ∞-categories and a Gabriel-Popescu theorem

A prestable ∞-category is _Grothendieck_ if it is presentable
and [[filtered colimits]] are [[left exact]].

There is a [[Gabriel-Popescu theorem]] for prestable ∞-categories:
the class of Grothendieck prestable ∞-categories
coincides with the class of accessible left exact localizations
of [[connective]] modules over a connective [[E1-ring|E_1-ring]].

## References

Lurie, [[Spectral Algebraic Geometry]], §A.3.

[[!redirects prestable ∞-category]]
[[!redirects prestable ∞-categories]]
[[!redirects prestable (∞,1)-categories]]


presymplectic reduction

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## Idea

The  generalization of [[symplectic reduction]] from [[symplectic manifolds]] to [[presymplectic manifolds]].

[[!include symplectic reduction - table]]

## Related concepts

* [[gauge reduction]], [[symplectic reduction]], [[Poisson reduction]], [[BV-BRST formalism]]

* [[quantization commutes with reduction]]

## References

* Francesco Bottacin, _A Marsden-Weinstein reduction for presymplectic manifolds_  ([pdf](http://www.math.unipd.it/~bottacin/papers/presympred.pdf))

* A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, _Reduction of Presymplectic Manifolds with Symmetry_ ([arXiv:math-ph/9911008](http://arxiv.org/abs/math-ph/9911008))

[[!redirects presymplectic reductions]]


presymplectic structure

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## Definition

A _presymplectic structure_ on a [[smooth manifold]] or more generally on a [[smooth space]] $X$  is simply a closed [[differential 2-form]] $\omega \in \Omega^2(X)$.

(In parts of the literature a pre-symplectic 2-form is required to have constant rank. However, other parts of the literature do not require this, e.g. [Bottacin, def. 3.1](#Bottacin). )

If the 2-form is happens to be _non-degenerate_ (have maximal rank) then it is a _[[symplectic structure]]_ on $X$.

Given a presymplectic structure, the [[quotient]] of $X$ by the [[flow]] of the [[vector fields]] in the [[kernel]] of $\omega$ is, if it exists in a reasonable way, a [[symplectic manifold]].

One speaks of closed 2-forms as presymplectic structures if one is interested in eventually forming this quotient and obtaining a symplectic structure.

The central application of this appears in the theory of [[quantization]] of [[action functionals]]. The [[covariant phase space]] of a [[local action functional]] is canonically presymplectic, and one is interested in its quotientient by symmetries to obtain a symplectic structure. This quotient generically is very ill behaved, though, when taken in the naive way. The [[BV-BRST formalism]] is all about forming this quotient "up to homotopy", such that it exists in a reasonable way. See [[derived critical locus]] for more on this.

The notion of presymplectic structure is a weakening of the notion of [[symplectic structure]] roughly orthogonal to the notion of [[Poisson structure]].

## Properties

### Ambient symplectic manifold

Under mild technical conditions, presymplectic manifolds arise as submanifolds of ambient symplectic manifolds. See ([EMR, theorem 3](#EMR)).

## Examples

* [[covariant phase space]]

## Related concepts

* [[symplectic structure]], [[almost symplectic structure]]

* [[geometric quantization]], [[geometric quantization of non-integral forms]]

[[!include symplectic reduction - table]]

## References

The generalization of [[symplectic reduction]] for presymplectic manifolds, _[[presymplectic reduction]]_ is discussed in

* Francesco Bottacin, _A Marsden-Weinstein  reduction theorem for presymplectic manifold_ ([pdf](http://www.math.unipd.it/~bottacin/papers/presympred.pdf))
  {#Bottacin}

* A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, _Reduction of Presymplectic Manifolds with Symmetry_ ([arXiv:math-ph/9911008](http://arxiv.org/abs/math-ph/9911008))
 {#EMR}

The [[geometric quantization]] of presymplectic manifolds by [[geometric quantization by push-forward]] is discussed in

* Ana Canas da Silva, [[Yael Karshon]], Susan Tolman, _Quantization of Presymplectic Manifolds and Circle Actions_, Trans. Amer. Math. Soc. 352 (2000), 525-552 ([arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008))

[[!redirects presymplectic structures]]

[[!redirects pre-symplectic form]]
[[!redirects presymplectic form]]
[[!redirects pre-symplectic forms]]
[[!redirects presymplectic forms]]

[[!redirects presymplectic smooth space]]
[[!redirects presymplectic smooth spaces]]

[[!redirects presymplectic manifold]]
[[!redirects presymplectic manifolds]]

[[!redirects pre-symplectic manifold]]
[[!redirects pre-symplectic manifolds]]


pretopological space

# Pretopological spaces
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## Idea

A _pretopological space_ is a slight generalisation of a [[topological space]] where the concept of _[[neighbourhood]]_ is taken as primary.  The [[extra structure]] on the [[underlying set]] of a pretopological space is called its _pretopology_, but this should not be confused with a [[Grothendieck pretopology]] (which is not even analogous).

## Definitions

Given a [[set]] $S$, let a __point__ in $S$ be an element of $S$, and let a __set__ in $S$ be a [[subset]] of $S$.  Given a [[relation]] $\stackrel{\circ}\in$ between points in $S$ and sets in $S$, say that the set $U$ is a __neighbourhood__ (or __$\stackrel{\circ}\in$-neighbourhood__ to be precise) of $x$ if $x \stackrel{\circ}\in U$ (which may also be written $x \in \stackrel{\circ}U$).

A __pretopology__ (or __pretopological structure__) on $S$ is such a relation $\stackrel{\circ}\in$ that satisfies these properties:

1.  Centred:  If $U$ is a neighbourhood of $x$, then $x$ belongs to $U$:
    $$ x \stackrel{\circ}\in U \;\Rightarrow\; x \in U .$$
2.  Nontrivial:  Every point $x$ has a neighbourhood.  In light of (4), the entire space is a neighbourhood of $x$:
    $$ x \stackrel{\circ}\in S .$$
    (Some references leave this out, but that seems to be an error.)
3.  Directed:  If $U$ and $V$ are neighbourhoods of $x$, then so is some set contained in their [[intersection]].  In the light of (4), it follows that their intersection is itself a neighbourhood:
    $$ x \stackrel{\circ}\in U \;\Rightarrow\; x \stackrel{\circ}\in V \;\Rightarrow\; x \stackrel{\circ}\in U \cap V .$$
    (Strictly speaking, the relation should not be called [[direction|directed]] unless it is also nontrivial.)
4.  Isotone:  If $U$ is a neighbourhood of $x$ and $U$ is contained in $V$, then $V$ is a neighbourhood of $x$:
    $$ x \stackrel{\circ}\in U \;\Rightarrow\; U \subseteq V \;\Rightarrow\; x \stackrel{\circ}\in V .$$

In other words, the collection of neighbourhoods of $x$ must be a [[filter]] that is refined by the free [[ultrafilter]] at $x$.  This filter is called the __neighbourhood filter__ of $x$.

A pretopology can also be given by a base or subbase.  A __[[base]]__ for a pretopology is any relation that satisfies (1--3), using the first version for each of (2,3); a __[[subbase]]__ is any relation that satisfies (1).  (It would not really be appropriate to use the symbol '$\stackrel{\circ}\in$' for a mere base or subbase; you\'d probably want to think of it as a family of sets indexed by the points, and use the term 'basic neighourhood' or 'subbasic neighbourhood'.)  You get a base (in fact one satisfying the stronger versions of 2,3) from a subbase by closing under finitary intersections; you get a pretopology from a base by taking supsersets.  (Really, this is just a special case of considering a base or subbase of a [[filter]].)

A __pretopological space__ is a set equipped with a pretopological structure.

A __continuous map__ from a pretopological space $S$ to a pretopological space $T$ is a [[function]] $f$ from $S$ to $T$ such that:

*  For every point $x$ in $S$, if $U$ is a neighbourhood of $f(x)$ in $T$, then the [[preimage]] $f^*(U)$ is a neighbourhood of $x$ in $S$.

In this way, pretopological spaces and continuous maps form a [[category]] $Pre Top$.

## Convergence structure

If $F$ is a [[filter]] on a pretopological space $S$, then $F$ __converges__ to a point $x$ (written $F \to x$) if $F$ refines (contains) the neighbourhood filter of $x$.

This relation satisfies the following properties:

*  Centred:  The free ultrafilter at $x$ (the collection of all sets that $x$ belongs to) converges to $x$:
   $$ \{ A \;|\; x \in A \} \to x .$$
*  Isotone:  If $F$ converges to $x$ and $G$ refines $F$, then $G$ converges to $x$:
   $$ F \to x \;\Rightarrow\; F \subseteq G \;\Rightarrow\; G \to x .$$
*  Infinitely directed:  The intersection of all filters that converge to $x$ itself converges to $x$:
   $$ \bigcap \{ F \;|\; F \to x \} \to x .$$

In this way, every pretopological space becomes a [[convergence space]].

In fact, we can recover the pretopological structure from the convergence structure as follows: $x \stackrel{\circ}\in U$ if and only if $U$ belongs to every filter that converges to $x$.  In other words, that intersection that appears in the infinite filtration condition is the neighbourhood filter of $x$.  Furthermore, this definition assigns a pretopological structure to any convergence space satsifying the conditions above, and a map between pretopological spaces is continuous if and only if it is continuous as a map between convergence spaces.  Thus, we can define a pretopological space as an infinitely directed convergence space, making $Pre Top$ a [[full subcategory]] of the category $Conv$ of convergence spaces.

Actually, we can do more.  The definition of $\stackrel{\circ}U$ from the convergence structure assigns a pretopological structure to *any* convergence space, although in general this pretopology defines a weaker notion of convergence (more filters converge to more points).  Thus, $Pre Top$ is also a [[reflective subcategory]] of $Conv$.

Every pretopological convergence satisfies the star property, so that $Pre Top$ is a full reflective subcategory of the category $Ps Top$ of [[pseudotopological spaces]].

## Examples

Every [[topological space]] is a pretopological space, using the usual definition of (not necessarily open) neighbourhood: $x \stackrel{\circ}\in U$ if there exists some open set $G$ such that $x \in G$ and $G \subseteq U$.  Also, a map between topological spaces is continuous if and only if it\'s continuous as a map between pretopological spaces.  In this way, the category [[Top]] of topological spaces becomes a full subcategory of $Pre Top$.

In fact, we can easily characterise the topological pretopologies, allowing us to define a topological space as a pretopological space satisfying this axiom:

*  If $U$ is a set, then let $\stackrel{\circ}U$ be the set of all points that $U$ is a neighbourhood of.  Then $\stackrel{\circ}U$ is a neighbourhood of each of its members.  That is,
   $$ x \stackrel{\circ}\in U \;\Rightarrow\; x \stackrel{\circ}\in \{ y \;|\; y \stackrel{\circ}\in U \} .$$

In the terms defined below, a topological space is a pretopological space in which every preinterior is open.

Here is an example of a nontopological pretopological space, although admittedly it is a bit artificial.  (This is based on Section 15.6 of [[HAF]].)  Consider a [[metric space]] $S$; according to the usual pretopology on $S$, $U$ is a neighbourhood of $x$ if there is a positive number $\epsilon$ such that $U$ contains the ball $ \{ y \;|\; d(x,y) \lt \epsilon \} $.  Now given a [[natural number]] $n$, we will give $S^n$ the _plus pretopology_: $U$ is a neighbourhood of $\vec{x} = (x_1,\ldots,x_n)$ if there is a positive number $\epsilon$ such that $U$ contains the $l^0$-ball $ \{ \vec{y} \;|\; \inf_i d(x_i,y_i) \lt \epsilon \} $.  (If $S$ is a line and $n = 2$, then this neighbourhood is a plus sign '+' with $(x_1,x_2)$ at the centre and cross bars of length $2 \epsilon$.)  Then $S^n$ is a pretopological space, but it is topological only if $n \leq 1$ or $S$ is a [[subsingleton]].

This example can probably be generalised to a [[uniform space]] $S$.  Possibly there is some interesting universal property of this 'plus product', although it seems to go from $Unif \times Unif$ to $Pre Top$, so maybe we need to work in a different category.  (There is a notion of [[uniform convergence space]] that generalises uniform spaces much like convergence spaces generalise topological spaces; perhaps the plus product takes place there.)

## Topological structure

Fix a pretopological space $S$.

The __preinterior__ of a set $A$ is the set $\stackrel{\circ}A$ or $A^\circ$ of all points that $A$ is a neighbourhood of:
$$ \stackrel{\circ}A = \{ x \;|\; x \stackrel{\circ}\in A \} .$$
A set $A$ is __open__ if it equals its preinterior.  The __interior__ $Int(A)$ of $A$ is the union of all of the open sets contained in $A$.  Note that we can immediately recover the pretopological structure from the preinterior operation (but not from the interior operation nor from the class of all open sets).

Similarly, the __preclosure__ of $A$ is the set $\bar{A}$ of all points that $A$ meets every neighbourhood of:
$$ \{ x \;|\; \forall{U},\; x \stackrel{\circ}\in U \;\Rightarrow\; A \cap U \neq \empty \} .$$
A set $A$ is __closed__ if it equals its preclosure.  The __closure__ $Cl(A)$ of $A$ is the intersection of all of the closed sets containing $A$.  Again, we can recover the pretopological structure from the preclosure operation; $x \stackrel{\circ}\in U$ iff $U$ meets every set $A$ such that $x \in \bar{A}$.  (This result seems to require [[excluded middle]].)

(Warning: not all references use these terms in the same way.  This terminology is based on the premise that a closure should be closed.)

The duality between (pre)interiors and open sets on the one hand and (pre)closures and closed sets on the other hand is (at least if you assume [[excluded middle]]) just what you would expect: the (pre)interior of a [[complement]] is the complement of the (pre)closure, and a set is open if and only if its complement is closed.  However, a preinterior is generally not open but larger than an interior; similarly, a preclosure is generally not closed but smaller than a closure.  The situation looks like this:
$$ A \supseteq \stackrel{\circ}A \supseteq (\stackrel{\circ}A)^{\circ} \supseteq \cdots \supseteq Int(A) ,$$
and
$$ A \subseteq \bar{A} \subseteq \overline{\bar{A}} \subseteq \cdots \subseteq Cl(A) .$$
In many cases this iteration stabilizes after finitely many terms.  The plus power $S^n$ seems to stabilise after $n$ iterations.  And in a topological space, of course, it only takes one step.

In general, however, there can be transfinitely many terms in these sequences.  For example, let $\Omega$ be any [[ordinal number]] (thought of as the [[well-ordered set]] of all smaller ordinal numbers) with the following pretopology:

* $0 \stackrel{\circ}\in U$ iff $U = \Omega$.
* $\alpha \stackrel{\circ}\in U$, where $\alpha \lt \Omega$ is a nonzero ordinal, iff $[\beta,\Omega) \subseteq U$ for some $\beta \lt \alpha$.

Let $A=[1,\Omega)$.  Then $\stackrel{\circ}A = [2,\Omega)$, $(\stackrel{\circ}A)^\circ = [3,\Omega)$, and so on, the process taking $\Omega$ steps to stabilize at $Int(A)=\emptyset$.

Note that an interior is open, and a closure is closed.  Indeed, the open sets in $S$ form a topological structure on $S$, giving the usual meanings of interior, closure, and closed set.  This topological structure does *not* (in general) give the original pretopology on $S$; instead, this makes $Top$ a [[reflective subcategory]] of $Pre Top$.

In the definition of pretopology, the neighbourhoods of each point may be given completely independently of any other point.  So the notion of topological space may also be seen as requiring some coherence between the neighbourhoods of nearby points.

[[!redirects pretopological space]]
[[!redirects pretopological spaces]]
[[!redirects pre-topological space]]
[[!redirects pre-topological spaces]]

[[!redirects preinterior]]
[[!redirects preinteriors]]
[[!redirects pre-interior]]
[[!redirects pre-interiors]]

[[!redirects preclosure]]
[[!redirects preclosures]]
[[!redirects pre-closure]]
[[!redirects pre-closures]]