category of G-sets

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###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

This entry is about the category of [[G-set|continuous G-sets]], for a [[topological group]] $G$. Continuous $G$-sets are sets $X$ with an action of $G$ that is continuous when $X$ is given the [[discrete topology]].

The category of continuous $G$-sets is a [[Grothendieck topos]], and is closely related to [[Fraenkel-Mostowski models]].

## Definition

Let $G$ be a [[topological group]].

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

The **category of continuous $G$-sets**, to be denoted $G Set$, is the [[category]] of sets $X$ equipped with a continuous (left) $G$-[[action]] $\mu \colon G \times X \to X$, where $X$ is given the [[discrete topology]], and the morphisms are the $G$-[[equivariant functions]].

=--

There is a simple characterization of when a $G$-action is continuous.

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

Let $G$ be a [[topological group]], and $X$ be a [[set]] equipped with a $G$ [[action]] $\mu \colon G \times X \to X$. Then the action is [[continuous function|continuous]] if and only if the [[stabilizer subgroup]] of each element is open.

=--

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

See [[G-sets#ContinuousCharacterization|G-sets]].

=--

## Properties

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

For a [[topological group]] $G$, we write $G^\delta$ for its underlying [[discrete group]]. Every $G$-set can be viewed as a $G^\delta$ set in the obvious way.

=--

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

The inclusion $i_G \colon G Set \to G^\delta Set$ has a [[right adjoint]] $r_G: G^\delta \to G Set$
such that for $X \in G Set$, the continuous $G$-set $r_G X$ is the subset of $X$ consisting of those points with open [[stabilizer|stabilizer subgroup]].

=--

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

This follows from the observation that if $f \colon X \to Y$ is a $G$-[[equivariant function]] between $G$-sets, then for each $x \in X$, the stabilizer of $f(x)$ includes the stabilizer of $x$.

=--

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

The [[forgetful functor]] $U \colon G^\delta Set \to Set$ [[created limit|creates]] all [[small limit|small]] [[limits]] and [[colimits]].

=--

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

This follows from the general fact that limits and colimits in [[presheaf categories]] are computed pointwise.

Alternatively, there is an obvious $G$-action we can put on the limits or colimits of the underlying sets of the $G$-sets.

The creating limits part also comes from the fact that the forgetful functor is [[monadic]].

=--

+-- {: .num_prop}
###### Proposition
The inclusion $i_G: G Set \to G^\delta Set$ [[created limit|creates]] all [[finite limits]] and all [[colimits]].
=--

+-- {: .proof}
###### Proof
This follows from the observation that the finite limits and colimits created by $U: G^\delta Set \to Set$ have a continuous action if each factor has a continuous action, using the fact that finite intersections and arbitrary unions of open sets are open.
=--

+-- {: .num_cor}
###### Corollary
The category $G Set$ has all [[finite limits]] and arbitrary [[colimits]].
=--

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

The [[adjunction]] $i_G \dashv r_G$ is a [[geometric morphism]].

=--

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

A map in $G Set$ is a [[monomorphism]] if and only if it is [[injective function|injective]]; [[epimorphism]] if and only if it is [[surjective function|surjective]].

=--

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

A map is monic if and only if its [[kernel pair]] is an [[isomorphism]], and similarly for epic, and the forgetful functor to $Set$ preserves all finite limits and colimits.

=--

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

The [[subobject classifier]] of $G Set$ is $\Omega = 2$, the two-point set with the [[trivial action|trivial]] $G$-action.

=--

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

The [[exponential object]] in $G^\delta Set$ is defined by $Y^X = \{\text{functions}\;X \to Y\}$, with the $G^\delta$-action given by
$$
  g \cdot f = g \circ f \circ g^{-1}.
$$
The exponential object $Y^X$ in $G Set$ is given by the subset of the functions $X \to Y$ that have an open [[stabilizer subgroup|stabilizer]], ie.
$$
  Y^X = r_G(i_G(Y)^{i_G(X)}).
$$

=--

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

Let $A, X, Y$ be $G^\delta$-sets. Given a map $f:A \times X \to Y$, we obtain $\hat{f}:A \to Y^X$ by
$$
  \hat{f}(a)(x) = f(a, x).
$$
We now check that $\hat{f}$ is $G^\delta$-invariant --- we have
$$
  (g \cdot \hat{f}(a))(x) = g \cdot \hat{f}(a)(g^{-1}\cdot x) = g \cdot f(a, g^{-1}x) = f(g \cdot a, x) = \hat{f}(g \cdot a)(x).
$$
So we get
$$
  g \cdot \hat{f}(a) = \hat{f}(g \cdot a).
$$
Conversely, given a $\hat{f}: A \to Y^X$, we obtain $f: A \times X \to Y$ by $f(a, x) = \hat{f}(a)(x)$, and we have
$$
  g \cdot f(a, x) = g \cdot \hat{f}(a)(x) = (g \cdot \hat{f}(a))(g\cdot x) = \hat{f}(g \cdot a)(g \cdot x) = f(g \cdot a, g \cdot x).
$$
So $f$ is $G^\delta$-invariant. It is clear that these operations are inverses to each other, and straightforward computations show that this is natural in $A$, $X$ and $Y$.

Then in general, if $A, X, Y$ are in fact $G$-sets, then we can compute
$$\begin{aligned}
  G Set\left(A, r_G(i_G(Y)^{i_G(X)})\right) &= G^\delta Set \left(i_G(A), i_G(Y)^{i_G(X)}\right)\\
  &= G^\delta Set\left(i_G(A) \times i_G(X), i_G(Y)\right)\\
  &= G^\delta Set\left(i_G(A \times X), i_G(Y)\right)\\
  &= G Set\left(A \times X, Y\right),
\end{aligned}$$
using the fact that $r_G(i_G(X)) = X$ for all $X \in G Set$.
=--

+-- {: .num_cor}
###### Corollary
The [[power object]] of $X \in G Set$ is given by the subsets of $X$ that have an open stabilizer.
=--

In conclusion we have:

+-- {: .num_theorem #GSetsIsAnElementaryTopos}
###### Theorem

The category $G Set$ is an [[elementary topos]].

=--

## Equivalent characterizations

### As a Grothendieck topos

+-- {: .num_theorem}
###### Theorem
Let $G$ be a topological group. Then the category $G Set$ is [[equivalence of categories|equivalent]] to the topos of [[sheaves]] on the [[atomic site|atomic]] [[site]] $(S(G), At)$, where the objects of $S(G)$ are the open subgroups of $G$, and the morphisms $H \to K$ are the left cosets $a K$ such that $H \subseteq a K a^{-1}$, and all non-empty [[sieves]] are covering.

Alternatively, it is the full subcategory of $G Set$ containing objects of the form $G/U$, where $U$ is an open subgroup.
=--

+-- {: .proof}
###### Proof
See [MacLane and Moerdijk, Chapter III.9](#MaclaneMoerdijk).
=--

More generally, by the [[comparison lemma]], we have

+-- {: .num_theorem}
###### Theorem
Let $G$ be a topological group, and $\mathcal{U}$ be a cofinal set of open subgroups, ie. every open subgroup contains a member of $\mathcal{U}$. Then the category $G Set$ is equivalent to the topos of sheaves on the atomic site $(S(G, \mathcal{U}), At)$, where the objects of $S(G, \mathcal{U})$ are the open subgroups in $\mathcal{U}$, and the morphisms $H \to K$ are the left cosets $a K$ such that $H \subseteq a K a^{-1}$, and all non-empty sieves are covering.

Alternatively, it is the full subcategory of $G Set$ containing objects of the form $G/U$, where $U \in \mathcal{U}$.
=--

In particular, if $G$ is a discrete group, then the [[trivial subgroup]] itself is a cofinal set of open subgroups. So $G Set$ is the category of sheaves on the category with only one object, whose morphisms are the elements of $G$. This is the usual characterization of $G Set$ as the [[functor category]] $Set^G$.

### As a comonad algebra
To be included.

### As a classifying topos
To be included.

## For internal group actions
 {#InternalGroupActions}

The above construction of the category $G Set$ -- of [[G-sets]] for a [[discrete group]] $G$ in the [[topos]] [[Set]] of [[sets]] -- generalizes to [[group objects]] $G \in Groups(\mathcal{E})$ in any [[topos]] $\mathcal{E}$:

The resulting category $G\mathcal{E}$ -- of [[objects]] of $\mathcal{E}$ equipped with [[internalization|internal]] $G$-[[action]] and with $G$-[[equivariant function|equivariant]] [[morphisms]] between them -- is itself a [[topos]], by the same proof as above Thm \ref{GSetsIsAnElementaryTopos}.

A particularly interesting case of this is an [[internal group]] $H$ in the topos $G Set$ itself, for $G$ a [[discrete group]] (a "$G$-[[equivariant group]]", see there for more):

Seen externally this is equivalently a discrete group $H$ equipped with a [[group homomorphism]] $G \to \Aut(H)$ to the [[automorphism group]] of $H$. Notice that this data induces the corresponding [[semidirect product group|semidirect product]] $H \rtimes G$. With that, we have:

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

Let $G$ be a [[discrete group]], and let $H$ be a [[group object]] in $G Set$. Then the category $H (G Set)$ is [[equivalence of categories|equivalent]] to $(H \rtimes G) Set$.

=--

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

This is a straightforward computation. Given an $X \in H (G Set)$, we write $\mu \colon H \times X \to X$ for the [[action]] morphism, and denote the action of $G$ merely by a dot. Then we define the action of $H \rtimes G$ by

$$
  ((h, g), x) \mapsto \mu(h, g \cdot x)
  \,.
$$

Conversely, given an object $X$ with an $H \rtimes G$ action, we give it a $G$ action by $g \cdot x = (e, g) \cdot x$, and an $H$-action by $\mu(h, x) = (h, e) \cdot x$.

=--

The case of [[topological groups]] is more complicated, because an internal topology $T$ on a space $X$ is an internally complete lattice $T \subseteq P(X)$, which is not necessarily closed under infinite external unions. However if we do the rather unnatural (?) thing of closing it under all external unions, then we make $X$ an external topological space. Then we have the following result:

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

Let $G$ be a [[topological group]], and let $H$ be a topological group object in $G Set$. Then $H (G Set)$ is equivalent to $(H \rtimes G) Set$, where $H \rtimes G$ is given the product topology.

=--

The **proof** is a straightforward check that the continuity conditions match up.

## Related concepts

* [[G-set]]

* [[topological G-space]]

* [[permutation representation]]

* [[Fraenkel-Mostowski model]]

* [[class equation]]

## References

Basic exposition of the topos of $G$-sets for [[discrete groups]]:

* [[Jaap van Oosten]], (eg. p. 32 in:) *Topos Theory* (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), [[vanOosten-ToposTheory.pdf:file]]]

Some elementary properties of continuous $G$-sets can be found in books such as

* {#MaclaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], *[[Sheaves in Geometry and Logic]]*, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]

The formal correspondence between permutation models of [[ZFA]] and toposes of continuous $G$-sets can be found in

* [[Michael Fourman]], _Sheaf models for set theory_, Journal of Pure and Applied Algebra, **19** (1980) pp 91-101, doi:[10.1016/0022-4049(80)90096-1](http://dx.doi.org/10.1016/0022-4049%2880%2990096-1), ([publisher pdf](http://www.sciencedirect.com/science/article/pii/0022404980900961/pdf?md5=0de18da810657bf1ba9de7399d224ec5&pid=1-s2.0-0022404980900961-main.pdf))

[[!redirects category of G sets]]
[[!redirects category of G set]]
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[[!redirects categories of G-sets]]
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[[!redirects category of continuous G-sets]]
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category of generalized elements

#Idea#

Given a collection of "parameterized objects", i.e. a [[functor]] $F : C \to D$, it is often of interest to consider the category whose objects are [[generalized elements]] of the objects of $D$ in the image of $F$, and whose morphisms are the maps between these generalized elements induced by the value of $F$ on morphisms in $C$.

For $D = $ [[Set]] and and with [[generalized element]] read as "ordinary element of a set" is yields the [[category of elements]] of the (co)[[presheaf]] $F : C \to D$.

Moreover, the description of of the [[category of elements]] of a presheaf in terms of a pullback of a [[generalized universal bundle]] generalizes directly to categories of generalized elements.

#Definition#

Let $D$ be a [[pointed object]] in [[Cat]], i.e. a [[category]] equipped with a choice $pt_D : {*} \to D$ of one of its objects.

Recall that a morphism $pt_D \to d$ in $D$ may be called a [[generalized element]] in $D$ "with domain of definition" being the object $pt_D$.

For instance if $D =$ [[Set]] the canonical choice is $pt_{Set} = {*}$ [[generalized the|the]] set with a single element. Generalized elements of a set "with domain of definition" ${*}$ are just the ordinary elements of a set.

Notice that the [[over category]] $({pt_D/D})$ is the category of generalized elements of $D$ with domain of definition $pt_D$:

* objects are such generalized elements $\delta : pt_D \to d$ of objects $d \in D$;

* morphisms $\delta \to \gamma$ are given whenever a morphism $f : d \to d'$ in $D$ takes the element $\delta$ to $\delta'$, i.e. whenever there is a commuting triangle
  $$
    \array{
      && pt_D
      \\
      & {}^\delta\swarrow && \searrow^{\delta'}
      \\
      d &&\stackrel{f}{\to}&& d'
    }
    \,.
  $$

Notice that the canonical projection $(pt_D/D) \to D$ from the [[over category]] that forgets the tip of these trangles may be regarded as the [[generalized universal bundle]] for the given pointed category $D$: it is the left composite vertical morphism in the pullback

$$
  \array{
    (pt_D/D) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    D^I &\stackrel{d_0}{\to}& D
    \\
    \downarrow^{d_1}
    \\
    D
  }
$$

(see also [[comma category]] for more on this perspective). So in fact such "categories of generalized elements" are precisely the [[generalized universal bundles]] in the 1-categorical context. And both are really fundamentally to be thought of as intermediate steps in the computation of [[weak limit|weak pullbacks]], as described now.

The above allows to generalize the notion of _category of generalized elements_ a bit further to that of generalized elements of functors with values in $D$: let $F : C \to D$ be a functor with codomain our category $D$ with point $pt_D$.

The **category of generalized elements** of $F$ is the [[pullback]] $El_{pt_D}(F) := C \times_D (pt_D/D)$

$$
  \array{
    El_{pt_D}(F) &\to& (pt_D/D)
    \\
    \downarrow && \downarrow
    \\
    C &\stackrel{F}{\to}& D
  }
  \,.
$$

This means:

* the objects of $El_{pt_D}(F)$ are all the generalized elements $\delta_c : pt_D \to F(c)$ for all $c \in C$;

* a morphism $\delta_c \to \delta_{c'}$ between two such generalized elements is a commuting triangle
  $$
    \array{
      && pt_D
      \\
      & {}^{\delta_c}\swarrow && \searrow^{\delta_{c'}}
      \\
      F(c) && \stackrel{F(f)}{\to} && F(c')
    }
    \,.
  $$
  for all morphisms $f : c \to c'$ in $C$.

#Examples#

##ordinary category of elements##

For $D = $ [[Set]] and $pt_{Set} = {*}$
the above reproduces the notion of [[category of elements]] of a [[presheaf]].

##Action Groupoid##

Given a vector space $V$, a [[group]] $G$ recall that a [[representation]] of $G$ on $V$

$$V\bullet\righttoleftarrow G$$

is canonically identified with a [[functor]]

$$
  \rho : \mathbf{B} G \to Vect
  \,.
$$

$$
  \rho : ({*} \stackrel{g}{\to} {*}) \mapsto
  (V \stackrel{\rho(g)}{\to} V)
  \,.
$$

The category [[Vect]] of $k$-vector spaces for some [[field]] $k$ has a standard point $pt_{Vect} \to Vect$, namely the field $k$ itself, regarded as the canonical 1-dimensional $k$-vector space over itself.

The corresponding [[over category]] of generalized elements of [[Vect]] $(pt_{Vect}/ Vect)$ has as objects pointed vector spaces and as morphisms linear maps of pointed vector spaces that map the chosen vectors to each other.

Now, as described in detail at [[action groupoid]] the category of generalized elements of the representation $\rho$ is the [[action groupoid]] $V//G$ of $G$ acting on $V$

$$
  \array{
   V//G &\to& Vect_*
   \\
   \downarrow && \downarrow
   \\
   \mathbf{B} G &\to& Vect
  }
   \,.
$$

As described there, $V//G \to \mathbf{B}G$ is the [[groupoid]] incarnation of the vector bundle that is associated via $\rho$ to the universal $G$-bundle on the [[classifying space]] $B G$.

##References##

* [Exploding a Category](http://golem.ph.utexas.edu/category/2008/06/an_exercise_in_groupoidificati.html#c017574)


[[!redirects category of interest]]
## Motivation

Many categories of algebraic objects have similar properties. Classically [[abelian categories]], abstracted from categories of modules were central to homological algebra, but many important contexts were non-abelian, some as central as that of groups. In these non-abelian contexts, research has centred on finding common frameworks to understand better both the resulting objects and morphisms and the interpretation of their homological algebra.

There are many types of algebraic object which are 'based' on some type of groups.  Examples include groups themselves, [[modules]] over a ring, including, of course, vector spaces, then, adding more structure, [[associative algebras]], [[commutative algebras]], [[Lie algebras]], [[Leibniz algebras]], [[Poisson algebras]], and then the dialgebras and trialgebras introduced by [[Loday]], as well as [[crossed module]]s of groups. We will call these examples of _group-based universal algebras_.

The similarities of the homological algebra of these settings suggested a study of what parts of that theory is common to all of these, and, conversely, what parts depend on particular properties of an example.

####Remark on terminology

Before giving the definition,  we note that categories of such group-based universal algebras are more often called _categories of interest_ in the literature.  This term was introduced by Orzech in the first paper on their theory and, although very uninformative, has become fairly standard. Our preference for the title given here is that, hopefully, it does say more of the context of the ideas than merely saying they are 'of interest'.

##Idea

In all these cases, the category, $\mathcal{C}$, of such 'algebras' is [[monadic]] over $Set$.  The monad, $\mathbf{T}=(T,\eta,\mu)$, involved  is such that $T(\emptyset)$ is a singleton, the category $\mathcal{C}$ is [[pointed]], with the algebra $(T(\emptyset),\mu_\emptyset)$, that is the free algebra on the empty set, as its [[zero object]], and is complete and cocomplete. In fact,  $\mathcal{C}$ is monadic over the category, $Set_*$, of _pointed sets_.

Again in all these cases, the forgetful functor, $U:\mathcal{C}\to Set_*$ factors through the category of groups and all the operations in the corresponding algebraic theory of $\mathcal{C}$ are finitary, so $\mathcal{C}$ can be thought of as a category of _groups with extra structure_, much as in the more general [[Omega-groups]].  We have, thus, that $\mathcal{C}$ is equivalent to a category $Grps^{\hat{\mathbf{T}}}$ for some monad $\hat{\mathbf{T}}$.

Within this abstract setting, one can formulate notions of subobject, center, etc, but to formulate in this setting the criteria for an object to be an ideal or normal subobject, it is useful to make the assumption that the theory of $\mathcal{C}$ can be presented by a set of operations and identities satisfying some extra conditions, most of which are versions of obvious ones from 'algebra'.

##Definition

(Revision in progress)

By a *group-based universal algebra*, we will mean an algebra for a theory, in the classical sense, which contains

* a unique constant, denoted $0$;
* a set, $\Omega$ of finitary operations;

and

* a set, $\mathbb{E}$, of idenities or equations,

which are to satisfy

* $\Omega=\Omega_0\cup \Omega_1\cup \Omega_2$ where $\Omega_i$ is the set of $i$-ary operations;

* $\Omega_0=\{0\}$; $\Omega_1$ contains an operation $-$, $\Omega_2$ contains an operation $+$, (but $+$ is not assumed to be commutative) and $\mathbb{E}$ contains the group laws for $0,-,+$.

We define $\Omega^\prime_i$ to be the set of 'extra' specified operations, so $\Omega^\prime_0 = \Omega_0\setminus \{0\}$, and so on.

We have the following conditions on the equations/identities:

* for any $*\in \Omega^\prime_2$, $\Omega^\prime_2$  also contains $\ast^{op}$, where $x\ast^{op}y= y\ast x$;
* for any $\omega \in \Omega^\prime_1$, $\mathbb{E}$ includes the identity $\omega(x+y)=\omega(x)+ \omega(y)$;
* for any $\ast \in \Omega^\prime_2$, $\mathbb{E}$ includes the distributive law: $x\ast (y+z)= x\ast y+x\ast z$;
* for any $\omega \in \Omega^\prime_1$ and $\ast \in \Omega^\prime_2$, $\mathbb{E}$ includes the identity $\omega(x)\ast y = \omega(x\ast y)$;
* for any $\ast \in \Omega^\prime_2$, $\mathbb{E}$ includes the identity $x+(y\ast z)=(y\ast z)+x$;
* for any $\ast, \circ \in \Omega^\prime_2$, there is a word $W$ such that $\mathbb{E}$ includes the identity:
$(x*y)\circ z = W(x\ast_1(y\circ_1 z),\ldots, x\ast_m(y\circ_m z), y\ast_{m+1}(x\circ_{m+1} z),\ldots, y\ast_{n}(x\circ_{n} z)$, where $\ast_1\ldots,\ast_n$ and $\circ_1,\ldots,  \circ_n$ are operations in $\Omega^\prime_2$.

## Properties

Any category of group-based universal algebras is a variety of $\Omega$-groups (in the sense of Higgins) and so is automatically a [[semi-abelian category]]. This provides a useful set of fairly concrete examples for study in the semi-abelian /  proto-modular context. In fact, they are [[protomodular category|strongly protomodular]], (and hence [[strongly semi-abelian]]). This means that in such categories you can deal with internal [[actions]] in more or less the same way as you do in the category of groups.

Moreover categories of group-based universal algebras are action accessible in the sense of Bourn-Janelidze, meaning that actions are not so far from being representable, allowing a Schreier-MacLane-like obstruction theory for the classification of extensions.

##Related entries

* [[Omega-group]]

* [[semi-abelian category]]

* [[protomodular category]]

* [[variety of algebras]]

##References

The original idea is in

* Grace Orzech, _Obstruction theory in algebraic categories I, II_, J. Pure Appl. Algebra __2__ (1972) 287-340, 315 - 340.

The point about actions is in

* [[G. Metere]], _A note on strong protomodularity, actions and quotients,_ Journal of Pure and Applied Algebra, 221 (2017) 75 - 88.

[[!redirects category of interest]]


category of Lie algebras

## Definition

The [[category]] whose objects are [[Lie algebras]] and morphisms
are homomorphisms of [[Lie algebras]].

## Properties

As a [[variety of algebras]], the category of Lie algebras
admits all small [[limits]] and [[colimits]].
The [[forgetful functor]] to the [[category of sets]]
[[creates]] all small [[limits]] and [[sifted colimits]].

## Related concepts

* [[category of groups]]


category of local models

# Category of local models
* table of contents
{: toc}

## Idea

A _category of local models_ is a category whose objects play the role of particularly well-controled test spaces in the sense of [[space and quantity]]. The major notions of spaces, such as [[topological space]]s, [[algebraic space]]s, [[smooth manifold]]s, are spaces modeled on a _category of local models in the sense of [[structured generalized space]]s.

## Definition

A **category of local models** is

* a small [[site]] $R$;

* a morphism of sites $U : R \to $ [[Top]];

* a set $L$ of [[diagram]]s $I_L \to R$ in $R$

* an object $A$ of $R$

* such that

  * $R$ is closed under [[limit]]s of shape in $L$;

  * $U$ is a _basis for its image_ in that...

  * $A$ generates $R$ under $L$-[[limit]]s and gluing (?).

The objects of $R$ are usually called **affine** spaces.
In particular the object $A$ is the **affine line**.

## Applications

* For every category of local models there is the corresponding notion of [[locally modeled monoid]]s. See the examples below.

## Examples

* $R = Rings^{op}$ is the category of local models for algebraic spaces; here $A = \mathbb{Z}[x]$;

* $R =$ [[CartSp]] is the category of local models for smooth manifolds and [[generalized smooth algebra]]s; here $A = \mathbb{R}$.

## References

section 1.1 of

* David Spivak, _Quasi-smooth derived manifolds_ ([pdf](http://math.berkeley.edu/~dspivak/thesis2.pdf))


category of maps

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

## Contents ##
* table of contents
{:toc}

## Definition ##

Given a [[dagger 2-poset]] $A$, the **category of maps** $Map(A)$ is the sub-2-poset whose objects are the objects of $A$ and whose morphisms are the [[map in a dagger 2-poset|maps]] of $A$. In every dagger 2-poset, given two maps $f:Map_A(a,b)$ and $g:Map_A(a,b)$, if $f \leq g$, then $f = g$. This means that the sub-2-poset $Map(A)$ is a [[category]] and trivially a 2-poset.

## Examples ##

* For the dagger 2-poset [[Rel]] of [[sets]] and [[relations]], the category of maps $Map(Rel)$ is equivalent to the category [[Set]] of sets and functions.

## See also ##

* [[map in a dagger 2-poset]]
* [[bicategory of maps]]
* [[2-poset of partial maps]]
* [[category of monic maps]]


category of monic maps

[[!redirects category of injective maps]]

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#### Higher category theory
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[[!include higher category theory - contents]]
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## Contents ##
* table of contents
{:toc}

## Definition ##

Given a [[dagger 2-poset]] $A$, the **category of monic maps** $MonoMap(A)$ is the [[subobject|sub]]-[[2-poset]] whose [[objects]] are the objects of $A$ and whose [[morphisms]] are the [[injective map in a dagger 2-poset|injective maps]] of $A$.

In every dagger 2-poset, given two injective maps $f:hom_A(a,b)$ and $g:hom_A(a,b)$, if $f \leq g$, then $f = g$. This means that the sub-2-poset $Map(A)$ is a [[category]] and trivially a 2-poset.

## Examples ##

* For the dagger 2-poset [[Rel]] of [[sets]] and [[relations]], the category of monic maps $MonoMap(Rel)$ is equivalent to the category of sets and [[injections]].

## See also ##

* [[map in a dagger 2-poset]]
* [[2-poset of partial maps]]
* [[category of maps]]


category of monoids

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###Context###
#### Monoidal categories
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[[!include monoidal categories - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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#### Monoid theory
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[[!include monoid theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

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

For $C$ a [[monoidal category]], the **category of [[monoid in a monoidal category|monoids]]** $Mon(C)$  in $C$ is the [[category]] whose

* [[objects]] are [[monoid in a monoidal category|monoids]] in $C$;

* [[morphisms]] are morphisms in $C$ of the underlying objects that respect the monoid structure, i.e., $(A,\nabla_{A},\eta_{A}), (B,\nabla_{B},\eta_{B})$ being two objects in $Mon(C)$, a morphism $f \colon A \rightarrow B$ is a morphism in $Mon(C)$ if these two diagrams commute:

\begin{tikzcd}
A \otimes A \arrow[r, "\nabla_{A}"] \arrow[d, "f \otimes f"'] & A \arrow[d, "f"] & I \arrow[rd, "\eta_{B}"'] \arrow[r, "\eta_{A}"] & A \arrow[d, "f"] \\
B \otimes B \arrow[r, "\nabla_{B}"']                          & B                &                                                 & B
\end{tikzcd}

Similarly for the **category of [[commutative monoid in a symmetric monoidal category|commutative monoids]]** $CMon(C)$, if $C$ is symmetric monoidal.

=--

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

If [[Assoc]] denotes the [[associative operad]] in $C$, then $Mon(C) = Alg_C Assoc$ is the category of [[algebras over an operad]] for $Assoc$.

=--

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

Every category of monoids comes with a [[forgetful functor]] $U \colon Mon(C) \to C$ which is [[faithful functor|faithful]] and [[conservative
functor|conservative]].  In many cases it is [[monadic functor|monadic]].

=--

## Properties

The properties of the [[category]] of monoids $Mon (C)$, especially with respect to [[colimit]]s, are markedly different according to whether or not the [[tensor product]] of $C$ preserves colimits in each variable.  (This is automatically the case if $C$ is [[closed monoidal category|closed]].)

Most "algebraic" situations have this property, but others do not.  For instance, the category of [[monads]] on a fixed category $A$ is $Mon (C)$, where $C= [A,A]$ is the category of [[endofunctors]] of $A$ with composition as its monoidal structure.  This monoidal product preserves colimits in one variable (since colimits in $[A,A]$ are computed pointwise), but not in the other (since most endofunctors do not preserve colimits).  So far, the material on this page focuses on the case where $\otimes$ does preserve colimits in both variables, although some of the references at the end discuss the more general case.

### Local presentability

+-- {: .num_theorem}
###### Theorem

Let $C$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with countable [[coproducts]] which is [[locally presentable category|locally presentable]].

Then

1. $U : Mon(C) \to C$ is a finitary [[monadic functor]].

1. If $C$ is a $\lambda$-[[locally presentable category]] then so is $Mon(C)$.

=--

This appears in ([Porst, page 7](#Porst)).

### Free and relative free monoids {#FreeMonoids}

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

Let $C$ be a [[monoidal category]] with countable [[coproduct]]s that are preserved by the [[tensor product]]. Then the forgetful functor $U_C$ has a [[left adjoint]] $F_C : C \to Mon(C)$. On an object $X \in C$ the underlying object of $F_C X$ is

$$
  U_C F_C X
  =
  \coprod_{n \in \mathbb{N}} X^{\otimes n}
  =
  I_C \coprod X \coprod (X \otimes X) \coprod \cdots
$$

in $C$, with the monoidal structure given by tensor product/juxtaposition.

=--

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

A morphism $f : F_C X \to A$ in $Mon(C)$ with components $f_k : X^{\otimes k} \to U_C A$ is entirely fixed by its component $\tilde f = f_1 : X \to U_C A$ on $X$, because by the homomorphism property and the special free nature of the product in $F_C X$

$$
  \array{
    X^{\otimes k} \otimes X^{\otimes (n-k)} &\stackrel{f_k \otimes f_{n-k}}{\to}&
    A \otimes A
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_A}}
    \\
    X^{\otimes n} &\stackrel{f_{n}}{\to}& A
  }
$$

it follows that

$$
  f_n :
  X^{\otimes n} \stackrel{f_1^{\otimes n}}{\to} A^{\otimes n} \stackrel{\mu_A}{\to} A
  \,.
$$

Conversely, every choice for $f_1$ extends to a morphism $f$ in $Mon(C)$ this way.

=--

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

Free algebras of the form $F(A)$ are called **[[tensor algebra]]**s, at least for $C = $ [[Vect]] and similar.

The elements of the free algebra $F(A)$ are somtimes called [[lists]], at least for $C = $ [[Set]] and similar.
=--

### Pushouts

We discuss forming [[pushouts]] in a category of monoids. The case

* [For commutative monoids](#PushoutOfCommutativeMonoids)

has a simple description. The case

* [For non-commutative monoids](#PushoutOfNoncommutativeMonoids)

is more involved.

#### Of commutative monoids
 {#PushoutOfCommutativeMonoids}

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

Suppose that $\mathcal{C}$ is

* a [[symmetric monoidal category]];

* with [[reflexive coequalizers]]

* that are preserved by the [[tensor product]] functors $A \otimes (-) \colon \mathcal{C} \to \mathcal{C}$ for all objects $A$ in $\mathcal{C}$.

Then for $f \colon A \to B$ and $g \colon A \to C$ two morphisms in the category $CMon(\mathcal{C})$ of _[[commutative monoid in a symmetric monoidal category|commutative monoids]]_ in $\mathcal{C}$, the underlying object in $\mathcal{C}$ of the [[pushout]] in $CMon(\mathcal{C})$ coincides with that of the pushout in the category $A$[[Mod]] of $A$-[[modules]]

$$
  U(B \coprod_A C) \simeq  B \otimes_A C
  \,.
$$

Here $B$ and $C$ are regarded as equipped with the canonical $A$-module structure induced by the morphisms $f$ and $g$, respectively.

=--

This appears for instance as ([Johnstone, page 478, cor. 1.1.9](#Johnstone)).

#### Of noncommutative monoids
 {#PushoutOfNoncommutativeMonoids}

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

If $C$ is [[cocomplete category|cocomplete]] and its tensor product preserves colimits on both sides, then the category $Mon(C)$ of monoids has all [[pushout]]s

$$
  \array{
    F(K) &\stackrel{F(f)}{\to}& F(L)
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow
    \\
    X &\to& P
  }
$$

along morphisms $F(f) : F(K) \to F(L)$, for $f : K \to L$ a morphism in $C$ and $F : C \to Mon(C)$ the free monoid functor from above.

Moreover, these pushouts in $Mon(C)$ are computed in $C$ as the [[colimit]] over a sequence

$$
  P \simeq \lim_{\to}( X := P_0 \to P_1 \to P_2 \to \cdots )
$$

of objects $(P_n)_{n \in \mathbb{N}}$, which are each given by pushouts in $C$ inductively as follows.

Assume $P_{n-1}$ has been defined. Write $Sub(\mathbf{n})$ for the [[poset]] of [[subset]]s of the $n$-element set $\mathbf{n}$ (this is the poset of paths along the edges of an $n$-dimensional cube). Define a [[diagram]]

$$
  K : Sub \mathbf{n} \to C
$$

by setting on subsets $S \subset \mathbf{n}$

$$
  K_S := X \otimes V_1 \otimes X \otimes V_2 \otimes  \cdots \otimes V_n \otimes X
$$

where

$$
  V_i := \left\{
     \array{
        K & if \, i \notin S
        \\
        L & if \, i \in S
     }
  \right.
$$

and by assigning to a morphism $S_1 \subset S_2$ the morphism which is the [[tensor product]] of identities on $X$, identities on $L$ and the given morphism $f : K \to L$.

Write $K^-$ for the same diagram minus the terminal object $S = \mathbf{n}$.

Now take $P_n$ to be the [[pushout]]

$$
  \array{
    \lim_{\to} K^- &\to& K \mathbf{n}
    \\
    \downarrow && \downarrow
    \\
    P_{n-1} &\to& P_n
  }
  \,,
$$

where the top morphism is the canonical one induced by the commutativity of the diagram $K$, and where the left morphism is defined in terms of components $K^-(S)$ of the colimit for $S \subset \mathbf{n}$ a proper subset by the tensor product morphisms of the form

$$
  (\cdots X \otimes K \otimes \cdots \otimes L \otimes \cdots)
  \stackrel{\cdots \otimes \mu_X \circ (Id \otimes p) \otimes \cdots \otimes Id_L \otimes \cdots}{\to}
  (X \otimes L)^{|S|} \otimes X
  \to P_{|S|} \to P_{n-1}
  \,.
$$

This gives the underlying object of the monoid $P$. Take the monoid structure on it as follows. The unit of $P$ is the composite

$$
 e_P : I_C \stackrel{e_X}{\to} X \to P
$$

with the unit of $X$. The product we take to be the image in the colimit of compatible morphisms $P_k \otimes P_k \to P_{k + l}$ defined by induction on $lk + l$ as follows. we observe that we have a pushout diagram

$$
  \array{
     Q_k \otimes (X \otimes L)^{\otimes l}
    \coprod_{Q_k \otimes Q_l}
    (X \otimes L)^{\otimes l} \otimes X \otimes Q_l
    &\to&
    (X \otimes L)^{\otimes k} \otimes X \otimes (X \otimes L)^{\otimes l} \otimes X
    \\
    \downarrow && \downarrow
    \\
    P_{k-1} \otimes P_l \coprod_{P_{k-1} \otimes P_{l-1}}
    P_k \otimes P_{l-1}
    &\to&
    P_k \otimes P_l
  }
  \,,
$$

where $Q_n := (\lim_{\to} K)_n$ is the colimit as in the above at stage $n$.

There is a morphism from the bottom left object to $P_{k+l}$ given by the induction assumption. Moreover we have a morphism from the top right object to $P_{k+1}$ obtained by first multiplying the two adjacent factors of $X$ and then applying the morphism $(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}$.
These are compatible and hence give the desired morphism $P_k \otimes P_k \to P_{k+l}$.

=--

This construction is spelled out for instance in the proof of [SchwedeShipley, lemma 6.2](#SchwedeShipley)

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

First we need to discuss that this definition is actually consistent, in that the morphism $\lim_\to K^- \to P_{n-1}$ is well defined and the monoid structure on $P$ is well defined.

(...)

That $X \to P$ is a morphism of monoids follows then essentially by the definition of the monoid structure on $P$.

Finally we need to check the universal property of the cocone $P$ obtained this way:

(...)

=--

### Filtered colimits
 {#FilteredColimits}

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

For $C$ a [[closed monoidal category|closed]] [[symmetric monoidal category]] the [[forgetful functor]]

$$
  U : CMon(C) \to C
$$

from commutative monoids to $C$ [[creates colimits|creates]] [[filtered colimit]]s.

=--

This appears for instance as ([Johnstone, C1.1 lemma 1.1.8](#Johnstone)).

### Structure induced from monoidal functors

If $F : C\to D$ is a [[lax monoidal functor]], then it induces canonically a functor between categories of monoids

$$
  Mon(F) : Mon(C) \to Mon(D)
  \,.
$$

This is one good way to remember the difference between *lax* and *colax* monoidal functors.

### Model structure

If $C$ is a [[monoidal model category]], then $Mon(C)$ may inherit itself the structure of a [[model category]]. See [[model structure on monoids in a monoidal model category]].

### Enrichment over $CMon$

Some categories are _implicitly_ [[enriched category|enriched]] over commutative monoids, in particular [[semiadditive categories]]. Also [[Ab]]-[[enriched categories]] (and hence in particular [[abelian categories]]) of course have an underlying $CMon$-enrichment.

## Related categories

* [[Grp]], [[Ab]]

* [[Ring]], [[CRing]]

* [[Rng]]

## References

A general discussion of categories of monoids in [[symmetric monoidal categories]] is in

* {#Porst} [[Hans Porst]], _On Categories of Monoids, Comonoids and bimonoids_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.4291&rep=rep1&type=pdf))

Free monoid constructions are discussed in

* [[Eduardo Dubuc]], _Free monoids_ Algebra J. 29, 208–228 (1974)

* [[Max Kelly]], _A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on_ Bull. Austral. Math. Soc. 22(1), 1–83 (1980)

* [[Stephen Lack]], _Note on the construction of free monoids_ Appl Categor Struct (2010) 18:17–29

The detailed discussion of pushouts along free monoid morphisms is in the proof of lemma 6.2 of

* [[Stefan Schwede]], [[Brooke Shipley]], _Algebras and modules in monoidal model categories_ Proc. London Math. Soc. (2000) 80(2): 491-511  ([pdf](http://www.math.uic.edu/~bshipley/monoidal.pdf))
{#SchwedeShipley}

Some remarks on commutative monoids are in section C1.1 of

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

Discussion of the [[closed monoidal category]] structure on a category of algebras of a [[commutative algebraic theory]] is in

* [[Peter Freyd]], _Algebra valued functors in general and tensor products in particular_, Colloq. Math. 14 (1966), 89-106.

[[!redirects categories of monoids]]

[[!redirects category of commutative monoids]]

[[!redirects Mon]]
[[!redirects AbMon]]
[[!redirects CMon]]

category: category


category of open subsets

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### Context
#### Topology
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[[!include topology - contents]]
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#### Topos Theory
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[[!include topos theory - contents]]
=--
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=--

# Category of open subsets
* table of contents
{: toc}

## Definition

Given a [[topological space]] $X$, the **category of open subsets** $Op(X)$ of $X$ is the [[category]] whose

* [[object | objects]] are the [[open subset | open subsets]] $U \hookrightarrow X$ of $X$;

* [[morphism | morphisms]] are the inclusions
$
  \array{
     V &&\hookrightarrow && U
     \\
     & \searrow && \swarrow
     \\
     && X
  }
$ of open subsets into each other.

## Properties

* The category $Op(X)$ is a [[poset]], in fact a [[frame]] (dually a [[locale]]): it is the [[frame of opens]] of $X$.

* The category $Op(X)$ is naturally equipped with the
structure of a [[site]], where a collection $\{U_i \to U\}_i$ of morphisms is a cover precisely if their [[union]] in $X$ equals $U$:
$$ \bigcup_i U_i = U .$$

  The [[category of sheaves]] on $Op(X)$ equipped with this site structure is typically referred to as the _[[category of sheaves on a topological space|category of sheaves on the topological space]]_ and denoted

  $$
    Sh(X) \;\coloneq\; Sh(Op(X))
    \,.
  $$

* The category $Op(X)$ is also a [[suplattice]].

## Related concepts

* [[spatial locale]]

* [[spatial topos]]

[[!redirects category of opens]]
[[!redirects categories of opens]]
[[!redirects category of open sets]]
[[!redirects categories of open sets]]
[[!redirects category of open subsets]]
[[!redirects categories of open subsets]]
[[!redirects category of open subspaces]]
[[!redirects categories of open subspaces]]

[[!redirects site of open subsets]]
[[!redirects site of opens]]

[[!redirects sites of open subsets]]
[[!redirects sites of opens]]

[[!redirects poset of open subsets]]
[[!redirects posets of open subsets]]

[[!redirects lattice of open subsets]]
[[!redirects lattices of open subsets]]

[[!redirects frame of open subsets]]
[[!redirects frames of open subsets]]


category of operators

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Every [[operad]] defines and is defined by a [[category]] -- its _category of operators_ -- whose

* [[object]]s are sequences consisting of colors of the operad;

* [[morphism]]s are tuples consisting of maps of sets between these sequences of colors, and a $k$-ary operation of the operad for each collection of $k$ source colors that are mapped to the same target color.

This is a universal construction: the category of operators is the free [[semicartesian monoidal category]] on the free semicartesian operad on the given operad.  It also generalizes in a straightforward way to "colored operads", i.e. [[multicategories]].

## Definition

### For symmetric colored operads

Let $FinSet_{*}$ be the category of [[finite set|finite]] [[pointed set]]s. Write $\langle n \rangle = \{*, 1,2, \cdots, n\}$ for [[generalized the|the]] pointed set with $n+1$ elements.

Let $A$ be a colored [[symmetric operad]] over [[Set]] ( hence a symmetric [[multicategory]]). Its **category of operators** is the [[category]]

* whose [[objects]] are finite sequences $(c_1, \cdots, c_n)$ of colors of $A$;

* whose [[morphisms]] $F : (c_1, \cdots, c_n) \to (d_1, \cdots, d_m)$ are given by a collection consisting of

  * a morphism $\phi : \langle n \rangle \to \langle m\rangle$ in $FinSet_*$;

  * for each $1 \leq i \leq m$ an operation

    $$
      f_i  \in Hom_A((c_k)_{k \in \phi^{-1}(i)}, d_i)
    $$

    from the objects whose indices are mapped to $i$ to the object $d_i$;

* composition is given componentwise by the composition in $FinSet_*$ and in $A$.

### $(\infty,1)$-Category of operators

The above definition has been [[vertical categorification|categorified]] to a notion of  [[(∞,1)-category]] of operators. See at _[[(∞,1)-operad]]_ for more.

## Properties

By construction, the category of operators $C_A$ of a symmetric colored operad is canonically equipped with a [[functor]] $p : C_A \to FinSet_*$.

From this functor, the original operad may be recovered up to canonical equivalence.

### Reconstruction of the operad

Given a functor $p : C_A \to FinSet_*$ from a category of operators of a symmetric colored operad, we reconstruct the operad $A$ as follows:

For $1 \leq i \leq n$ let

$$
  \rho^i : \langle n \rangle \to \langle 1 \rangle
$$

be the map that sends all elements to the point, except the element $i$.

Write $A_{n} := p^{-1}(\langle n\rangle)$ for the [[fiber]] of $p$ over $\langle n\rangle$. $A_1$ is the [[category]] underlying the operad $A$: the category whose morphisms are the unary operations of the operad.

The morphisms $\rho^i$ in $FinSet_*$ induce a functor

$$
  \prod_i \rho^i_* : A_n \to (A_1)^n
$$

which is an [[isomorphism]] that identifies $A_n$ with the $n$-fold cartesian [[product]] of the category $A_1$ with itself.

The morphisms $h \in Hom_A((c_1, \cdots, c_n), d)$ of $A$ are recovered as the collection of morphisms in $C_A$ from $(c_1, \cdots, c_n)$ to $(d)$ that cover the morphism $\langle n\rangle \to \langle 1\rangle$ in $FinSet_*$ whose [[preimage]] of the point contains just the point.

### Relation to underlying multicategories

Forming categories of operators is left 2-adjoint to forming the [[representable multicategory|underlying multicategory]] of a semi-cartesian monoidal category.  (For a left adjoint to the underlying multicategory of an arbitrary monoidal category, see instead  [[props]].)  For the moment, see there for more details.

## Related concepts

* The ([[syntactic category]] of) a (homogeneous) [[globular theory]] is the category of operators of a [[globular operad]].

## References

The notion originates in

* [[Peter May]], [[Robert Thomason]], _The uniqueness of infinite loop space machines_ , Topology  17(3) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/22.pdf))

A discussion of the general logic behind the notion is at

* [[Mike Shulman]], _Generalized operads in classical algebraic topology_
([blog](http://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html))

This summarizes aspects of

* G. Cruttwell, [[Mike Shulman]], _A unified framework for generalized multicategories_ ([arXiv:0907.2460](http://arxiv.org/abs/0907.2460))

See [example 11.20](http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.2460v1.pdf#page=70) there.

[[!redirects categories of operators]]

[[!redirects (∞,1)-category of operators]]
[[!redirects (infinity,1)-category of operators]]
[[!redirects (∞,1)-categories of operators]]
[[!redirects (infinity,1)-categories of operators]]


category of partial endofunctions

[[!redirects category of partial functions]]

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

#Contents#
* table of contents
{:toc}

## Idea ##

The essentially algebraic structure of partial endofunctions on a set $S$, and specific cases for when $S$ is an [[abelian group]], [[commutative ring]], and [[field]] respectively.

## Definition ##

### In a set ###

Given a set $S$, the **category of [[partial function|partial]] [[endofunctions]]** in $S$, or just **category of partial functions**, is the [[concrete category]] $Part(S)$ with objects called [[subset]]s $A \in Ob(Part(S))$ with the set of elements for each subset $El(A)$, and the set of morphisms consist of functions $Hom(A, \Im(S)) \coloneqq (A \to S)$ for each subset $A \in Ob(Part(S))$, where $\Im(S)$ is the improper subset, as well as the set of monomorphisms $Hom(A, B)$ consisting of the subset inclusions for subsets $A \in Ob(Part(S))$ and $B:Ob(Part(S))$.

There exist a global operator representing **composition of partial functions**

$$(-)\circ_{Part(S)}(-): \sum_{A:Ob(Part(S))} \sum_{B:Ob(Part(S))} Hom(A, \Im(S)) \times Hom(B, \Im(S)) \to Hom(A \cap B, \Im(S))$$

where

* for partial functions $f \in Hom(A, \Im(S))$, $g \in Hom(B, \Im(S))$, and $h \in Hom(C, \Im(S))$, given the canonical isomorphism $i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f \circ_{Part(S)} (g \circ_{Part(S)} h)) = ((f \circ_{Part(S)} g) \circ_{Part(S)} h)$

* for partial function $f \in Hom(A, \Im(S))$ and subset $B \subseteq A$, there is a function $g \in Hom(B, \Im(S))$ such that $g = f \circ_{Part(S)} i_{B,A}$ for canonical injection $i_{B,A} \in Hom(B,A)$,

* for partial function $f \in Hom(A, \Im(S))$ and superset $B \supseteq A$, there is a function $h \in Hom(B, \Im(S))$ such that $h \circ_{Part(S)} i_{A,B} = f $ for canonical injection $i_{A,B} \in Hom(A,B)$,

* for partial function $f \in Hom(A, \Im(S))$, $f = f \circ_{Part(S)} id_S$ and $f = id_S \circ_{Part(S)} f$ for the identity function $id_S:Hom(\Im(S), \Im(S))$

### In an abelian group ###

If $S$ is a [[abelian group]], then for every subset $A \in Ob(Part(S))$, $Hom(A, \Im(S))$ is a [[abelian group]], and in addition to the global operators corresponding to composition of partial functions, there exist global operators representing **addition of partial functions** and **negation of partial functions**,

$$(-)+(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \times \{B \in Ob(Part(S)) \vert Hom(B, \Im(S))\} \to \{(A,B) \in Ob(Part(S)) \times Ob(Part(S)) \vert Hom(A \cap B, \Im(S))\}$$

$$-(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \to Hom(A, \Im(S)))$$

where

* for partial functions $f \in Hom(A, \Im(S))$ and $g \in Hom(B, \Im(S))$ there is a partial function $f + g \in Hom(A \cap B, \Im(S))$ and a partial function $g + f \in Hom(B \cap A, \Im(S))$ such that given the canonical isomorphism $i_c \in Hom(A \cap B, B \cap A)$,  $i_c \circ (f + g) = (g + f)$

* for partial functions $f \in Hom(A, \Im(S))$, $g \in Hom(B, \Im(S))$, and $h \in Hom(C, \Im(S))$, given the canonical isomorphism $i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f + (g + h)) = ((f + g) + h)$

* for partial function $f \in Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B \in Ob(Part(S))$, given the local additive unit $0_{B,\Im{S}} \in Hom(B, \Im(S)$, $f + 0_{B,\Im{S}} = f$ and $0_{B,\Im{S}} + f = f$

* for partial function $f \in Hom(A, \Im(S))$, there is a partial function $-f \in Hom(A, \Im(S))$ representing negation where the negation of $F$ is the local additive inverse of $f$: $-f = -_{A,S}f$

### In a commutative ring ###

If $S$ is a [[commutative ring]], then for every subset $A \in Ob(Part(S))$, $Hom(A, \Im(Part(S)))$ is a $S$-[[commutative algebra]], and in addition to the global operators corresponding to composition, addition, and negation of partial functions, there exist a global operator representing **multiplication of partial functions**

$$(-)\cdot(-): \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \times \{B \in Ob(Part(S)) \vert Hom(B, \Im(S))\} \to \{(A,B) \in Ob(Part(S)) \times Ob(Part(S)) \vert Hom(A \cap B, \Im(S))\}$$

where

* for partial functions $f \in Hom(A, \Im(S))$ and $g \in Hom(B, \Im(S))$ there is a partial function $f \cdot g \in Hom(A \cap B, \Im(S))$ and a partial function $g \cdot f \in Hom(B \cap A, \Im(S))$ such that given the canonical isomorphism $i_c \in Hom(A \cap B, B \cap A)$,  $i_c \circ (f \cdot g) = (g \cdot f)$

* for partial functions $f \in Hom(A, \Im(S))$, $g \in Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_a \in Hom(A \cap (B \cap C), (A \cap B) \cap C)$, $i_a \circ (f \cdot (g \cdot h)) = ((f \cdot g) \cdot h)$

* for partial function $f \in Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B:Ob(Part(S))$, given the local multiplicative unit $1_{B,\Im{S}} \in Hom(B, \Im(S)$, $f \cdot 1_{B,\Im{S}} = f$ and $1_{B,\Im{S}} \cdot f = f$

* for partial function $f \in Hom(A, \Im(S))$, and supersets $B \supseteq A$ for $B:Ob(Part(S))$, given the local additive unit $0_{B,\Im{S}} \in Hom(B, \Im(S)$, $f \cdot 0_{B,\Im{S}} = 0_{A,\Im{S}}$ and $0_{B,\Im{S}} \cdot f = 0_{A,\Im{S}}$

* for partial functions $f \in Hom(A, \Im(S))$, $g \in Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_l \in Hom(A \cap (B \cap C), (A \cap B) \cap (A \cap C)$, $i_a \circ (f \cdot (g + h)) = (f \cdot g) + (f \cdot h)$

* for partial functions $f \in Hom(A, \Im(S))$, $g \in Hom(B, \Im(S))$, and $h:Hom(C, \Im(S))$, given the canonical isomorphism $i_r \in Hom((A \cap B) \cap C, (A \cap C) \cap (B \cap C)$, $i_a \circ ((f + g) \cdot h)) = (f \cdot h) + (g \cdot h)$

### In a field ###

If $S$ is a [[Heyting field]], then for every subset $A \in Ob(Part(S))$, $Hom(A, \Im(S))$ is a $S$-[[commutative algebra]], with global operators corresponding to composition, addition, negation, and multiplication of partial functions. Let

$$Hom_{\#0}(A, \im(S)) \coloneqq \{f \in Hom(A, \Im(S)). \vert \forall x \in El(A).f(x) # 0\}$$

be the type of all functions whose evaluations at each element are apart from zero on the entire domain. There exists a global operator representing the **reciprocal of partial functions**:

$$\frac{1}{(-)}: \{A \in Ob(Part(S)) \vert Hom(A, \Im(S))\} \to Hom_{\#0}(A, \Im(S)$$

where

* for partial function $f \in Hom(A, \Im(S))$,
$$f \cdot \frac{1}{f} = id_{\im(f)_{\#0}}$$
and
$$\frac{1}{f} \cdot f = id_{\im(f)_{\#0}}$$

and the set $\im(f)_{\#0}$ is defined as

$$\im(f)_{\#0} \coloneqq \{x \in El(A)\vert f(x) # 0\}$$

## See also ##

* [[partial function]]

* [[rational function]]

## References ##

* {#Richman12} [[Fred Richman]], *Algebraic functions, calculus style*. Communications in Algebra, Volume 40, Issue 7, July 2012, Pages 2671-2683 [[doi:10.1080/00927872.2011.584337](https://doi.org/10.1080/00927872.2011.584337)]

[[!redirects reciprocal of partial functions]]


category of partial equivalence relations > history

see [[category of PERs]]


category of PERs

# The category of PERs

* table of contents
{: toc}

## Idea

The **category of PERs** over a [[partial combinatory algebra]] $A$ is a particularly concrete and simple subcategory of its [[realizability topos]], which is especially well-suited to modelling "computable" objects relative to $A$.

It is based on the idea that a "computable set" $X$ relative to $A$ is specified by saying which elements of $A$ are "codes" representing which elements of $X$, such that no element of $A$ can code for more than one element of $X$ (but not all elements of $A$ need to code for any element of $X$).  Thus this sort of "computable set" (sometimes called a **modest set**) is equivalently a partition of a subset of $A$, i.e. a [[partial equivalence relation]] on $A$.

## Definition

Let $A$ be a [[partial combinatory algebra]].  If $E$ is a [[partial equivalence relation]] on $A$, we write $A/E$ for its set of equivalence classes.  Now the [[category]] $PER(A)$ has:

* as objects, the [[partial equivalence relations]] on $A$.

* as morphisms $E\to E'$, for PERs $E$ and $E'$, the set-theoretic functions $\phi:A/E \to A/E'$ with the [[property]] of being "tracked", i.e. such that there exists an element $f\in A$ such that whenever $(a,a)\in E$ we have $\phi([a]) = [f\cdot a]$ (hence in particular $(f\cdot a,f\cdot a)\in E'$).  Equivalently, it is the quotient of the set of elements $f\in A$ such that $(a,b)\in E$ implies $(f\cdot a,f\cdot b)\in E'$ by the relation $f\sim g$ meaning that $(f\cdot a,g\cdot a)\in E'$ whenever $(a,a)\in E$.

## Properties

* $PER(A)$ is a [[locally cartesian closed category]] and a [[Heyting category]].

* $PER(A)$ is a full subcategory of the category of [[assemblies]] over $A$, which is in turn a full subcategory of the [[realizability topos]] over $A$.

## In first-order logic

Given a theory in first-order logic with equality (categorically, this can be taken as a [[first-order hyperdoctrine]]), its category of partial equivalence relations represents all the [[subquotient]]s the theory can "see" of its given sorts and all the functions the theory can "see" between them.

Specifically, the objects of this category are pairs $(X, R)$ where $X$ is a sort in the first-order theory and $R$ is a binary predicate which the theory proves to be a [[partial equivalence relation]] on $X$. A morphism between objects $(X, R)$ and $(Y, S)$ can be taken to be a binary predicate $F \in P(X \times Y)$ such that the theory proves $F$ is essentially the graph of a function between the specified subquotients (In detail: ...). Composition is given in the obvious way (In detail: ...).

In the case where the original hyperdoctrine was in fact a [[tripos]], the resulting category of PERs will be a [[topos]] (while the converse isn't quite true, the condition of being a tripos is just slightly stronger than necessary for this to happen).

[[!redirects categories of PERs]]
[[!redirects category of partial equivalence relations]]
[[!redirects categories of partial equivalence relations]]
[[!redirects modest set]]
[[!redirects modest sets]]


category of presheaves

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

#Contents#
* table of contents
{:toc}

## Definition

For $C$ a [[small category]], its **category of presheaves** is the [[functor category]]

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

from the [[opposite category]] of $C$ to [[Set]].

An object in this category is a [[presheaf]]. See there for more details.

## Properties

### General
 {#GeneralProperties}

For $\mathcal{C}$ any [[category]], consider $PSh(\mathcal{C})$ its category of [[Set]]-valued presheaves.

\begin{proposition}\label{LimitsOfPresheavesAreComputedObjectwise}
  $PSh(\mathcal{C})$ has all [[limits]] and [[colimits]] (over [[small diagrams]]), and these are computed objectwise: For $\mathcal{I}$ any [[diagram]] (a [[small category]]) we have:
$$
  X_{\bullet} \,\colon\, \mathcal{I} \longrightarrow PSh(\mathcal{C})
  \;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;
  \underset{c \in Obj(\mathcal{C})}{\forall}
  \;\;\;\;
  \big(
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} X_i
  \big)
  (c)
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
  \big(
     X_i(c)
  \big)
  \;\;\;\;\text{and}\;\;\;\;
  \big(
  \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} X_i
  \big)
  (c)
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim}
  \big(
     X_i(c)
  \big)
$$
\end{proposition}
\begin{proof}
  The defining [[universal property]] readily follows from that of the objectwise (co-)limits.
\end{proof}

\begin{proposition}\label{EpimorphismsOfPresheavesAreDetectedObjectwise}
  A morphism $X \xrightarrow{\;f\;} Y$ of presheaves $X,Y \,\in\, PSh(\mathcal{C})$ is an [[epimorphism]] or [[monomorphism]] precisely if it is so over each [[object]] of $\mathcal{C}$, hence precisely if it is object-wise a [[surjection]] or [[injection]], respectively.
\end{proposition}
\begin{proof}
  Using for instance the characterization of epimorphisms by [[pushouts]] ([this Prop.](epimorphism#BasicCharacterizationOfEpimorphisms)) and of monomorphism by [[pullbacks]] ([this Prop.](monomorphism#BasicCharacterizationOfMonomorphisms)), the statement follows by Prop. \ref{LimitsOfPresheavesAreComputedObjectwise}.
\end{proof}

Now assume that $\mathcal{C}$ is a [[small category]].

\begin{proposition}
  $PSh(\mathcal{C})$ is a [[cartesian closed category]].
\end{proposition}
\begin{proof}
  This is spelled out at *[[closed monoidal structure on presheaves]]*.
\end{proof}

\begin{proposition}
  $PSh(\mathcal{C})$ is a [[topos]].
\end{proposition}
\begin{proof}
  This is the base case of *[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]*.
\end{proof}

* The category of presheaves $PSh(\mathcal{C})$ is the [[free cocompletion]] of $\mathcal{C}$.

* the [[Yoneda lemma]] says that the [[Yoneda embedding]] $j \colon \mathcal{C} \to PSh(\mathcal{C})$ is -- in particular -- a [[full and faithful functor]].

* The construction of forming (co)-presheaves extends to a [[2-functor]]

  $$
    [-,Set] : Cat \to Topos
  $$

  from the [[2-category]] [[Cat]] to the [[2-category]] [[Topos]]. (See at [[geometric morphism]] the section [Between presheaf toposes](geometric+morphism#BetweenPresheafToposes) for details.)

* A [[reflective subcategory]] of a category of presheaves is a [[locally presentable category]] if it is closed under $\kappa$-[[directed colimit]]s for some [[regular cardinal]] $\kappa$ (the embedding is an [[accessible functor]]).

* A [[geometric embedding|sub-topos]] of a category of presheaves is a [[Grothendieck topos]]: a [[category of sheaves]] (see there for details).

### (Co-)Monadicity over Families

There is a functor from presheaves to families of sets $U : Psh(C) \to Set^{Ob(C)} \cong Set/Ob(C)$ given by "forgetting" the functorial action on morphisms of $C$. This functor is both [[monadic functor|monadic]] and [[comonadic functor|comonadic]].

### Functoriality

See [[functoriality of categories of presheaves]].

### Characterization

The following [[Giraud theorem|Giraud like]] theorem stems from [[Marta Bunge|Marta Bunge's]] dissertation (1966)

+-- {: .num_theorem #bunge_theorem}
###### Theorem

A category $E$ is equivalent to a presheaf topos if and only if it is [[cocomplete]], [[well-powered category|well-powered]], [[well-powered category|co-well-powered]], [[atomic category|atomic]], and [[regular category|regular]].

=--

This characterisation is proven for [[enriched categories|enriched]] presheaf categories in Theorem 4.16 of [Bunge 1969](#Bunge69) (Corollary 4.19 for the unenriched statement).

A second characterization using [[exact completion|exact completions]] can be found in Carboni-Vitale ([1998](#CarboniVitale98)) or Centazzo-Vitale ([2004](#Centazzo-Vitale04)):

+-- {: .num_theorem #carboni_vitale_theorem}
###### Theorem

A category $E$ is equivalent to a presheaf topos if and only if it is [[locally small category|locally small]], [[extensive category|extensive]], [[exact category|exact]] and has a small set of projective and indecomposable [[separator|generators]].

=--

([1998](#CarboniVitale98)) has also an interesting comparison to a classical characterization of categories [[monadic category|monadic]] over Set.

### Presheaves on over-categories and over-categories of presheaves {#RelWithOvercategories}

Let $C$ be a [[category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[over category]] of $C$ over $c$. Write
$PSh(C/c) = [(C/c)^{op}, Set]$ for the category of [[presheaf|presheaves]] on $C/c$ and write
$PSh(C)/Y(c)$ for the [[over category]] of [[presheaf|presheaves]] on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the [[Yoneda embedding]].

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

There is an [[equivalence]] of categories

$$
  e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)
  \,.
$$

=--

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

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf
$F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

A weak inverse of $e$ is given by the functor
$$
  \bar e : PSh(C)/Y(c) \to PSh(C/c)
$$
which sends
$
  \eta : F' \to Y(C))
$ to $F \in PSh(C/c)$ given by
$$
  F : (f : d \to c) \mapsto F'(d)|_c
  \,,
$$
where $F'(d)|_c$ is the [[pullback]]
$$
  \array{
     F'(d)|_c &\to& F'(d)
     \\
     \downarrow && \downarrow^{\eta_d}
     \\
     pt &\stackrel{f}{\to}& C(d,c)
  }
  \,.
$$

=--

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

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $C$, i.e. suppose that it factors through the forgetful functor from the [[over category]] to $C$:
$$
  F : (C/c)^{op} \to C^{op} \to Set
  \,.
$$

Then
$
  F'(d) = \sqcup_{f \in C(d,c)} F(f)
  = \sqcup_{f \in C(d,c)} F(d)
  \simeq
  C(d,c) \times F(d)
$
and hence $F ' = Y(c) \times F$ with respect to the [[closed monoidal structure on presheaves]].

=--

See also [[functors and comma categories]].

For the analog statement in [[(∞,1)-category]] theory see

* (∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories

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

Consider $\int_C Y(c)$ , the [[category of elements]] of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.

Hence, the above proposition \ref{representable_case} can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

=--

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

Let $P:C^{op}\to Set$ be a presheaf. Then there is an [[equivalence of categories]]

$$
  PSh(\int_C P) \simeq PSh(C)/P
  \,.
$$

=--

On objects this takes $F : (\int_C P)^{op} \to Set$ to
$$i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)$$
with obvious projection to $P$. The inverse takes $f : Q \to P$ to
$$i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.$$

For a proof see [Kashiwara-Schapira (2006, Lemma 1.4.12, p. 26)](#KS06). For a more general statement involving slices of Grothendieck toposes see [Mac Lane-Moerdijk (1994, p.157)](#MacLaneMoerdijk).

In particular, this equivalence shows that _slices of presheaf toposes are presheaf toposes_.

### Artin gluings

The statements of the preceding [subsection](#RelWithOvercategories) may be generalized further:

+-- {: .num_theorem}
###### Theorem
Let $T: Set^{C^{op}} \to Set^{D^{op}}$ be a functor that preserves [[wide pullbacks]]. Then the [[Artin gluing]] $(Set^{D^{op}} \downarrow T)$ is also a presheaf topos.
=--

+-- {: .proof}
###### Proof
The functor $T$ is [[parametric right adjoint|familially representable]]: there is a functor $D^{op} \to Fam(Set^{C^{op}})$ into the small [[free coproduct completion|coproduct cocompletion]] of $Set^{C^{op}}$, taking $d \in Ob(D)$ to a formal coproduct $\sum_{x \in T(1)(d)} W_{(d, x)}$ of presheaves $W_{(d, x)}: C^{op} \to Set$, such that $T(F)$ is given by the formula

$$T(F)(d) = \sum_{x \in T(1)(d)} Set^{C^{op}}(W_{(d, x)}, F).$$

The Artin gluing itself is then describable as the [[collage]] of the [[profunctor]] $W: C \nrightarrow (y_D \downarrow T(1))$ defined by the formula

$$W(c; (d, x: D(-, d) \to T(1))) = W_{(d, x)}(c).$$
For details, see for example Appendix C.3 of [Leinster](#Lein). The result is due to [Carboni and Johnstone](#cj).
=--

In passing, we note that the small coproduct cocompletion $Fam(Set^{C^{op}})$ is itself a presheaf topos: there are [[equivalence of categories|equivalences]]

$$Fam(Set^{C^{op}}) \simeq Set^{C^{op}} \downarrow \Delta \simeq Set^{C_+^{op}}$$

where $\Delta: Set \to Set^{C^{op}}$ is the [[diagonal functor]], and $C_+$ is the result of freely adjoining an [[initial object]] to $C$, i.e., the [[ordinal sum of categories]] $1 +_\sigma C$ of categories ($1$ being [[terminal object|terminal]]), aka the [[cone]] of $C$.

### Finite presheaves

A _finite presheaf_ on a category $C$ is a functor $C^{op}\to FinSet$ valued in the [[FinSet|category of finite sets]]. Categories of finite presheaves will hardly be [[Grothendieck toposes]] for want of infinite limits but they still can turn out to be [[elementary toposes]] as e.g. in the case of $FinSet$ itself.

By going through the proof that ordinary categories of presheaves are toposes, one observes that the constructions stay within finite presheaves when applied to a finite category $C$ i.e. one with only a finite set of morphisms. Hence, one has the following

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

Let $C$ a finite category. Then the category of finite presheaves $[C^{op},FinSet]$ is a topos. $\qed$
=--

Note, that the category $[G,FinSet]$ of finite $G$-sets is a topos even when the group $G$ is infinite! In this case it is crucial that $\Omega =\{\emptyset , G\}$ in $[G,Set]$ is a finite set.

(Cf. [Borceux (1994, p.299)](#Borceux3))

### Models in presheaf toposes

See at _[[models in presheaf toposes]]_.

## Related concepts

For [[(∞,1)-category]] theory see [[(∞,1)-category of (∞,1)-presheaves]].

[[!include locally presentable categories - table]]

## References

Original discussion:

* [[Marta Bunge]], *Categories of Set-Valued Functors*, PhD thesis, University of Pennsylvania (1966) [[[Bunge-SetValuedFunctors.pdf:file]]]

https://ncatlab.org/nlab/files/Bunge-SetValuedFunctors.pdf

* {#SGA4} [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), *Théorie des Topos et Cohomologie Etale des Schémas - [[SGA 4]]*, LNM 269 Springer (1972)

Basic exposition:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

Introduction:

* [[Marie La Palme Reyes]], [[Gonzalo E. Reyes]], [[Houman Zolfaghari]], _Generic figures and their glueings.  A constructive approach to functor categories_, Polimetrica (2008) [[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf), ISBN:8876990046]

See also most accounts of general [[topos theory]], such as:

* {#Borceux3} [[Francis Borceux]], *Handbook of Categorical Algebra 3 : Categories of Sheaves*, Cambridge UP 1994.

* {#KS06} [[Masaki Kashiwara]], [[Pierre Schapira]], _Categories and Sheaves_ , Springer Heidelberg 2006.

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer Heidelberg 1994.

The characterizations of categories of presheaves are discussed in

* {#Bunge69} [[Marta Bunge]], *Relative functor categories and categories of algebras*, J. of Algebra 11, Issue 1 (1969), 64--101. [web](https://www.sciencedirect.com/science/article/pii/0021869369901021), [MR236238](https://mathscinet.ams.org/mathscinet-getitem?mr=236238) doi

* {#CarboniVitale98} [[Aurelio Carboni|A. Carboni]], [[Enrico Vitale|E. M. Vitale]], *Regular and exact completions*, JPAA 125 (1998), 79-116.

* {#CentazzoVitale04} C. Centazzo, [[Enrico Vitale|E. M. Vitale]], *Sheaf theory*, pp. 311-358 in Pedicchio, Tholen (eds.), *Categorical Foundations*, Cambridge UP 2004. ([draft](https://perso.uclouvain.be/enrico.vitale/chapter7.pdf))

The result about Artin gluings of presheaf toposes is due to Carboni and Johnstone,

* {#cj} [[Aurelio Carboni]] and [[Peter Johnstone]], *Connected limits, familial representability and Artin glueing*, Mathematical Structures in Computer Science, Volume 5, Issue 4 (December 1995), 441-459. ([link](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/connected-limits-familial-representability-and-artin-glueing/54E93903F7D7321B98B64AE7CB3E7AE0))

and is explained in section C.3 of Tom Leinster's book,

* {#Lein} [[Tom Leinster]], *Higher Categories, Higher Operads*, ([https://arxiv.org/abs/math/0305049](https://arxiv.org/abs/math/0305049)).

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

[[!redirects presheaf topos]]
[[!redirects presheaf toposes]]
[[!redirects presheaf topoi]]

[[!redirects Presheaves]]


category of representations

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

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of all [[representation]]s of [[algebra|algebraic]] structures $C$ of some kind.

## Definition

### Of groups, algebras, groupoids, algebroids, etc.

A typical such algebraic structure is a [[category]]. We may think of [[groups]] $G$ , and [[associative algebras]] $A$ as special cases of this by passing to their [[delooping]] $C = \mathbf{B}G$ or $C = \mathbf{B}A$. More generally $C$ may be a [[groupoid]] or [[algebroid]].

For all these cases, a [[representation]] of $C$ on [[objects]] in another [[category]] $D$ (for instance [[Set]] for [[permutation representations]] or  [[Vect]] for [[linear representations]]) is nothing but a [[functor]] $C \to D$.

In this case the **representation category** $Rep(C)$ is nothing but the [[functor category]]

$$
  Rep(C) = Func(C,D)
  \,.
$$

Notably when $G$ is a [[group]], an ordinary linear representation is a [[functor]] $\mathbf{B}G \to Vect$ from the [[delooping]] [[groupoid]] of $G$ to [[Vect]], and so the representation category is

$$
  Rep(G) = Func(\mathbf{B}G,Vect)
  \,.
$$

Often $C$ and $D$ are regarded as equipped with some extra [[structure]] (for instance [[topology]], [[smooth structure]]) and then the functors above are required to respect that structure.

### Higher and internal representations

In the context of [[homotopy theory]] and [[higher category theory]] there are analogous definitions of [[∞-representations]].

For $G$ an [[∞-group]] and $\mathbf{B}G$ its [[delooping]] [[∞-groupoid]], an $\infty$-representation on objects of some [[(∞,1)-category]] $D$ (such as that of [[(∞,n)-vector spaces]]) is the [[(∞,1)-category of (∞,1)-functors]]

$$
  Rep(G) = Func(\mathbf{B}G, D)
  \,.
$$

If $D  \simeq$ [[∞Grpd]] this are [[∞-permutation representations]] and by the [[(∞,1)-Grothendieck construction]] any such corresponds to an [[associated ∞-bundle]]

$$
  V \to V//G \to \mathbf{B}G
$$

over $\mathbf{B}G$ in such a way that we have an [[equivalence of (∞,1)-categories]]

$$
  Rep(G, \infty Grpd)
  \simeq
  \infty Grpd_{/\mathbf{B}G}
$$

with the [[over-(∞,1)-category]] of [[∞-groupoids]] over $\mathbf{B}G$.

This way of looking at categories of representations generalizes to every [[(∞,1)-topos]] $\mathbf{H}$ of [[homotopy dimension]] 0.

In this context any morphism $\rho : Q \to \mathbf{B}G$ encodes a representation of $G$ on the [[homotopy fiber]] $V$ of $\rho$, identifying $Q$ as $V//G$.

The assumption that $\mathbf{H}$ has [[homotopy dimension]] 0 guarantees that the homotopy fiber exists (since a global point $* \to \mathbf{B}G$ exists) and is well defined up to [[equivalence in an (∞,1)-category]].

## Properties

### Tannakian reconstruction

Representation categories come with [[forgetful functor]]s that send a representation to the underlying object that carries the representation.

For instance for group representations the canonical inclusion ${*} \to \mathbf{B}G$ induces the functor $Func(\mathbf{B}G,Vect) \to Func(*,Vect)$, hence

$$
  F : Rep(G) \to Vect
$$

that sends a representation to its underlying [[vector space]]. The [[Tannakian reconstruction theorem]] says that the group $G$ may be recovered essentially as the group of [[automorphism]]s of the fiber functor $F$.

## Related concepts

* [[GSet]]

* [[tensor product of representations]]

## References

The lecture notes

* _Monoidal Categories_ MIT course (2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec01.pdf))

list some basic examples of [[monoidal category|monoidal]] representation categories from page 7 on.

A standard textbook on [[representation theory]] of [[compact space|compact]] [[Lie groups]] is

* Theodor Bröcker, [[Tammo tom Dieck]], _Representations of compact Lie groups_ Graduate Texts in Mathematics, Springer (1985)

The stable [[triangulated category]] of representations of a [[finite group]]:

* [[Fernando Muro]], around slide 105 of _Triangulated categories_, 2009 [pdf](http://personal.us.es/fmuro/files/slides/htag.pdf)

category: category

[[!redirects categories of representations]]

[[!redirects GRep]]
[[!redirects G-Rep]]

[[!redirects representation category]]
[[!redirects representation categories]]


category of sheaves

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

#Contents#
* table of contents
{:toc}

## Idea

This entry is about the properties and the characterization of the category $Sh(S)$ of ([[set]]-valued) [[sheaf|sheaves]] on a ([[small category|small]]) [[site]] $S$, which is a [[Grothendieck topos]].  Among other things it gives a definition and a characterization of the notion of [[sheaf]] itself, but for more details on [[sheaf|sheaves]] themselves see there.

## Definition

Let $(C,J)$ be a [[site]]: a (small) [[category]] equipped with a [[coverage]].

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

The **category of sheaves** on $(C,J)$ is the [[full subcategory]] of the [[category of presheaves]]

$$
  i : Sh_J(C) \hookrightarrow
  PSh(C)
$$

of those presheaves which are [[sheaves]] with respect to $J$.

=--

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

Every category of sheaves is a [[reflective subcategory]]

$$
  (L \dashv i) : Sh_J(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  Psh(C)
  \,,
$$

hence a [[subtopos]] of the [[presheaf topos]].  Moreover, every such subtopos arises in this way: there is a [[bijection]] between [[Grothendieck topologies]] on $C$ and equivalence classes of [[geometric embeddings]] into $PSh(C)$.

=--

This appears for instance as ([Johnstone, corollary C.2.1.11](#Johnstone)).  See also [[Lawvere-Tierney topology]].

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

Details on the first statement are at [[sheafification]]. A full proof for the second statement is at [[(∞,1)-category of (∞,1)-sheaves]] (there proven in [[(∞,1)-category theory]], but the proof is verbatim the same in [[category theory]]).

=--

## Equivalent characterizations

### As localizations

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

The category of sheaves is equivalent to the [[homotopy category]] of the [[category with weak equivalences]] $PSh(C)$ with the weak equivalences given by $W = $[[local isomorphisms]]

$$
  Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[\text{local isomorphisms}]^{-1}
  \,.
$$

The converse is also true: for every [[left exact functor]] $L : PSh(S) \to PSh(S)$ (preserving finite limits) which is [[adjoint functor|left adjoint]] to the inclusion of its image, there is a Grothendieck topology on $S$ such that the image of $L$ is the category of sheaves on $S$ with respect to that topology.

=--

We spell out proofs of some of the above claims.

Let $C$ be a [[small category]] and

$$
  i : \mathcal{E} \hookrightarrow [C^{op}, Set]
$$

a [[reflective subcategory]], hence a [[full subcategory]] with a [[left adjoint]] $L : [C^{op}, Set] \to \mathcal{E}$, such that moreover $L$ preserves [[finite limit]]s.

Write $W := L^{-1}(isos) \subset Mor([C^{op}, Set])$ for the [[class]] of [[morphism]]s in $[C^{op}, Set]$ that are sent to [[isomorphism]]s by $L$.

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

A [[presheaf]] $A \in [C^{op}, Set]$ is in $\mathcal{E}$ (meaning: in the [[essential image]] of $i$) precisely if for all $f : X \to Y$ in $W$ the induced function

$$
  Hom(f,A) : Hom(Y,A) \to Hom(X,A)
$$

is a [[bijection]].

=--

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

If $A \simeq i \hat A$, then by the $(L \dashv i)$-[[adjunction]] [[isomorphism]] we have

$$
  Hom(f, i \hat A) \simeq \mathcal{E}(L(f), A)
  \,.
$$

But by assumption $L(f)$ is an [[isomorphism]], so the claim is immediate.

Conversely, if for all $f$ the function $Hom(f,A)$ is a bijection, define $\hat A := L(A)$ and let $ \epsilon_A : A \to i L(A)$ be the
$(L \dashv i)$-[[unit of an adjunction|unit]].

By the assumption that $i$ is a [[full and faithful functor]] and basic properties of [[adjoint functor]]s we have that the counit

$$
  L i \to Id
$$

is a [[natural isomorphism]]. By the [[zig-zag law]] the composite

$$
  L A \stackrel{L \epsilon_A}{\to} L i L A \stackrel{\simeq}{\to} L A
$$

is the [[identity]] and therefore $L \epsilon$ is an [[isomorphism]] and so $\epsilon_A$ is in $W$, under our assumption on $A$.

Using this it follows that

$$
  Hom(\epsilon_A, A) : Hom(i L A, A) \stackrel{\simeq}{\to} Hom(A,A)
$$

is an [[isomorphism]]. Write $k_A : i L A \to A$ for the preimage of $id_A$ under this isomorphism, which is therefore a [[left inverse]] of $\epsilon_A$. This immediately implies that also $k_A$ is in $W$, and so we can enter the same argument with $k_A$ to find that it has a left inverse itself. But this means that $k_A$ is in fact an [[isomorphism]] and hence so is $\epsilon A$, which thus exhibits $A$ as being in the essential image of $i$.

=--

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

A [[morphism]] $f : X \to Y$ is in $W$ precisely if for every morphism $z : j(c) \to Y$ with [[representable functor|representable]] [[domain]], the [[pullback]] $z^* f$ in

$$
  \array{
     X \times_Y j(c) &\to& X
     \\
     {}^{\mathllap{z^* f}}\downarrow && \downarrow^{\mathrlap{f}}
     \\
     j(c) &\stackrel{z}{\to}& Y
  }
$$

is in $W$.

=--

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

Assume first that $f$ is in $W$. Since by assumption $L$ preserves finite limits, it follows that

$$
  \array{
     L(X \times_Y j(c)) &\to& L X
     \\
     {}^{\mathllap{L(z^* f)}}\downarrow && \downarrow^{\mathrlap{L f}}
     \\
     L(j(c)) &\stackrel{L z}{\to}& L Y
  }
$$

is still a [[pullback]] diagram in $\mathcal{E}$ and hence that $L(z^* f)$ is the pullback of the [[isomorphism]] $L f$ and thus itself an isomorphism. Therefore $z^* f$ is in $W$.

Conversely, suppose that all these pullbacks are in $W$. Then use the "[[co-Yoneda lemma]]" to write the presheaf $Y$ as a [[colimit]] over all representables mapping into it

$$
  {\lim_\to}_{j(c_i) \stackrel{z_i}{\to} Y} j(c)  \stackrel{\simeq}{\to} Y
  \,.
$$

Forming the [[pullback]] along $f$, using that in a [[topos]] (such as our [[presheaf topos]]) colimits are preserved by pullbacks, we get

$$
  \array{
    {\lim_\to}_i f^* j(c_i) &\stackrel{\simeq}{\to}& X
    \\
    {}^{\mathllap{{\lim_\to}_i z_i^* f}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    {\lim_\to}_i j(c_i)
    &\stackrel{\simeq}{\to}&
    Y
  }
  \,.
$$

Since $L$ preserves all colimits and finite limits, we also get

$$
  \array{
    {\lim_\to}_i L(f^* j(c_i)) &\stackrel{\simeq}{\to}& L(X)
    \\
    {}^{\mathllap{{\lim_\to}_i L(z_i^* f)}}\downarrow && \downarrow^{\mathrlap{L(f)}}
    \\
    {\lim_\to}_i L(j(c_i))
    &\stackrel{\simeq}{\to}&
    L(Y)
  }
  \,.
$$

Since by assumption now all $L(z_i^* f )$ are isomorphisms, also ${\lim_\to}_i L(z_i^* f) $ is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also $L(f)$ is and hence $f$ is in $W$.

=--

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

The collection of [[sieve]]s in $W$, hence the collection of [[monomorphism]]s in $W$ whose [[codomain]] is a [[representable functor|representable]], constitute a [[Grothendieck topology]] on $C$.

=--

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

We check the list of axioms, given at [[Grothendieck topology]]:

1. _Pullbacks of covering sieves are covering_ :

   First of all, the pullback of a sieve along a morphism of representables is still a sieve, because [[monomorphism]]s are (as discussed there) stable under pullback.

   Next, since $L$ preserves [[finite limit]]s, $L$ applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in $W$.

1. _The maximal sieve is covering._ Clear: $L$ applied to an [[isomorphism]] is an [[isomorphism]].

1. _Two sieves cover precisely if their intersection covers._ This is again due to the [[pullback]]-stability of elements of $W$, due to the preservation of finite limits by $L$.

1. _If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering_ .

   This is the same argument as in the second part of the proof of
   prop. \ref{WeakEquivalencesDetectedOnRepresentableCodomains}.

=--

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

$\mathcal{E}$ is a [[Grothendieck topos]].

=--

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

By prop. \ref{LocalizationInducesGrothendieckTopology} and
prop. \ref{CharacterizationOfLocalObjects} we are reduced to showing that an object $A$ is in $\mathcal{E}$ already if for all _[[monomorphisms]]_ $f$ in $W$ the function $Hom(f,A)$ is a bijection.

(...)

=--

### As toposes
 {#AsToposes}

Categories of sheaves are examples of [[categories]] that are [[topos]]es: they are the [[Grothendieck topos]]es characterized among all toposes as those satisfying [[Grothendieck topos|Giraud's axioms]].

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

[[Grothendieck topos|Sheaf toposes]] are equivalently the [[geometric embedding|subtoposes]] of [[presheaf toposes]].

=--

This appears for instance as
([Johnstone, corollary C.2.1.11](#Johnstone)).

### As accessible reflective subcategories
 {#AsAccessibleReflectiveSubcategories}

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

[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]

=--

See also at _[[reflective sub-(∞,1)-category]]_.

## Properties

### Dependence on the site

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

For $(L \dashv i) : \mathcal{E} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}$ a category of sheaves and $(C,J)$ a [[site]] such that we have an equivalence of categories $\mathcal{E} \simeq Sh_J(C)$ we say that $C$ is a **site of definition** for the [[topos]] $\mathcal{E}$.

=--

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

There are always different [[site]]s $(C,J)$ whose categories of sheaves are [[equivalence of categories|equivalent]]. First of all for fixed $C$ and given a [[coverage]] $J$, the category of sheaves depends only on the [[Grothendieck topology]] generated by $J$. But there may be site structures also on inequivalent categories $C$ that have equivalent categories of sheaves.

=--

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

For $(C,J)$ a [[site]] with [[coverage]] $J$ and $D \to C$ any [[subcategory]], the **induced coverage** $J_D$ on $D$ has as [[covering]] [[sieve]]s the intersections of the covering sieves of $C$ with the morphisms in $D$.

=--

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

Let $(C,J)$ be a [[site]] (possibly [[large site|large]]). A [[subcategory]] $D \to C$ (not necessarily full) is called a **[[dense sub-site]]** with the [induced coverage](#InducedCoverage) $J_D$ if

1. every object $U \in C$ has a [[covering]] $\{U_i \to U\}$ in $J$ with all $U_i$ in $D$;

1. for every morphism $f : U \to d$ in $C$ with $d \in D$ there is a [[covering]] family $\{f_i : U_i \to U\}$ such that the composites $f \circ f_i$ are in $D$.

=--

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

If $D$ is a [[full subcategory]] then the second condition is automatic.

=--

+-- {: .num_theorem}
###### Theorem
**(comparison lemma)**

Let $(C,J)$ be a (possibly [[large site|large]]) [[site]] with $C$ a [[locally small category]] and let $f : D \to C$ be a [[small category|small]] [dense sub-site](#DenseSubSite). The pair of [[adjoint functor]]s

$$
  (f^* \dashv f_*)
    :
  PSh(D)
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
   PSh(C)
$$

with $f^*$ given by precomposition with $f$ and $f_*$ given by right [[Kan extension]] induces an [[equivalence of categories]] between the categories of sheaves

$$
  (f_* \dashv f^*) :
  Sh_{J_D}(D)
  \underoverset
    {\underset{f_*}{\to}}{\overset{f^*}
    {\leftarrow}}
    {\simeq}
  Sh_J{C}
  \,.
$$

=--

This appears as ([Johnstone, theorm C2.2.3](#Johnstone)).

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

* Let $X$ be a [[locale]] with [[frame]] $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the [[join]] of the $U_i$ is $U$). Let $bOp(X)$ be a [[basis for the topology]] of $X$: a complete join-[[semilattice]] such that every object of $Op(X)$ is the [[join]] of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.

  * For $X$ a [[locally contractible space]], $Op(X)$ its [[category of open subsets]] and $cOp(X)$ the full subcategory of [[contractible]] open subsets, we have that $cOp(X)$ is a dense sub-site.

* For $C = TopManifold$  the category of all [[paracompact topological space|paracompact]] [[topological manifold]]s equipped with the [[open cover]] coverage, the category [[CartSp]]${}_{top}$ is a dense sub-site: every paracompact manifold has a [[good open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]].

* Similaryl for $C = $ [[Diff]] the category of [[smooth manifold]]s equipped with the [[good open cover]] [[coverage]], the full subcategory [[CartSp]]${}_{smooth}$ is a dense sub-site.

=--

### Limits and colimits

Beware that, in general, [[constant functor|constant presheaves]] need not be [[sheaves]]. But the presheaf constant on the [[singleton]] set is always a sheaf:

\begin{example}\label{TerminalSheaf}
**(terminal sheaf)**
\linebreak
For every [[site]] $\mathcal{S}$, the [[terminal object]] in its [[category of sheaves]] $Sh(\mathcal{S})$ is, as a [[presheaf]], the [[constant functor]] with value the [[singleton set]].
\end{example}
\begin{proof}
  Recalling the defining [[geometric embedding]] of any [[Grothendieck topos]] into the corresponding [[category of presheaves]]
  $$
    Sh(\mathcal{S})
    \underoverset
      {\underset{i}{\hookrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\; \bot \;\;\;}
    PSh(\mathcal{S})
    \,,
  $$
  both functors [[preserved limit|preserve]] the [[terminal object]] (this being the [[limit]] over the [[empty diagram]]): The [[direct image]]  because it is a [[right adjoint]] and [[right adjoints preserve limits]], and the [[inverse image]] because [[sheafification preserves finite limits]].

But the terminal presheaf is clearly the functor $\const_\ast \,\in\, PSh(\mathcal{S})$ which is [[constant functor|constant]] on the singleton set. Therefore $L(const_\ast)$ is the terminal sheaf. But then its [[underlying]] presheaf $i\big(L(const_\ast)\big)$ must again be the terminal presheaf, hence still the functor constant on the singleton set.
\end{proof}

### Epi- mono- and isomorphisms
 {#EpiMonoIsomorphisms}

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

Let $\mathcal{C}$ be a [[small site]] and let $Sh(\mathcal{C})$ be its category of sheaves. Let $f \colon X \to Y$ be a homomorphism of sheaves, hence a morphism in $Sh(\mathcal{C})$. Then:

1. $f$ is a [[monomorphism]] or [[isomorphism]] precisely if it is so _globally_ in that for each object $U \in \mathcal{C}$ in the site, then the component $f_U \colon X(U) \to Y(U)$ is an [[injection]] or [[bijection]] of [[sets]], respectively.

1.  $f$ is an [[epimorphism]] precisely if it is so _locally_, in that: for all $U \in C$ and every element $y \in Y(U)$ there is a [[covering]] $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ the element $Y(p_i)(y)$ is in the image of $f(U_i) : X(U_i) \to Y(U_i)$.

But if $\{x_i\}_{i \in I}$ is a [[set]] of [[points of a topos]] for $Sh(\mathcal{C})$ such that these are _enough points_ ([def.](point+of+a+topos#EnoughPoints)) then the morphism $f$ is epi/mono/iso precisely it is is so an all [[stalks]], hence precisely if

$$
  x_i^\ast f \;\colon\; x_i^\ast X \longrightarrow x_i^\ast Y
$$

is a surjection/injection/bijection of sets, respectively, for all $i \in I$.

=--

### Exactness properties

Every sheaf topos satisfies the following [[exactness properties]]. it is an

* [[extensive category]];

* [[adhesive category]];

* [[exhaustive category]].

## Related concepts

* [[coverage]]

  [[trivial coverage]]

* **category of sheaves**

* [[category of sheaves on a topological space]]

* [[(∞,1)-category of (∞,1)-sheaves]].

[[!include locally presentable categories - table]]

## References

* {#Borceux94} [[Francis Borceux]], vol 3 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* {#Johnstone} [[Peter Johnstone]], sections A.4 and C.2 in _[[Elephant|Sketches of an Elephant]]_

* {#StacksProject} [[The Stacks Project]], _Sites and sheaves_ ([pdf](http://stacks.math.columbia.edu/download/sites.pdf))

The characterization of sheaf toposes and Grothendieck topologies in terms of left exact [[reflective subcategory|reflective subcategories]] of a presheaf category is also in

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.

There is also a direct proof in [[Shane Kelly]], _What is the relationship between Grothendieck pretopologies and Grothendieck topologies?_ ([web] (https://www.ms.u-tokyo.ac.jp/~kelly/pdfs/grothendieckTop.pdf)) .

The characterization of $Sh(S)$ as the [[homotopy category]] of $PSh(S)$ with respect to local isomorphisms is emphasized at the beginning of the text

* [[Bertrand Toen]], _Stacks and non-abelian cohomology_ ([web](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html)) .

Details are in

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_ (2006)

It's a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:

* exercise 16.7 shows that [[sheafification]] inverts precisely the [[local isomorphism]]s, so that in particular every [[local isomorphism]] between [[sheaf|sheaves]] is an [[isomorphism]];

* lemma 16.3.2 states that the unit of the [[adjoint functor|adjunction]] $Id_{PSh(S)} \rightarrow i \circ \bar{(-)} : PSh(S) \to PSh(S)$ is componentwise a [[local isomorphism]];

* using this corollary 7.2.2 says that $Sh(S) \simeq Ho_{PSh(S)}$ with the [[homotopy category]] $Ho_{PSh(S)}$ formed using [[local isomorphism]]s as weak equivalences.

Letcure notes:

* [[Pierre Schapira]], *An Introduction to Categories and Sheaves*, lecture notes (2023) [[pdf](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf)]

[[!redirects sheaf category]]
[[!redirects categories of sheaves]]
[[!redirects topos of sheaves]]
[[!redirects topoi of sheaves]]
[[!redirects toposes of sheaves]]