Contents

topos theory

category theory

# Contents

## 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

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

###### Proposition

$PSh(\mathcal{C})$ has all limits and colimits (over small diagrams), and these are computed objectwise: For $\mathcal{I}$ any diagram (a small category) we have:

$X_{\bullet} \,\colon\, \mathcal{I} \longrightarrow PSh(\mathcal{C}) \;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\; \underset{c \in Obj(\mathcal{C})}{\forall} \;\;\;\; \big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} X_i \big) (c) \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} \big( X_i(c) \big) \;\;\;\;\text{and}\;\;\;\; \big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} X_i \big) (c) \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} \big( X_i(c) \big)$

###### Proof

The defining universal property readily follows from that of the objectwise (co-)limits.

###### Proposition

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.

###### Proof

Using for instance the characterization of epimorphisms by pushouts (this Prop.) and of monomorphism by pullbacks (this Prop.), the statement follows by Prop. .

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

###### Proposition

$PSh(\mathcal{C})$ is a cartesian closed category.

###### Proof

This is spelled out at closed monoidal structure on presheaves.

###### Proposition

$PSh(\mathcal{C})$ is a topos.

### Characterization

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

###### Theorem

A category $E$ is equivalent to a presheaf topos if and only if it is cocomplete, atomic, and regular.

A proof as well as a second characterization using exact completions can be found in Carboni-Vitale (1998) or Centazzo-Vitale (2004). The first paper has also an interesting comparison to a classical characterization of categories monadic over Set.

### Presheaves on over-categories and over-categories of presheaves

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 presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the Yoneda embedding.

###### Proposition

There is an equivalence of categories

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

$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.$
###### 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.

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

###### 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 can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

###### 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). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).

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

### Artin gluings

The statements of the preceding subsection may be generalized further:

###### 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

The functor $T$ is familially representable: there is a functor $D^{op} \to Fam(Set^{C^{op}})$ into the small 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. The result is due to Carboni and Johnstone.

In passing, we note that the small coproduct cocompletion $Fam(Set^{C^{op}})$ is itself a presheaf topos: there are 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), aka the cone of $C$.

### Finite presheaves

A finite presheaf on a category $C$ is a functor $C^{op}\to FinSet$ valued in the 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

###### 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.

### Models in presheaf toposes

See at models in presheaf toposes.

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

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$$\phantom{A}$toposes$\phantom{A}$locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

A classical (advanced) reference is exposé 1 of

An elementary introduction to presheaf toposes emphasizing finite underlying categories $C$ is

• M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.

Standard references are

The characterizations of categories of presheaves are discussed in

• A. Carboni, E. M. Vitale, Regular and exact completions , JPAA 125 (1998) pp.79-116.

• C. Centazzo, E. M. Vitale, Sheaf theory , pp.311-358 in Pedicchio, Tholen (eds.), Categorical Foundations , Cambridge UP 2004. (draft)

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

• 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)

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