nLab geometry of physics - basic notions of topos theory

Basic notions of Topos theory

We have explained in Remark how presheaves on a category $\mathcal{C}$ may be thought of as generalized spaces probe-able by the objects of $\mathcal{C}$, and that two consistency conditions on this interpretation are provided by the Yoneda lemma (Prop. ) and the resulting Yoneda embedding (Prop. ). Here we turn to a third consistency condition that one will want to impose, namely a locality or gluing condition (Remark below), to be called the sheaf condition (Def. below).

More in detail, we had seen that any category of presheaves $[\mathcal{C}^{op}, Set]$ is the free cocompletion of the given small category $\mathcal{C}$ (Prop. ) and hence exhibits generalized spaces $\mathbf{X} \in [\mathcal{C}^{op}, Set]$ as being glued or generated form the “ordinary spaces” $X \in \mathcal{C}$. Further conditions to be imposed now will impose relations among these generators, such as the locality relation embodied by the sheaf-condition.

It turns out that these relations are reflected by special properties of an adjunction (Def. ) that relates generalized spaces to ordinary spaces:

generalized spaces via generators and relations:

$\phantom{A}$free cocompletion$\phantom{A}$ $\phantom{A}=$presheaves$\phantom{A}$	$\phantom{A}$loc. presentable category$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{AAAA}$
$\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq} [\mathcal{C}^{op},Set]$	$\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]$	$\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]$
$\phantom{A}$Prop. $\phantom{A}$	$\phantom{A}$Def. $\phantom{A}$	$\phantom{A}$Prop. $\phantom{A}$
$\phantom{A}$simplicial presheaves$\phantom{A}$	$\phantom{A}$combinatorial model category$\phantom{A}$	$\phantom{A}$model topos$\phantom{A}$
$\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq_{Qu}} [\mathcal{C}^{op},sSet_{Qu}]_{proj}$	$\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$	$\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$
$\phantom{A}$Example	$\phantom{A}$Def.	$\phantom{A}$Def.

$\,$

Remark

(sheaf condition as local-to-global principle for generalized spaces)

If the objects of $\mathcal{C}$ are thought of as spaces of sorts, as in Remark , then there is typically a notion of locality in these spaces, reflected by a notion of what it means to cover a given space by (“smaller”) spaces (a coverage, Def. below).

But if a space $X \in \mathcal{C}$ is covered, say by two other spaces $U_1, U_2 \in \mathcal{C}$, via morphisms

$$\array{ U_1 && && U_2 \\ & {}_{\mathllap{i_1}}\searrow && \swarrow_{\mathrlap{i_2}} \\ && X }$$

then this must be reflected in the behaviour of the probes of any generalized space $\mathbf{Y}$ (in the sense of Remark ) by these test spaces:

For ease of discussion, suppose that there is a sense in which these two patches above intersect in $X$ to form a space $U_1 \cap_X U_2 \in \mathcal{C}$. Then locality of probes should imply that the ways of mapping $U_1$ and $U_2$ into $\mathbf{Y}$ such that these maps agree on the intersection $U_1 \cap_X U_2$, should be equivalent to the ways of mapping all of $X$ into $\mathbf{Y}$.

$$\text{locality} \;:\; \left\{ \array{ \text{maps from}\,U_1\,\text{and}\,U_2\,\text{to}\,\mathbf{Y} \\ \text{that coincide on}\,U_1 \cap_X U_2 } \right\} \;\simeq\; \left\{ \text{maps from}\,X\,\text{into}\,\mathbf{Y} \right\}$$

One could call this the condition of locality of probes of generalized spaces probeable by objects of $\mathcal{C}$. But the established terminology is that this is the sheaf condition (6) on presheaves over $\mathcal{C}$. Those presheaves which satisfy this condition are called the sheaves (Def. below).

Remark

Warning

Most (if not all) introductions to sheaf theory insist on motivating the concept from the special case of sheaves on topological spaces (Example below). This is good motivation for what Grothendieck called “petit topos”-theory. The motivation above, instead, naturally leads to the “gros topos”-perspective, as in Example below, which is more useful for discussing the synthetic higher supergeometry of physics. In fact, this is the perspective of functorial geometry that has been highlighted since Grothendieck 65, but which has maybe remained underappreciated.

$\,$

We now first introduce the sheaf-condition (Def. ) below in its traditional form via “matching families” (Def. below). Then we show (Prop. below) how this is equivalently expressed in terms of Cech groupoids (Example below). This second formulation is convenient for understanding and handling various constructions in ordinary topos theory (for instance the definition of cohesive sites) and it makes immediate the generalization to higher topos theory.

$\,$

Descent

Here we introduce the sheaf-condition (Def. below) in its component-description via matching families (Def. below). Then we consider some of the general key properties of the resulting categories of sheaves, such as notably their “convenience”, in the technical sense of Prop. below.

$\,$

Definition

(coverage and site)

Let $\mathcal{C}$ be a small category (Def. ). Then a coverage on $\mathcal{C}$ is

• for each object $X \in \mathcal{C}$ a set of indexed sets of morphisms into $X$

  $$\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$$

  called the coverings of $X$,

such that

• for every covering $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of $X$ and every morphism $Y \overset{f}{\longrightarrow} X$ there exists a refining covering $\left\{ V_j \overset{\iota_j}{\to} Y \right\}_{j \in J}$ of $Y$, meaning that for each $j \in J$ there exists $i \in I$ and a morphism $V_j \overset{\iota_{j,i}}{\to} U_i$ such that

  (1)$$f \circ \iota_j \;=\; \iota_i \circ \iota_{j,i} \phantom{AAAAAAA} \array{ V_j &\overset{\iota_{j,i}}{\longrightarrow}& U_i \\ {}^{\mathllap{ \iota_j }}\big\downarrow && \big\downarrow{}^{\mathrlap{ \iota_i}} \\ Y &\underset{f}{\longrightarrow}& X }$$

A small category $\mathcal{C}$ equipped with a coverage is called a site.

Example

(canonical coverage on topological spaces)

The category Top of (small) topological spaces (Example ) carries a coverage (Def. ) whose coverings are the usal open covers of topological spaces.

The condition (1) on a coverage is met, since the preimages of open subsets under a continuous function $f$ are again open subsets, so that the preimages of an open cover consistitute an open cover of the domain, such that the commuting diagram-condition (1) is immediage.

Similarly, for $X \in Top$ a fixed topological space, there is the site $Op(X)$ whose underlying category is the category of opens of $X$, which is the thin category (Example ) of open subsets of $X$ and subset inclusions, and whose coverings are again the open covers.

Example

(differentiably good open covers of smooth manifolds)

The category SmthMfd of smooth manifold (Example ) carries a coverage (Def. ), where for $X \in SmthMfd$ any smooth manifold of dimension $D \in \mathbb{N}$, its coverings are collections of smooth functions from the Cartesian space $\mathbb{R}^D$ to $X$ whose image is the inclusion of an open ball.

Hence these are the usual open covers of $X$, but with the extra condition that every patch is diffeomorphic to a Cartesian space (hence to a smooth open ball).

One may further constrain this and ask that also all the non-empty finite intersections of these open balls are diffeomorphic to open balls. These are the differentiably good open covers.

To see that these coverings satisfy the condition (1): The plain pullback of an open cover along any continuous function is again an open cover, just not necessarily by patches diffeomorphic to open balls. But every open cover may be refined by one that is (see at good open cover), and this is sufficient for (1).

Example is further developed in the chapters smooth sets and on smooth homotopy types.

Definition

(matching family – descent object)

Let $\mathcal{C}$ be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example ) over $\mathcal{C}$.

Given an object $X \in \mathcal{C}$ and a covering $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it (Def. ) we say that a matching family (of probes of $\mathbf{Y}$) is a tuple $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and pairs of morphisms $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying

(2)$$\iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }$$

we have

(3)$$\mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \,.$$

We write

(4)$$Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big) \subset \underset{i}{\prod} \mathbf{Y}(U_i) \;\in\; Set$$

for the set of matching families for the given presheaf and covering.

This is also called the descent object of $\mathbf{Y}$ for descent along the covering $\{U_i \overset{\iota_i}{\to}X\}$.

Example

(matching families that glue)

Let $\mathcal{C}$ be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example ) over $\mathcal{C}$.

Given an object $X \in \mathcal{C}$ and a covering $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it (Def. ), then every element

$$\phi \;\in\; \mathbf{Y}(X)$$

induces a matching family (Def. ) by

$$\big( \mathbf{Y}(\iota_i)(\phi) \big)_{i \in I} \,.$$

(That this indeed satisfies the matching condition follows immediately by the functoriality of $\mathbf{Y}$.)

This construction provides a function of the form

(5)$$\mathbf{Y}(X) \longrightarrow Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big)$$

The matching families in the image of this function are hence those tuples of probes of $\mathbf{Y}$ by the patches $U_i$ of $X$ which glue to a global probe out of $X$.

Definition

(sheaves and sheaf toposes)

Let $\mathcal{C}$ be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example ) over $\mathcal{C}$.

The presheaf $\mathbf{Y}$ is called a sheaf if for every object $X \in \mathcal{C}$ and every covering $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of $X$ all matching families glue uniquely, hence if the comparison morphism (5) is a bijection

(6)$$\mathbf{Y}(X) \overset{\simeq}{\longrightarrow} Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big) \,.$$

The full subcategory (Example ) of the category of presheaves over a given site $\mathcal{C}$, on those that are sheaves is the category of sheaves, denoted

(7)$$Sh(\mathcal{C}) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} [\mathcal{C}^{op}, Set] \,.$$

A category which is equivalent (Def. ) to a category of sheaves is called a sheaf topos, or often just topos, for short.

For $\mathbf{H}_1$ and $\mathbf{H}_2$ two such sheaf toposes, a homomorphism $f \;\colon\; \mathbf{H}_1 \to \mathbf{H}_2$ between them, called a geometric morphism is an adjoint pair of functors (Def. )

(8)$$\mathbf{H}_1 \underoverset {\underset{ \phantom{AA} f_\ast \phantom{AA} }{\longrightarrow}} \overset{ \phantom{AA} f^\ast \phantom{AA} }{\longleftarrow} {} \mathbf{H}_2$$

such that

• the left adjoint $f^\ast$, called the inverse image, preserves finite products.

Hence there is a category Topos, whose objects are sheaf toposes and whose morphisms are geometric morphisms.

Example

(global sections geometric morphism)

Let $\mathbf{H}$ be a sheaf topos (Def. ). Then there is a geometric morphism (8) to the category of sets (Example ), unique up to natural isomorphism (Def. ):

$$\mathbf{H} \underoverset {\underset{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow}} {\overset{L}{\hookleftarrow}} {\bot} Set \,.$$

Here $\Gamma$ is called the global sections-functor.

Proof

Notice that every set $S \in Set$ is the coproduct, indexed by itself, of the terminal object $\ast \in Set$ (the singleton):

$$S \;\simeq\; \underset{s \in S}{\coprod} \ast \,.$$

Since $L$ is a left adjoint, it preserves this coproduct (Prop. ). Moreover, since $L$ is assumed to preserve finite products, and since the terminal object is the empty product (Example ), it also preserves the terminal object. Therefore $L$ is fixed, up to natural isomorphism, to act as

$$\array{ L(S) & \simeq L \left( \underset{s \in S}{\coprod} \ast \right) \\ & \simeq \underset{s \in S}{\coprod} L(\ast) \\ & \simeq \underset{s \in S}{\coprod} \ast } \,.$$

This shows that $L$ exists and uniquely so, up to natural isomorphism. This implies the essential uniqueness of $\Gamma$ by uniqueness of adjoints (Prop. ).

Example

(trivial coverage)

For $\mathcal{C}$ a small category (Def. ), the trivial coverage on it is the coverage (Def. ) with no covering families at all, meaning that the sheaf condition (Def. ) over the resulting site is empty, in that every presheaf is a sheaf for this coverage.

Hence the category of presheaves $[\mathcal{C}^{op},Set]$ (Example ) over a site $\mathcal{C}_{triv}$ with trivial coverage is already the corresponding category of sheaves, hence the corresponding sheaf topos:

$$Sh\left( \mathcal{C}_{triv}\right) \;\simeq\; [\mathcal{C}^{op}, Set] \,.$$

Example

(sheaves on the terminal category are plain sets)

Consider the terminal category $\ast$ (Example ) equipped with its trivial coverage (Example ). Then there is a canonical equivalence of categories (Def. ) between the category of sheaves on this site (Def. ) and the category of sets (Example ):

$$Sh(\ast) \;\simeq\; Set \,.$$

Hence the category of sets is a sheaf topos.

Example

(sheaves on a topological space – spatial petit toposes)

In the literature, the concept of (pre-)sheaf (Def. ) is sometimes not defined relative to a site, but relative to a topological space. But the latter is a special case: For $X$ a topological space, consider its category of open subsets $Op(X)$ from Example , with coverage given by the usual open covers. Then a “sheaf on this topological space” is a sheaf, in the sense of Def. , on this site of opens. One writes

$$Sh(X) \;\coloneqq\; Sh(Op(X)) \overset{\phantom{AA}}{\hookrightarrow} [Op(X)^{op}, Set] \,,$$

for short. The sheaf toposes arising this way are also called spatial toposes.

Proposition

(localic reflection)

The construction of categories of sheaves on a topological space (Example ) extends to a functor from the category Top of topological spaces and continuous functions between them (Example ) to the category Topos of sheaf toposes and geometric morphisms between them (Example ).

$$Sh(-) \;\colon\; Top \longrightarrow Topos \,.$$

Moreover, when restricted to sober topological spaces, this becomes a fully faithful functor, hence a full subcategory-inclusion (Def. )

$$Sh(-) \;\colon\; SoberTop \overset{\phantom{AAA}}{\hookrightarrow} Topos \,.$$

More generally, this holds for locales (i.e. for “sober topological spaces not necessarily supported on points”), in which case it becomes a reflective subcategory-inclusion (Def. )

$$Locale \underoverset {\underset{\phantom{AA} Sh(-) \phantom{AA} }{\hookrightarrow}} {\overset{\phantom{AAAA}}{\longleftarrow}} {\bot} Topos$$

This says that categories of sheaves on topological spaces are but a reflection of soper topological spaces (generally: locales) and nothing more, whence they are also called petit toposes.

Example

(abelian sheaves)

In the literature, sometimes sheaves are understood by default as taking values not in the category of sets, but in the category of abelian groups. Combined with Example this means that some authors really mean “sheaf of abelian groups of the site of opens of a topological space”, when they write just “sheaf”.

But for $\mathcal{S}$ any mathematical structure, a sheaf of $\mathcal{S}$-structured sets is equivalently an $\mathcal{S}$-structure internal to the category of sheaves according to Def. . In particular sheaves of abelian groups are equivalently abelian group objects in the category of sheaves of sets as discussed here.

Example

(smooth sets)

Consider the site SmthMfd of all smooth manifolds, from Example . The category of sheaves over this (Def. ) is equivalent to the category of smooth sets, discussed in the chapter geometry of physics – smooth sets:

$$Sh(SmthMfd) \;\simeq\; SmoothSet \,.$$

This is a gros topos, in a sense made precise by Def. below (a cohesive topos).

Remark

(ordinary spaces and their coverings are generators and relations for generalized spaces)

Given a site $\mathcal{C}$ (Def. ), then its presheaf topos $[\mathcal{C}^{op}, Set]$ (Example ) is the free cocompletion of the category $\mathcal{C}$ (Prop. ), hence the category obtained by freely forming colimits (“gluing”) of objects of $\mathcal{C}$.

In contrast, the full subcategory inclusion $Sh(\mathcal{C}) \hookrightarrow [\mathcal{C}^{op}, Set]$ enforces relations between these free colimits.

Therefore in total we may think of a sheaf topos $Sh(\mathcal{C})$ as obtained by generators and relations from the objects of its site $\mathcal{C}$:

• the objects of $\mathcal{C}$ are the generators;

• the coverings of $\mathcal{C}$ are the relations.

Proposition

(sheafification and plus construction)

Let $\mathcal{C}$ be a site (Def. ). Then the full subcategory-inclusion (7) of the category of sheaves over $\mathcal{C}$ (Def. ) into the category of presheaves (Example ) has a left adjoint (Def. ) called sheafification

$$Sh(\mathcal{C}) \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{ L }{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Set] \,.$$

An explicit formula for sheafification is given by applying the following “plus construction” twice:

$$L(\mathbf{Y}) \simeq (\mathbf{Y}^+)^+ \,.$$

Here the plus construction

$$(-)^+ \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow [\mathcal{C}^{op}, Set]$$

is given by forming equivalence classes of sets of matching families (Def. ) for all possible covers (Def. )

$$\mathbf{Y}^+(X) \;\coloneqq\; \left\{ \{U_i \overset{\iota_i}{\to} X\} \; \text{covering} \;, \phi \in Match\left( \{U_i\}, \mathbf{Y} \right) \right\}/\sim$$

under the equivalence relation which identifies two such pairs if the two covers have a joint refinement such that the restriction of the two matching families to that joint refinement coincide.

$\,$

Example

(induced coverage)

Let $\mathcal{C}$ be a site (Def. ). Then a full subcategory (Def. )

$$\mathcal{D} \hookrightarrow \mathcal{C}$$

becomes a site itself, whose coverage consists of those coverings $\{U_i \overset{\iota_i}{\to} Y\}$ in $\mathcal{C}$ that happen to be in $\mathcal{D} \hookrightarrow \mathcal{C}$.

Definition

(dense subsite)

Let $\mathcal{C}$ and $\mathcal{D}$ be sites (Def. ) with a a full subcategory-inclusion (Def. )

$$\mathcal{D} \hookrightarrow \mathcal{C}$$

and regard $\mathcal{D}$ as equipped with the induced coverage (Def. ).

This is called a dense subsite-inclusion if every object $X \in \mathcal{C}$ has a covering $\{U_i \overset{\iota_i}{\to} X\}_i$ such that for all $i$ the patches are in the subcategory:

$$U_i \;\in\; \mathcal{D} \hookrightarrow \mathcal{C} \,.$$

Proposition

(comparison lemma)

Let $\mathcal{D} \overset{\iota}{\hookrightarrow} \mathcal{C}$ be a dense subsite inclusion (def. ). Then precomposition with $\iota$ induces an equivalence of categories (Def. ) between their categories of sheaves (Def. ):

$$\iota^\ast \;\colon\; Sh(\mathcal{C}) \overset{\simeq}{\longrightarrow} Sh(\mathcal{D})$$

Proposition

(recognition of epi-/mono-/isomorphisms of sheaves)

Let $\mathcal{C}$ be a site (Def. ) with $Sh(\mathcal{C})$ its category of sheaves (Def. ).

Then a morphisms $f \;\colon\; \mathbf{X} \to \mathbf{Y}$ in $Sh(\mathcal{C})$ is

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

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

Proposition

(epi/mono-factorization through image)

Let $Sh(\mathcal{C})$ be a category of sheaves (Def. ). Then every morphism $f \;\colon\; \mathbf{X} \to \mathbf{Y}$ factors as an epimorphism followed by a monomorphism (Def. ) uniquely up to unique isomorphism:

$$f \;\colon\; \mathbf{X} \overset{epi}{\longrightarrow} im(f) \overset{mono}{\longrightarrow} \mathbf{Y} \,.$$

Theobject $im(f)$, as a subobject of $\mathbf{Y}$, is called the image of $f$.

In fact this is an orthogonal factorization system, in that for every commuting square where the left morphism is an epimorphism, and the right one a monomorphism, there exists a unique lift:

(9)$$\array{ A &\overset{\phantom{AAA}}{\longrightarrow}& B \\ {}^{\mathllap{epi}}\big\downarrow &{}^{\exists!}\nearrow& \big\downarrow^{\mathrlap{mono}} \\ C &\underset{\phantom{AAA}}{\longrightarrow}& D }$$

This implies that this is a functorial factorization, in that for every commuting square

$$\array{ \mathbf{X}_1 &\overset{f_1}{\longrightarrow}& \mathbf{Y}_1 \\ \big\downarrow && \big\downarrow \\ \mathbf{X}_2 &\underset{f_2}{\longrightarrow}& \mathbf{Y}_2 }$$

there is an induced morphism of images such that the resulting rectangular diagram commutes:

$$\array{ \mathbf{X}_1 &\overset{epi}{\longrightarrow}& im(f_1) &\overset{mono}{\longrightarrow}& \mathbf{Y}_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ \mathbf{X}_2 &\overset{epi}{\longrightarrow}& im(f_2) &\overset{mono}{\longrightarrow}& \mathbf{Y}_2 }$$

$\,$

We discuss some of the key properties of sheaf toposes:

Proposition

(sheaf toposes are cosmoi)

Let $\mathcal{C}$ be a site (Def. ) and $Sh(\mathcal{C})$ its sheaf topos (Def. ). Then:

1. All limits exist in $Sh(\mathcal{C})$ (Def. ), and they are computed as limits of presheaves, via Example :

   $$\iota\left( \underset{\underset{d}{\longleftarrow}}{\lim} \mathbf{X}_d \right) \;\simeq\; \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d)$$

2. All colimits exist in $Sh(\mathcal{C})$ (Def. ) and they are given by the sheafification (Def. ) of the same colimits computed in the category of presheaves, via Example :

   $$\underset{\underset{d}{\longrightarrow}}{\lim} \mathbf{X}_d \;\simeq\; L\left( \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d) \right)$$

3. The cartesian (Example ) closed monoidal category-structure (Def. ) on the category of presheaves $[\mathcal{C}^{op}, Set]$ from Example restricts to sheaves:

   $$Sh(\mathcal{C}) \underoverset {\underset{[\mathbf{X}, -]}{\longrightarrow}} {\overset{\mathbf{X} \times (-)}{\longleftarrow}} {} Sh(\mathcal{C})$$

   In particular, for $\mathbf{X}, \mathbf{Y} \in Sh(\mathcal{C})$ two sheaves, their internal hom $[\mathbf{X}, \mathbf{Y}] \in Sh(\mathcal{C})$ is a sheaf given by

   $$[\mathbf{X}, \mathbf{Y}] \;\colon\; U \;\mapsto\; Hom_{Sh(\mathcal{C})}( y(U) \mathbf{X}, \mathbf{Y} ) \,,$$

   where $y(U)$ is the presheaf represented by $U \in \mathcal{C}$ (Example ).

This may be summarized by saying that every sheaf topos (in particular every category of presheaves, by Example ) is a cosmos for enriched category theory (Def. ).

Definition

(local epimorphism)

Let $\mathcal{C}$ be a site (Def. ). Then a morphism of presheaves over $\mathcal{C}$ (Example )

$$\mathbf{Y} \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{S}^{op}, Set]$$

is called a local epimorphism if for every object $U \in \mathcal{C}$, every morphism $y(U) \longrightarrow \mathbf{X}$ out of its represented presheaf (Example ) has the local lifting property through $f$ in that there is a covering $\big\{ U_i \overset{\iota_i}{\to} U \big\}$ (Def. ) and a commuting diagram of the form

$$\array{ y(U_i) &\overset{\phantom{AA}\exists\phantom{AA}}{\longrightarrow}& \mathbf{Y} \\ {}^{\mathllap{y(\iota_i)}} \Big\downarrow && \Big\downarrow{}^{\mathrlap{ f }} \\ y(U) &\underset{\phantom{AAAA}}{\longrightarrow}& \mathbf{X} }$$

$\,$

Codescent

In order to understand the sheaf condition (6) better, it is useful to consider Cech groupoids (Def. below). These are really presheaves of groupoids (Def. below), a special case of the general concept of enriched presheaves. The key property of the Cech groupoid is that it co-represents the sheaf condition (Prop. below). It is in this incarnation that the concept of sheaf seamlessly generalizes to homotopy theory via “higher stacks”.

$\,$

Definition

(presheaves of groupoids)

For $\mathcal{C}$ a small category (Def. ) consider the functor category (Example ) from the opposite category of $\mathcal{C}$ (Example ) to the category Grpd of small groupoids (Example )

$$[\mathcal{C}^{op}, Grpd] \,.$$

By Example we may regard Grpd as a cosmos for enriched category theory. Since the inclusion $Set \hookrightarrow Grpd$ (Example ) is a strong monoidal functor (Def. ) of cosmoi (Example ), the plain category $\mathcal{C}$ may be thought of as a Grpd-enriched category (Def. ) and hence a functor $\mathcal{C}^{op} \to Grpd$ is equivalently a Grpd-enriched functor (Def. ).

This means that the plain category of functors $[\mathcal{C}^{op}, Grpd]$ enriches to Grpd-enriched category of Grpd-enriched presheaves (Example ).

Hence we may speak of presheaves of groupoids.

Remark

(presheaves of groupoids as internal groupoids in presheaves)

From every presheaf of groupoids $\mathbf{Y} \in [\mathcal{C}^{op}, Grpd]$ (Def. ), we obtain two ordinary presheaves of sets (Def. ) called the

• presheaf of objects

  $$Obj_{\mathbf{Y}(-)} \in [\mathcal{C}^{op}, Set]$$

• the presheaf of morphisms

  $$Mor_{\mathbf{Y}(-)} \;\coloneqq\; \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod} Hom_{{\mathbf{Y}(-)}} \;\colon\; [\mathcal{C}^{op}, Set]$$

In more abstract language this assignment constitutes an equivalence of categories

(10)$$\array{ [\mathcal{C}^{op}, Grpd] &\overset{\simeq}{\longrightarrow}& Grpd\left( [\mathcal{C}^{op}, Grpd]\right) \\ \mathbf{Y} &\mapsto& \left( \phantom{AAA} \array{ \underset{ Mor_{\mathbf{Y}(-)} }{ \underbrace{ \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod} Hom_{{\mathbf{Y}(-)}} }} \\ {}^{\mathllap{ \array{(x \overset{f}{\to}y) \\ \mapsto\\ x} } }\Big\downarrow \;\;\;\; \Big\uparrow^{\mathrlap{ \array{ x \\ \mapsto \\ x \overset{id_x}{\to} x } }} \phantom{x \overset{id_x}{\to} } \Big\downarrow^{ \mathrlap{ \array{ (x \overset{f}{\to} y) \\ \mapsto \\ y } } } \\ Obj_{\mathbf{Y}(-)} } \phantom{AAA} \right) } \,.$$

from presheaves of groupoids to internal groupoids- in the category of presheaves over $\mathcal{C}$ (Def. ).

Example

(presheaves of sets form reflective subcategory of presheaves of groupoids)

Let $\mathcal{C}$ be a small category (Def. ). There is the reflective subcategory-inclusion (Def. ) of the category of presheaves over $\mathcal{C}$ (Example ) into the category of presheaves of groupoids over $\mathcal{C}$ (Def. )

$$[\mathcal{C}^{op}, Set] \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{\pi_0}{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Grpd]$$

which is given over each object of $\mathcal{C}$ by the reflective inclusion of sets into groupoids (Example ).

Example

(Cech groupoid)

Let $\mathcal{C}$ be a site (Def. ), and $X \in \mathcal{C}$ an object of that site. For each covering family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given coverage, the Cech groupoid is the presheaf of groupoids (Def. )

$$C(\{U_i\}) \;\in\; [\mathcal{C}^{op}, Grpd] \;\simeq\; Grpd\left( [\mathcal{C}^{op}, Set] \right)$$

which, regarded as an internal groupoid in the category of presheaves over $\mathcal{C}$, via (10), has as presheaf of objects the coproduct

$$Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod} y(U_i)$$

of the presheaves represented (under the Yoneda embedding, Prop. ) by the covering objects $U_i$, and as presheaf of morphisms the coproduct over all fiber products of these:

$$Mor_{C(\{U_i\})} \;\coloneqq\; \underset{i,j}{\coprod} y(U_i) \times_{y(X)} y(U_j) \,.$$

This means equivalently that for any $V \in \mathcal{C}$ the groupoid assigned by $C(\{U_i\})$ has as set of objects pairs consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects

$$\kappa_i \longrightarrow \kappa_j$$

precisely if

(11)$$\iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }$$

Condition (11) for morphisms in the Cech groupoid to be well-defined is verbatim the condition (2) in the definition of matching families. Indeed, Cech groupoids serve to conveniently summarize (and then generalize) the sheaf condition (Def. ):

Proposition

(Cech groupoid co-represents matching families – codescent)

For Grpd regarded as a cosmos (Example ), and $\mathcal{C}$ a site (Def. ), let

$$\mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]$$

be a presheaf on $\mathcal{C}$ (Example ), regarded as a Grpd-enriched presheaf via Example , let $X \in \mathcal{C}$ be any object and $\{U_i \overset{\iota_i}{\to} X\}_i$ a covering family (Def. ) with induced Cech groupoid $C(\{U_i\}_i)$ (Example ).

Then there is an isomorphism

$$[\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;\simeq\; Match\left( \{U_i\}_i, \, \mathbf{Y} \right)$$

between the hom-groupoid of Grpd-enriched presheaves (Def. ) and the set of matching families (Def. ).

Since hence the Cech-groupoid co-represents the descent object, it is sometimes called the codescent object along the given covering.

Moreover, under this identification the canonical morphism

(12)$$C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} y(X)$$

induces the comparison morphism (5)

$$\array{ [\mathcal{C}^{op}, Grpd]\left( y(X), \, \mathbf{Y} \right) & \simeq & \mathbf{Y}(X) \\ {}^{ \mathllap{ [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y}) } }\downarrow && \downarrow \\ [\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) &\simeq& Match\left( \{U_i\}_i, \, \mathbf{Y} \right) } \,.$$

In conclusion, this means that the presheaf $\mathbf{Y}$ is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism:

(13)$$\mathbf{Y} \,\text{is a sheaf} \phantom{AAAA} \Leftrightarrow \phantom{AAAA} \left[ C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}I}}{\to} y(X) \;,\; \mathbf{Y} \right] \, \text{is iso, for all covering families} \, \{U_i \to X\}$$

One also says in this case that $\mathbf{Y}$ is a local object with respect to Cech covers/

Proof

By (?) the hom-groupoid is computed as the end

$$[\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;=\; \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] \,,$$

where, by Example , the “integrand” is the functor category (here: a groupoid) from the Cech groupoid at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.

Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the functors $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those functions on sets of objects

$$\underset{i}{\coprod} y(U_i)(V) = Obj_{C\left(\{U_i\}_i\right)(V)} \longrightarrow Obj_{\mathbf{Y}(V)} = \mathbf{Y}(V)$$

which respect the equivalence relation induced by the morphisms in the Cech groupoid at $V$.

Hence the hom-groupoid is a subset of the end of these function sets:

$$\begin{aligned} \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] & \hookrightarrow \int_{V \in \mathcal{C}} \left[ \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \int_{V \in \mathcal{C}} \underset{i}{\prod} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \int_{V \in \mathcal{C}} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \mathbf{Y}(U_i) \end{aligned}$$

Here we used: first that the internal hom-functor turns colimits in its first argument into limits (Prop. ), then that limits commute with limits (Prop. ), hence that in particular ends commute with products , and finally the enriched Yoneda lemma (Prop. ), which here is, via Example , just the plain Yoneda lemma (Prop. ). The end result is hence the same Cartesian product set that also the set of matching families is defined to be a subset of, in (4).

This shows that an element in $\int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right]$ is a tuple $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment

$$\array{ C\left(\{U_i\}_i\right)(V) & \longrightarrow & \mathbf{Y}(V) \\ (V \overset{\kappa_i}{\to} U_i) &\mapsto& \kappa_i^\ast \phi_i = \mathbf{Y}(\kappa_i)(\phi_i) }$$

constitutes a functor of groupoids.

By definition of the Cech groupoid, and since the codomain is a just set regarded as a groupoid, this is the case precisely if

$$\mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \phantom{AAAA} \text{for all}\, i,j \,,$$

which is exactly the condition (3) that makes $(\phi_i)_i$ a matching family.

$\,$

Local presentation

We now discuss a more abstract characterization of sheaf toposes, in terms of properties enjoyed by the adjunction that relates them to the corresponding categories of presheaves.

Definition

(locally presentable category)

A category $\mathbf{H}$ (Def. ) is called locally presentable if there exists a small category $\mathcal{C}$ (Def. ) and a reflective subcategory-inclusion of $\mathcal{C}$ into its category of presheaves (Example )

$$\mathbf{H} \underoverset {\underset{\text{acc}}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Set]$$

such that the inclusion functor is an accessible functor in that it preserves $\kappa$-filtered colimits for some regular cardinal $\kappa$.

Proposition

(Giraud's theorem)

A sheaf topos (Def. ) is equivalently a locally presentable category (Def. ) with

1. universal colimits,

2. effective quotients,

3. disjoint coproducts.

Proposition

(sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes)

Let $(\mathcal{C}, \tau)$ be a site (Def. ). Then the full subcategory inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ of its sheaf topos (Def. ) into its category of presheaves is a reflective subcategory inclusion (Def. )

$$Sh(\mathcal{C},\tau) \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} PSh(\mathcal{C})$$

such that:

1. the inclusion $\iota$ is an accessible functor, thus exhibiting $Sh(\mathcal{C},\tau)$ as a locally presentable category (Def. )

2. the reflector $L \colon PSh(\mathcal{C}) \to Sh(\mathcal{C})$ (which is sheafification, Prop. ) is left exact (“lex”) in that it preserves finite limits.

Conversely, every sheaf topos arises this way. Hence sheaf toposes $\mathbf{H}$ are equivalently the left exact-reflectively full subcategories of presheaf toposes over some small category $\mathcal{C}$:

(14)$$\mathbf{H} \underoverset {\underset{\phantom{AA}acc\phantom{AA}}{\hookrightarrow}} {\overset{\phantom{AA}lex\phantom{AA}}{\longleftarrow}} {\bot} PSh(\mathcal{C})$$

(e.g. Borceux 94, prop. 3.5.4, cor. 3.5.5, Johnstone, C.2.1.11)

Remark

(left exact reflections of categories of presheaves are locally presentable categories)

In the characterization of sheaf toposes as left exact reflections of categories of presheaves in Prop. , the accessibility of the inclusion, equivalently the local presentability (Def. ) is automatically implied (using the adjoint functor theorem), as indicated in (14).