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geometry of physics - basic notions of topos theory

Basic notions of Topos theory

Basic notions of Topos theory

We have explained in Remark how presheaves on a category ๐’ž\mathcal{C} may be thought of as generalized spaces probe-able by the objects of ๐’ž\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 [๐’ž op,Set][\mathcal{C}^{op}, Set] is the free cocompletion of the given small category ๐’ž\mathcal{C} (Prop. ) and hence exhibits generalized spaces Xโˆˆ[๐’ž op,Set]\mathbf{X} \in [\mathcal{C}^{op}, Set] as being glued or generated form the โ€œordinary spacesโ€ Xโˆˆ๐’žX \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:

A\phantom{A}free cocompletionA\phantom{A}
A=\phantom{A}=presheavesA\phantom{A}
A\phantom{A}loc. presentable categoryA\phantom{A}A\phantom{A}sheaf toposAAAA\phantom{AAAA}
AHโ‰ƒโŸถAAAโŸต[๐’ž op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq} [\mathcal{C}^{op},Set]AHโŠฅโ†ชaccessibleโŸต[๐’ž op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]AHโŠฅโ†ชaccessibleโŸตleft exact[๐’ž op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]
A\phantom{A}Prop. A\phantom{A}A\phantom{A}Def. A\phantom{A}A\phantom{A}Prop. A\phantom{A}
A\phantom{A}simplicial presheavesA\phantom{A}A\phantom{A}combinatorial model categoryA\phantom{A}A\phantom{A}model toposA\phantom{A}
AHโ‰ƒ QuโŸถAAAโŸต[๐’ž op,sSet Qu] proj\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq_{Qu}} [\mathcal{C}^{op},sSet_{Qu}]_{proj}AHโŠฅ Quโ†ชaccessibleโŸต[๐’ž op,sSet Qu] proj\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}AHโŠฅ Quโ†ชaccessibleโŸตleft exact[๐’ž 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}
A\phantom{A}Example A\phantom{A}Def. A\phantom{A}Def.

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Remark

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

If the objects of ๐’ž\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 (โ€œsmallerโ€) spaces (a coverage, Def. below).

But if a space Xโˆˆ๐’žX \in \mathcal{C} is covered, say by two other spaces U 1,U 2โˆˆ๐’žU_1, U_2 \in \mathcal{C}, via morphisms

U 1 U 2 i 1โ†˜ โ†™ i 2 X \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 Y\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 XX to form a space U 1โˆฉ XU 2โˆˆ๐’žU_1 \cap_X U_2 \in \mathcal{C}. Then locality of probes should imply that the ways of mapping U 1U_1 and U 2U_2 into Y\mathbf{Y} such that these maps agree on the intersection U 1โˆฉ XU 2U_1 \cap_X U_2, should be equivalent to the ways of mapping all of XX into Y\mathbf{Y}.

locality:{maps fromU 1andU 2toY that coincide onU 1โˆฉ XU 2}โ‰ƒ{maps fromXintoY} \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 ๐’ž\mathcal{C}. But the established terminology is that this is the sheaf condition (6) on presheaves over ๐’ž\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 โ€œpetit toposโ€-theory. The motivation above, instead, naturally leads to the โ€œgros toposโ€-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.

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We now first introduce the sheaf-condition (Def. ) below in its traditional form via โ€œmatching familiesโ€ (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.

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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 โ€œconvenienceโ€, in the technical sense of Prop. below.

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Definition

(coverage and site)

Let ๐’ž\mathcal{C} be a small category (Def. ). Then a coverage on ๐’ž\mathcal{C} is

  • for each object Xโˆˆ๐’žX \in \mathcal{C} a set of indexed sets of morphisms into XX

    {U iโ†’ฮน iX} iโˆˆI \left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}

    called the coverings of XX,

such that

  • for every covering {U iโ†’ฮน iX} iโˆˆI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of XX and every morphism YโŸถfXY \overset{f}{\longrightarrow} X there exists a refining covering {V jโ†’ฮน jY} jโˆˆJ\left\{ V_j \overset{\iota_j}{\to} Y \right\}_{j \in J} of YY, meaning that for each jโˆˆJj \in J there exists iโˆˆIi \in I and a morphism V jโ†’ฮน j,iU iV_j \overset{\iota_{j,i}}{\to} U_i such that

    (1)fโˆ˜ฮน j=ฮน iโˆ˜ฮน j,iAAAAAAAV j โŸถฮน j,i U i ฮน jโ†“ โ†“ ฮน i Y โŸถf X 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 ๐’ž\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 ff 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 XโˆˆTopX \in Top a fixed topological space, there is the site Op(X)Op(X) whose underlying category is the category of opens of XX, which is the thin category (Example ) of open subsets of XX 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 XโˆˆSmthMfdX \in SmthMfd any smooth manifold of dimension Dโˆˆโ„•D \in \mathbb{N}, its coverings are collections of smooth functions from the Cartesian space โ„ D\mathbb{R}^D to XX whose image is the inclusion of an open ball.

Hence these are the usual open covers of XX, 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 โ€“ descent object)

Let ๐’ž\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Yโˆˆ[๐’ž op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over ๐’ž\mathcal{C}.

Given an object Xโˆˆ๐’žX \in \mathcal{C} and a covering {U iโ†’ฮน iX} iโˆˆI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of it (Def. ) we say that a matching family (of probes of Y\mathbf{Y}) is a tuple (ฯ• iโˆˆY(U i)) iโˆˆI(\phi_i \in \mathbf{Y}(U_i))_{i \in I} such that for all i,jโˆˆIi,j \in I and pairs of morphisms U iโ†ฮบ iVโ†’ฮบ jU jU_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j satisfying

(2)ฮน iโˆ˜ฮบ i=ฮน jโˆ˜ฮบ jAAAAAAAA V ฮบ iโ†™ โ†˜ ฮบ j U i U j ฮน iโ†˜ โ†™ ฮน j X \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)Y(ฮบ i)(ฯ• i)=Y(ฮบ j)(ฯ• j). \mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \,.

We write

(4)Match({U i} iโˆˆI,Y)โŠ‚โˆiY(U i)โˆˆSet 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 Y\mathbf{Y} for descent along the covering {U iโ†’ฮน iX}\{U_i \overset{\iota_i}{\to}X\}.

Example

(matching families that glue)

Let ๐’ž\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Yโˆˆ[๐’ž op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over ๐’ž\mathcal{C}.

Given an object Xโˆˆ๐’žX \in \mathcal{C} and a covering {U iโ†’ฮน iX} iโˆˆI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of it (Def. ), then every element

ฯ•โˆˆY(X) \phi \;\in\; \mathbf{Y}(X)

induces a matching family (Def. ) by

(Y(ฮน i)(ฯ•)) iโˆˆI. \big( \mathbf{Y}(\iota_i)(\phi) \big)_{i \in I} \,.

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

This construction provides a function of the form

(5)Y(X)โŸถMatch({U i} iโˆˆI,Y) \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 Y\mathbf{Y} by the patches U iU_i of XX which glue to a global probe out of XX.

Definition

(sheaves and sheaf toposes)

Let ๐’ž\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Yโˆˆ[๐’ž op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over ๐’ž\mathcal{C}.

The presheaf Y\mathbf{Y} is called a sheaf if for every object Xโˆˆ๐’žX \in \mathcal{C} and every covering {U iโ†’ฮน iX} iโˆˆI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of XX all matching families glue uniquely, hence if the comparison morphism (5) is a bijection

(6)Y(X)โŸถโ‰ƒMatch({U i} iโˆˆI,Y). \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 ๐’ž\mathcal{C}, on those that are sheaves is the category of sheaves, denoted

(7)Sh(๐’ž)โ†ชAAฮนAA[๐’ž op,Set]. 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 H 1\mathbf{H}_1 and H 2\mathbf{H}_2 two such sheaf toposes, a homomorphism f:H 1โ†’H 2f \;\colon\; \mathbf{H}_1 \to \mathbf{H}_2 between them, called a geometric morphism is an adjoint pair of functors (Def. )

(8)H 1โŸถAAf *AAโŸตAAf *AAH 2 \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

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

Example

(global sections geometric morphism)

Let H\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. ):

HโŠฅโŸถAAฮ“AAโ†ฉLSet. \mathbf{H} \underoverset {\underset{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow}} {\overset{L}{\hookleftarrow}} {\bot} Set \,.

Here ฮ“\Gamma is called the global sections-functor.

Proof

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

Sโ‰ƒโˆsโˆˆS*. S \;\simeq\; \underset{s \in S}{\coprod} \ast \,.

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

L(S) โ‰ƒL(โˆsโˆˆS*) โ‰ƒโˆsโˆˆSL(*) โ‰ƒโˆsโˆˆS*. \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 LL exists and uniquely so, up to natural isomorphism. This implies the essential uniqueness of ฮ“\Gamma by uniqueness of adjoints (Prop. ).

Example

(trivial coverage)

For ๐’ž\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 [๐’ž op,Set][\mathcal{C}^{op},Set] (Example ) over a site ๐’ž triv\mathcal{C}_{triv} with trivial coverage is already the corresponding category of sheaves, hence the corresponding sheaf topos:

Sh(๐’ž triv)โ‰ƒ[๐’ž op,Set]. 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(*)โ‰ƒSet. Sh(\ast) \;\simeq\; Set \,.

Hence the category of sets is a sheaf topos.

Example

(sheaves on a topological space โ€“ 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 XX a topological space, consider its category of open subsets Op(X)Op(X) from Example , with coverage given by the usual open covers. Then a โ€œsheaf on this topological spaceโ€ is a sheaf, in the sense of Def. , on this site of opens. One writes

Sh(X)โ‰”Sh(Op(X))โ†ชAA[Op(X) op,Set], 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(โˆ’):TopโŸถTopos. 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(โˆ’):SoberTopโ†ชAAATopos. Sh(-) \;\colon\; SoberTop \overset{\phantom{AAA}}{\hookrightarrow} Topos \,.

More generally, this holds for locales (i.e. for โ€œsober topological spaces not necessarily supported on pointsโ€), in which case it becomes a reflective subcategory-inclusion (Def. )

LocaleโŠฅโ†ชAASh(โˆ’)AAโŸตAAAATopos 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 โ€œsheaf of abelian groups of the site of opens of a topological spaceโ€, when they write just โ€œsheafโ€.

But for ๐’ฎ\mathcal{S} any mathematical structure, a sheaf of ๐’ฎ\mathcal{S}-structured sets is equivalently an ๐’ฎ\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 โ€“ smooth sets:

Sh(SmthMfd)โ‰ƒSmoothSet. 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 ๐’ž\mathcal{C} (Def. ), then its presheaf topos [๐’ž op,Set][\mathcal{C}^{op}, Set] (Example ) is the free cocompletion of the category ๐’ž\mathcal{C} (Prop. ), hence the category obtained by freely forming colimits (โ€œgluingโ€) of objects of ๐’ž\mathcal{C}.

In contrast, the full subcategory inclusion Sh(๐’ž)โ†ช[๐’ž op,Set]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(๐’ž)Sh(\mathcal{C}) as obtained by generators and relations from the objects of its site ๐’ž\mathcal{C}:

  • the objects of ๐’ž\mathcal{C} are the generators;

  • the coverings of ๐’ž\mathcal{C} are the relations.

Proposition

(sheafification and plus construction)

Let ๐’ž\mathcal{C} be a site (Def. ). Then the full subcategory-inclusion (7) of the category of sheaves over ๐’ž\mathcal{C} (Def. ) into the category of presheaves (Example ) has a left adjoint (Def. ) called sheafification

Sh(๐’ž)โŠฅโ†ชAAฮนAAโŸตL[๐’ž op,Set]. 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 โ€œplus constructionโ€ twice:

L(Y)โ‰ƒ(Y +) +. L(\mathbf{Y}) \simeq (\mathbf{Y}^+)^+ \,.

Here the plus construction

(โˆ’) +:[๐’ž op,Set]โŸถ[๐’ž op,Set] (-)^+ \;\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. )

Y +(X)โ‰”{{U iโ†’ฮน iX}covering,ฯ•โˆˆMatch({U i},Y)}/โˆผ \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.

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Example

(induced coverage)

Let ๐’ž\mathcal{C} be a site (Def. ). Then a full subcategory (Def. )

๐’Ÿโ†ช๐’ž \mathcal{D} \hookrightarrow \mathcal{C}

becomes a site itself, whose coverage consists of those coverings {U iโ†’ฮน iY}\{U_i \overset{\iota_i}{\to} Y\} in ๐’ž\mathcal{C} that happen to be in ๐’Ÿโ†ช๐’ž\mathcal{D} \hookrightarrow \mathcal{C}.

Definition

(dense subsite)

Let ๐’ž\mathcal{C} and ๐’Ÿ\mathcal{D} be sites (Def. ) with a a full subcategory-inclusion (Def. )

๐’Ÿโ†ช๐’ž \mathcal{D} \hookrightarrow \mathcal{C}

and regard ๐’Ÿ\mathcal{D} as equipped with the induced coverage (Def. ).

This is called a dense subsite-inclusion if every object Xโˆˆ๐’žX \in \mathcal{C} has a covering {U iโ†’ฮน iX} i\{U_i \overset{\iota_i}{\to} X\}_i such that for all ii the patches are in the subcategory:

U iโˆˆ๐’Ÿโ†ช๐’ž. U_i \;\in\; \mathcal{D} \hookrightarrow \mathcal{C} \,.
Proposition

(comparison lemma)

Let ๐’Ÿโ†ชฮน๐’ž\mathcal{D} \overset{\iota}{\hookrightarrow} \mathcal{C} be a dense subsite inclusion (def. ). Then precomposition with ฮน\iota induces an equivalence of categories (Def. ) between their categories of sheaves (Def. ):

ฮน *:Sh(๐’ž)โŸถโ‰ƒSh(๐’Ÿ) \iota^\ast \;\colon\; Sh(\mathcal{C}) \overset{\simeq}{\longrightarrow} Sh(\mathcal{D})
Proposition

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

Let ๐’ž\mathcal{C} be a site (Def. ) with Sh(๐’ž)Sh(\mathcal{C}) its category of sheaves (Def. ).

Then a morphisms f:Xโ†’Yf \;\colon\; \mathbf{X} \to \mathbf{Y} in Sh(๐’ž)Sh(\mathcal{C}) is

  1. a monomorphism (Def. ) or isomorphism (Def. ) precisely if it is so globally in that for each object Uโˆˆ๐’žU \in \mathcal{C} in the site, then the component f U:X(U)โ†’Y(U)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 UโˆˆCU \in C there is a covering {p i:U iโ†’U} iโˆˆI\{p_i : U_i \to U\}_{i \in I} such that for all iโˆˆIi \in I and every element yโˆˆY(U)y \in \mathbf{Y}(U) the element f(p i)(y)f(p_i)(y) is in the image of f(U i):X(U i)โ†’Y(U i)f(U_i) : \mathbf{X}(U_i) \to \mathbf{Y}(U_i).

Proposition

(epi/mono-factorization through image)

Let Sh(๐’ž)Sh(\mathcal{C}) be a category of sheaves (Def. ). Then every morphism f:Xโ†’Yf \;\colon\; \mathbf{X} \to \mathbf{Y} factors as an epimorphism followed by a monomorphism (Def. ) uniquely up to unique isomorphism:

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

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

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)A โŸถAAA B epiโ†“ โˆƒ!โ†— โ†“ mono C โŸถAAA D \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

X 1 โŸถf 1 Y 1 โ†“ โ†“ X 2 โŸถf 2 Y 2 \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:

X 1 โŸถepi im(f 1) โŸถmono Y 1 โ†“ โ†“ โ†“ X 2 โŸถepi im(f 2) โŸถmono Y 2 \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 ๐’ž\mathcal{C} be a site (Def. ) and Sh(๐’ž)Sh(\mathcal{C}) its sheaf topos (Def. ). Then:

  1. All limits exist in Sh(๐’ž)Sh(\mathcal{C}) (Def. ), and they are computed as limits of presheaves, via Example :

    ฮน(limโŸตdX d)โ‰ƒlimโŸตdฮน(X d) \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(๐’ž)Sh(\mathcal{C}) (Def. ) and they are given by the sheafification (Def. ) of the same colimits computed in the category of presheaves, via Example :

    limโŸถdX dโ‰ƒL(limโŸตdฮน(X d)) \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 [๐’ž op,Set][\mathcal{C}^{op}, Set] from Example restricts to sheaves:

    Sh(๐’ž)โŸถ[X,โˆ’]โŸตXร—(โˆ’)Sh(๐’ž) Sh(\mathcal{C}) \underoverset {\underset{[\mathbf{X}, -]}{\longrightarrow}} {\overset{\mathbf{X} \times (-)}{\longleftarrow}} {} Sh(\mathcal{C})

    In particular, for X,YโˆˆSh(๐’ž)\mathbf{X}, \mathbf{Y} \in Sh(\mathcal{C}) two sheaves, their internal hom [X,Y]โˆˆSh(๐’ž)[\mathbf{X}, \mathbf{Y}] \in Sh(\mathcal{C}) is a sheaf given by

    [X,Y]:Uโ†ฆHom Sh(๐’ž)(y(U)X,Y), [\mathbf{X}, \mathbf{Y}] \;\colon\; U \;\mapsto\; Hom_{Sh(\mathcal{C})}( y(U) \mathbf{X}, \mathbf{Y} ) \,,

    where y(U)y(U) is the presheaf represented by Uโˆˆ๐’žU \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 ๐’ž\mathcal{C} be a site (Def. ). Then a morphism of presheaves over ๐’ž\mathcal{C} (Example )

YโŸถAAfAAXโˆˆ[๐’ฎ op,Set] \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 Uโˆˆ๐’žU \in \mathcal{C}, every morphism y(U)โŸถXy(U) \longrightarrow \mathbf{X} out of its represented presheaf (Example ) has the local lifting property through ff in that there is a covering {U iโ†’ฮน iU}\big\{ U_i \overset{\iota_i}{\to} U \big\} (Def. ) and a commuting diagram of the form

y(U i) โŸถAAโˆƒAA Y y(ฮน i)โ†“ โ†“ f y(U) โŸถAAAA X \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 โ€œhigher stacksโ€.

\,

Definition

(presheaves of groupoids)

For ๐’ž\mathcal{C} a small category (Def. ) consider the functor category (Example ) from the opposite category of ๐’ž\mathcal{C} (Example ) to the category Grpd of small groupoids (Example )

[๐’ž op,Grpd]. [\mathcal{C}^{op}, Grpd] \,.

By Example we may regard Grpd as a cosmos for enriched category theory. Since the inclusion Setโ†ชGrpdSet \hookrightarrow Grpd (Example ) is a strong monoidal functor (Def. ) of cosmoi (Example ), the plain category ๐’ž\mathcal{C} may be thought of as a Grpd-enriched category (Def. ) and hence a functor ๐’ž opโ†’Grpd\mathcal{C}^{op} \to Grpd is equivalently a Grpd-enriched functor (Def. ).

This means that the plain category of functors [๐’ž op,Grpd][\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 Yโˆˆ[๐’ž op,Grpd]\mathbf{Y} \in [\mathcal{C}^{op}, Grpd] (Def. ), we obtain two ordinary presheaves of sets (Def. ) called the

  • presheaf of objects

    Obj Y(โˆ’)โˆˆ[๐’ž op,Set] Obj_{\mathbf{Y}(-)} \in [\mathcal{C}^{op}, Set]
  • the presheaf of morphisms

    Mor Y(โˆ’)โ‰”โˆx,yโˆˆObj Y(โˆ’)Hom Y(โˆ’):[๐’ž op,Set] 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)[๐’ž op,Grpd] โŸถโ‰ƒ Grpd([๐’ž op,Grpd]) Y โ†ฆ (AAAโˆx,yโˆˆObj Y(โˆ’)Hom Y(โˆ’)โŸMor Y(โˆ’) (xโ†’fy) โ†ฆ xโ†“โ†‘ x โ†ฆ xโ†’id xxxโ†’id xโ†“ (xโ†’fy) โ†ฆ y Obj Y(โˆ’)AAA). \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 ๐’ž\mathcal{C} (Def. ).

Example

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

Let ๐’ž\mathcal{C} be a small category (Def. ). There is the reflective subcategory-inclusion (Def. ) of the category of presheaves over ๐’ž\mathcal{C} (Example ) into the category of presheaves of groupoids over ๐’ž\mathcal{C} (Def. )

[๐’ž op,Set]โŠฅโ†ชAAAAโŸตฯ€ 0[๐’ž op,Grpd] [\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 ๐’ž\mathcal{C} by the reflective inclusion of sets into groupoids (Example ).

Example

(Cech groupoid)

Let ๐’ž\mathcal{C} be a site (Def. ), and Xโˆˆ๐’žX \in \mathcal{C} an object of that site. For each covering family {U iโ†’ฮน iX}\{ U_i \overset{\iota_i}{\to} X\} of XX in the given coverage, the Cech groupoid is the presheaf of groupoids (Def. )

C({U i})โˆˆ[๐’ž op,Grpd]โ‰ƒGrpd([๐’ž op,Set]) 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 ๐’ž\mathcal{C}, via (10), has as presheaf of objects the coproduct

Obj C({U i})โ‰”โˆiy(U i) 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 iU_i, and as presheaf of morphisms the coproduct over all fiber products of these:

Mor C({U i})โ‰”โˆi,jy(U i)ร— y(X)y(U j). Mor_{C(\{U_i\})} \;\coloneqq\; \underset{i,j}{\coprod} y(U_i) \times_{y(X)} y(U_j) \,.

This means equivalently that for any Vโˆˆ๐’žV \in \mathcal{C} the groupoid assigned by C({U i})C(\{U_i\}) has as set of objects pairs consisting of an index ii and a morphism Vโ†’ฮบ iU iV \overset{\kappa_i}{\to} U_i in ๐’ž\mathcal{C}, and there is a unique morphism between two such objects

ฮบ iโŸถฮบ j \kappa_i \longrightarrow \kappa_j

precisely if

(11)ฮน iโˆ˜ฮบ i=ฮน jโˆ˜ฮบ jAAAAAAAA V ฮบ iโ†™ โ†˜ ฮบ j U i U j ฮน iโ†˜ โ†™ ฮน j X \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 โ€“ codescent)

For Grpd regarded as a cosmos (Example ), and ๐’ž\mathcal{C} a site (Def. ), let

Yโˆˆ[๐’ž op,Set]โ†ช[๐’ž op,Grpd] \mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]

be a presheaf on ๐’ž\mathcal{C} (Example ), regarded as a Grpd-enriched presheaf via Example , let Xโˆˆ๐’žX \in \mathcal{C} be any object and {U iโ†’ฮน iX} i\{U_i \overset{\iota_i}{\to} X\}_i a covering family (Def. ) with induced Cech groupoid C({U i} i)C(\{U_i\}_i) (Example ).

Then there is an isomorphism

[๐’ž op,Grpd](C({U i} i),Y)โ‰ƒMatch({U i} i,Y) [\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({U i} i)โŸถp {U i} iy(X) C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} y(X)

induces the comparison morphism (5)

[๐’ž op,Grpd](y(X),Y) โ‰ƒ Y(X) [๐’ž op,Grpd](p {U i} i,Y)โ†“ โ†“ [๐’ž op,Grpd](C({U i} i),Y) โ‰ƒ Match({U i} i,Y). \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 Y\mathbf{Y} is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism:

(13)Yis a sheafAAAAโ‡”AAAA[C({U i} i)โ†’p {U i}Iy(X),Y]is iso, for all covering families{U iโ†’X} \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 Y\mathbf{Y} is a local object with respect to Cech covers/

Proof

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

[๐’ž op,Grpd](C({U i} i),Y)=โˆซ Vโˆˆ๐’ž[C({U i} i)(V),Y(V)], [\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 โ€œintegrandโ€ is the functor category (here: a groupoid) from the Cech groupoid at a given VV to the set (regarded as a groupoid) assigned by Y\mathbf{Y} to VV.

Since Y(V)\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({U i} i)(V)โŸถY(V)C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V), which are equivalently those functions on sets of objects

โˆiy(U i)(V)=Obj C({U i} i)(V)โŸถObj Y(V)=Y(V) \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 VV.

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

โˆซ Vโˆˆ๐’ž[C({U i} i)(V),Y(V)] โ†ชโˆซ Vโˆˆ๐’ž[โˆiy(U i)(V),Y(V)] โ‰ƒโˆซ Vโˆˆ๐’žโˆi[y(U i)(V),Y(V)] โ‰ƒโˆiโˆซ Vโˆˆ๐’ž[y(U i)(V),Y(V)] โ‰ƒโˆiY(U i) \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 โˆซ Vโˆˆ๐’ž[C({U i} i)(V),Y(V)] \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] is a tuple (ฯ• iโˆˆY(U i)) i(\phi_i \in \mathbf{Y}(U_i))_i, subject to some condition. This condition is that for each Vโˆˆ๐’žV \in \mathcal{C} the assignment

C({U i} i)(V) โŸถ Y(V) (Vโ†’ฮบ iU i) โ†ฆ ฮบ i *ฯ• i=Y(ฮบ i)(ฯ• i) \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

Y(ฮบ i)(ฯ• i)=Y(ฮบ j)(ฯ• j)AAAAfor alli,j, \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 (ฯ• i) i(\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 H\mathbf{H} (Def. ) is called locally presentable if there exists a small category ๐’ž\mathcal{C} (Def. ) and a reflective subcategory-inclusion of ๐’ž\mathcal{C} into its category of presheaves (Example )

HโŠฅโ†ชaccโŸตAALAA[๐’ž op,Set] \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 ฮบ\kappa-filtered colimits for some regular cardinal ฮบ\kappa.

Proposition

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

Let (๐’ž,ฯ„)(\mathcal{C}, \tau) be a site (Def. ). Then the full subcategory inclusion i:Sh(๐’ž,ฯ„)โ†ชPSh(๐’ž)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(๐’ž,ฯ„)โŠฅโ†ชฮนโŸตAALAAPSh(๐’ž) Sh(\mathcal{C},\tau) \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} PSh(\mathcal{C})

such that:

  1. the inclusion ฮน\iota is an accessible functor, thus exhibiting Sh(๐’ž,ฯ„)Sh(\mathcal{C},\tau) as a locally presentable category (Def. )

  2. the reflector L:PSh(๐’ž)โ†’Sh(๐’ž)L \colon PSh(\mathcal{C}) \to Sh(\mathcal{C}) (which is sheafification, Prop. ) is left exact (โ€œlexโ€) in that it preserves finite limits.

Conversely, every sheaf topos arises this way. Hence sheaf toposes H\mathbf{H} are equivalently the left exact-reflectively full subcategories of presheaf toposes over some small category ๐’ž\mathcal{C}:

(14)HโŠฅโ†ชAAaccAAโŸตAAlexAAPSh(๐’ž) \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).

Last revised on September 20, 2018 at 03:47:14. See the history of this page for a list of all contributions to it.