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geometric embedding

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  • last time we had seen that every continuous map f:XYf : X \to Y of topological spaces induces an adjoint pair f *f *f^* \dashv f_* of functors of the corresponding presheaf categories PSh(X)f *f *PSh(Y) PSh(X) \stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\to}} PSh(Y) called direct image and inverse image

  • conversely, one finds that a functor into or out of a presheaf category which has a left adjoint behaves in many respects as if it came from an underlying “geometric” morphism of spaces;

  • here we begin with investigating such geometric morphisms of categories for the special simple case they behave like inclusions: geometric embeddings.

  • the categories of sheaves that we are going to be interested in turn out to be simply precisely the categories geometrically embedded in presheaf categories.

  • we now look at the simple definition of geometric embeddings and then step by step work out all the consequences and constructions that follow. this will lead us to sheaves, derived categories and eventually to abelian sheaf cohomology

Some properties of adjoint functors

In the following we will work out some consequences of the existence of certain adjoint functors. This will make use of some standard facts about adjoint functors. Since we haven’t had much of a chance to look at these standard facts so far, we start by deriving/recalling them.

Suppose two functors L:CDL : C \to D and R:DCR : D \to C are adjoint in that there is a natural isomorphism

Φ:Hom D(L(),)Hom E(,R()). \Phi : Hom_D(L(-),-) \stackrel{\simeq}{\to} Hom_E(-, R(-)) \,.
Lemma

The existence of the hom-set isomorphism Φ\Phi is equivalent to the existence of a unit-counit adjunction:

  • a natural transformation η:Id DRL\eta : Id_D \to R \circ L – called the unit

  • a natural transformation ϵ:LRId C \epsilon : L \circ R \to Id_C – called the counit;

such that for all objects cCc \in C and dDd \in D

  • (L(c)L(η c)LRL(c)ϵ L(c)L(c))=Id c(L(c) \stackrel{L(\eta_c)}{\to} L\circ R \circ L(c) \stackrel{\epsilon_{L(c)}}{\to} L(c)) = Id_c

  • (R(d)η R(d)RLR(d)R(ϵ d)R(d))=Id d(R(d) \stackrel{\eta_{R(d)}}{\to} R \circ L \circ R(d) \stackrel{R(\epsilon_{d})}{\to} R(d)) = Id_d.

Proof

Given a unit-counit adjunction (η,ϵ):LR(\eta,\epsilon) : L \dashv R define a natural transformation Φ:Hom D(L(),)Hom C(,R())\Phi : Hom_D(L(-),-) \to Hom_C(-,R(-)) by setting

Φ:(L(c)fd)(cη cRL(c)R(f)R(d)). \Phi : (L(c) \stackrel{f}{\to} d) \mapsto (c \stackrel{\eta_c}{\to} R \circ L (c) \stackrel{R(f)}{\to} R(d)) \,.

That this extends to a natural transformation follows from the naturality of η\eta.

The claim is that this transformation is invertible with inverse constructed similarly:

Φ 1:(cgR(d))(L(c)L(g)LR(d)ϵ dd). \Phi^{-1} : (c \stackrel{g}{\to} R(d)) \mapsto (L(c) \stackrel{L(g)}{\to} L\circ R(d) \stackrel{\epsilon_d}{\to} d) \,.

The defining properties of (η,ϵ)(\eta,\epsilon) imply that this is indeed the inverse to Φ\Phi.

Conversely, given the hom-isomorphism Φ\Phi, define for all objects cCc \in C and dDd \in D

(cη cRL(c)):=Φ(L(c)IdL(c)) (c \stackrel{\eta_c}{\to} R \circ L(c)) := \Phi( L(c) \stackrel{Id}{\to} L(c))
(LR(d)ϵ dd):=Φ 1(R(d)IdR(d)). (L \circ R (d) \stackrel{\epsilon_d}{\to} d) := \Phi^{-1}( R(d) \stackrel{Id}{\to} R(d)) \,.

For every L(c)fdL(c) \stackrel{f}{\to} d it follows by the naturality of Φ\Phi

Hom D(L(c),d) Φ Hom C(c,R(d)) f R(f) Hom D(L(c),L(c)) Φ Hom C(c,RL(c)) \array{ Hom_D(L(c),d) &\stackrel{\Phi}{\to}& Hom_C(c,R(d)) \\ \uparrow^{f \circ -} && \uparrow^{R(f) \circ -} \\ Hom_D(L(c),L(c)) &\stackrel{\Phi}{\to}& Hom_C(c, R \circ L(c)) }

that

Φ(L(c)fd)=R(f)Φ(Id L(c))=:R(f)η c. \Phi(L(c) \stackrel{f}{\to} d) = R(f)\circ \Phi(Id_{L(c)}) =: R(f) \circ \eta_c \,.

Similarly one finds that for all cgR(d)c \stackrel{g}{\to} R(d) we have

Φ 1(cgR(d))=ϵ cL(f). \Phi^{-1}(c \stackrel{g}{\to} R(d)) = \epsilon_c \circ L(f) \,.

Applying the first identity to (L(c)fd):=(LR(d)ϵ dd)(L(c) \stackrel{f}{\to} d) := (L\circ R(d) \stackrel{\epsilon_d}{\to} d) yields the defining equation for the counit, applying the second identity to (cgR(d)):=(cη cR(L(c)))(c \stackrel{g}{\to} R(d)) := (c \stackrel{\eta_c}{\to} R(L(c))) yields the defining equation for the unit.

Proposition

Let (LR)(L \dashv R) be an adjunction as above. Then

Proof

This follows directly once one observes that the following diagrams commute

Hom C(R(d),R(d)) R d,d Φ 1 Hom D(d,d) Hom D(ϵ d,d) Hom D(LR(d),d) \array{ && Hom_C(R(d), R(d')) \\ & {}^{R_{d,d'}}\nearrow && \searrow^{\Phi^{-1}} \\ Hom_D(d,d') &&\stackrel{Hom_D(\epsilon_d, d')}{\to}&& Hom_D(L \circ R(d), d') }

and

Hom C(c,c) Hom C(c,η c) Hom C(c,RL(c) L c,c Φ Hom D(L(c),L(c)). \array{ Hom_C(c,c') &&\stackrel{Hom_C(c, \eta_{c'})}{\to}&& Hom_C(c, R \circ L(c') \\ & {}_{L_{c,c'}}\searrow && \nearrow_{\Phi} \\ && Hom_D(L(c), L(c')) } \,.

The first diagram asserts that for dfdd \stackrel{f}{\to} d' a morphism, the two possible ways to get a morphism of the form R(d)R(d)R (d) \to R(d') coincide, namely

Φ(LR(d)ϵ ddfd)=(R(c)R(f)R(d)). \Phi(L \circ R (d) \stackrel{\epsilon_d}{\to} d \stackrel{f}{\to} d') = (R(c) \stackrel{R(f)}{\to} R(d')) \,.

This follows by chasing Id R(d)Id_{R(d)} through the naturality diagram for Φ\Phi:

Hom C(R(d),R(d)) Φ 1 Hom D(LR(d),d) Hom C(R(d),R(f)) Hom D(LR(d),f) Hom C(R(d),R(d)) Φ Hom D(LR(d),d). \array{ Hom_C(R(d), R(d)) &\stackrel{\Phi^{-1}}{\to}& Hom_D(L \circ R(d),d) \\ \downarrow^{Hom_C(R(d), R(f))} && \downarrow^{Hom_D(L \circ R(d), f)} \\ Hom_C(R(d), R(d')) &\stackrel{\Phi}{\leftarrow}& Hom_D(L \circ R(d), d') } \,.

Analogously for the second statement.

Geometric morphisms

We want to begin to investigate categories that nicely sit inside categories of presheaves. The right notion of morphism for that turns out to be what is called a geometric morphism between categories that have finite limits. The term is usually used more specifically for morphisms between categories that have finite limits and a bit more extra properties: these are called topoi. The categories of presheaves and sheaves that we care about will be of this type, but for quite a while we will care only about the fact that they have finite limits. Still, for completeness, we briefly give the formal definition of topos, just to fix the term.

Topoi

We had seen that categories of presheaves have many nice properties when it comes to universal constructions: they behave in these abstract aspects essentially the way we are familiar with from the category Set of all sets. Notably

Since for many constructions one keeps always referring to these properties, it is worthwhile to abstract them and define

Definition

A topos EE is a category that

  1. has finite limits
  2. has a power object.

This already implies in particular that EE

We shall not need or dwell on the second condition here, except for noting that

  • this is the same kind of condition we already used when setting up ETCS;

  • all categories of presheaves and sheaves satisfy this condition.

The notion of topos, simple as it is, has tremendous implications is large areas of mathematics. Notably, it appears both as a fundament of logic as well as of geometry. See

Here we are interested in the geometric aspects. In our original motivation for sheaves, cohomology and higher stacks we had advertized categories of sheaves as and (infinity,1)-category of (infinity,1)-sheaves as places where generalized spaces live. One way to characterize this is to realize that these categories are topoi or (infinity,1)-topoi, respectively.

To fully realize this relevance of the notion of topos, however, it is of course not sufficient to have just the notion of the concept itself, we also need the right notion of morphisms.

Definition

Morphisms between topoi are functors that are geometric morphisms.

If FF and EE are toposes, a geometric morphism f:EFf:E\to F consists of an pair of adjoint functors (f *,f *)(f^*,f_*)

f *:FE f_* : F \to E
f *:EF, f^* : E \to F \,,

such that the left adjoint f *:EFf^*:E \to F preserves finite limits.

Recall that, as a left adjoint, f *f^* necessarily already preserves all finite colimits.

Examples

At this point we do not really have nontrivial examples of this notion. Instead, we will turn this now around and use this definition to construct examples of topoi. Notably, we will consider now geometric morphisms which are inclusions and then eventually define sheaves to be objects of topoi geometrically included in sheaf topoi.

geometric embedding

A geometric embedding is the right notion of embedding or inclusion of topoi FEF \hookrightarrow E.

Notably the inclusion Sh(S)PSh(S)Sh(S) \hookrightarrow PSh(S) of a category of sheaves into its presheaf topos or more generally the inclusion Sh jEESh_j E \hookrightarrow E of sheaves in a topos EE into EE itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.

Another perspective is that a geometric embedding FEF \hookrightarrow E is the localizations of EE at the class WW or morphisms that the left adjoint EFE \to F sends to isomorphisms in FF.

Definition

For FF and EE two topoi, a geometric morphism

FfEFf *f *E F \stackrel{f}{\to} E \;\;\;\; F \stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}} E

is a geometric embedding if the following equivalent conditions are satisfied

Remark

Given a full and faithful functor F:DCF : D \to C we can and will assume in the following that it is injective on objects. For if it is not, let DD' be a full subcategory of DD with only one representative of each collection of objects with the same image under FF. Then notice that the inclusion DDD' \hookrightarrow D, which by construction is injective on objects and full and faithful, is also essentially surjective: for a,aa,a' two ojects of DD with F(a)=F(a)F(a) = F(a'), the preimage of Id F(a)Id_{F(a)} F a,a 1Id F(a):aaF^{-1}_{a,a'}Id_{F(a)} : a \to a' is clearly an isomorphism. Hence the inclusion DDD' \stackrel{\simeq}{\hookrightarrow} D is an equivalence of categories and we can consider without restriction of generality the full and faithful and injective on objects functor DDFCD' \hookrightarrow D \stackrel{F}{\to} C.

Consequences

A large amount of structure and concepts that will accompany us from now on follows from this simple definition. We will now incrementally extract consequences implied by a geometric morphism and introduce the necessary concepts to organize the.

Write η:Id Ef *f *\eta : Id_E \to f_* f^* for the unit of the adjunction.

Since f *f_* is fully faithful we will identify objects and morphism of FF with their images in EE. To further trim down the notation write ()¯:=f *\bar {(-)} := f^* for the left adjoint.

Definition

A category with weak equivalences is a category CC equipped with a subcategory WCW \subset C

  • which contains all isomorphisms of CC;

  • which satisfies “two-out-of-three”: for f,gf, g any two composable morphisms of CC, if two of {f,g,gf}\{f, g, g \circ f\} are in WW, then so is the third.

Definition

Write WW for the class of morphism in EE that are sent to isomorphism under f *f^*,

W=(f *) 1{g:cdMor(E)}. W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\} \,.
Proposition

EE equipped with the class WW is a category with weak equivalences, in that WW satisfies 2-out-of-3.

Proof

Follows since isomorphisms satisfy 2-out-of-3.

Definition

A left multiplicative system in a category CC is a collection WW of morphisms in CC such that

  • all isomorphisms are in WW;

  • WW is closed under composition;

  • given A w C B\array{ && A \\ && \downarrow^w \\ C &\to& B } with wWw \in W there exists D A w w C B\array{ D &\to& A \\ \downarrow^{w'} && \downarrow^w \\ C &\to& B } with ww' in WW.

  • given AfgBwCA \stackrel{\stackrel{g}{\to}}{\stackrel{f}{\to}} B \stackrel{w}{\to} C with wf=wgw \circ f = w \circ g there is DwAfgBD \stackrel{w'}{\to} A \stackrel{\stackrel{g}{\to}}{\stackrel{f}{\to}} B such that fw=gwf \circ w' = g \circ w'.

Proposition

The collection WMor(E)W \subset Mor(E) defined above is a left multiplicative system.

Proof

This follows using the fact that f *f^* is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

  • every isomorphism is in WW;

  • WW is closed under composition.

It remains to check the following points:

Given any

a w b h c \array{ && a \\ && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with wWw \in W, we have to show that there is

d a w w b h c \array{ d &\to& a \\ \downarrow^{w'} && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with wWw' \in W.

To get this, take this to be the pullback diagram, w:=h *ww' := h^* w. Since f *f^* preserves pullbacks, it follows that

d¯ a¯ w¯ w¯ b¯ h¯ c¯ \array{ \bar d &\to& \bar a \\ \downarrow^{\bar w'} && \downarrow^{\bar w} \\ \bar b &\stackrel{\bar h}{\to}& \bar c }

is a pullback diagram in FF with w¯=h¯ *w¯\bar w' = \bar h^* \bar w. But by assumption w¯\bar w is an isomorphism. Therefore w¯\bar w' is an isomorphism, therefore ww' is in WW.

Finally for every

asrbwc a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c

with wWw \in W such that the two composites coincide, we need to find

dwasrb d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b

with wWw' \in W such that the composites again coincide.

To get this, take ww' to be the equalizer of the two morphisms. Sending everything with f *f^* to FF we find from

a¯s¯r¯bw¯c \bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c

that r¯=s¯\bar r = \bar s, since w¯\bar w is an isomorphism. This implies that w¯\bar w' is the equalizer

d¯w¯as¯r¯b \bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b

of two equal morphism, hence an identity. So ww' is in WW.

Proposition

For every object aEa \in E

  • the unit η a:aa¯\eta_a : a \to \bar a is in WW;

  • if aa is already in FF then the unit is already an isomorphism.

Proof

This follows from the zig-zag identities of the adjoint functors.

η Id E E ()¯ F E ()¯ F Id F= ()¯ E Id F ()¯ \array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F \\ &&& \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow^{\bar{(-)}} \\ E &\Downarrow^{Id}& F \\ & \searrow \nearrow_{\bar{(-)}} }

and

η Id E F E ()¯ F E Id F= F Id E \array{ &&& \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow \\ F &\Downarrow^{Id}& E \\ & \searrow \nearrow }

In components they say that

  • for every aEa \in E we have (a¯η¯ aa¯¯a¯)=Id a¯(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}

  • for every aFa \in F we have (aη aa¯a)=Id a(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a) = Id_a

This implies the claim.

Definition

An object aEa \in E is a WW-local object if for every g:cdg : c \to d in WW the map

g *:Hom E(d,a)Hom E(c,a) g^* : Hom_E(d,a) \stackrel{\simeq}{\to} Hom_E(c,a)

obtained by precomposition is an isomorphism.

Conversely, a morphism f:cdf : c \to d is WW-local if for every WW-local object aa the morphism f *f^* is an isomorphism.

Notice that every object is local with respect to isomorphisms. So a WW-local object is one which regards morphisms in WW as isomorphisms as far as maps into it are concerned. Conversely, a WW-local morphism is one which behaves like an isomorphism as far as maps into WW-local objects are concerned.

Every morphism in WW is WW-local. The collection WW of morphisms is called saturated if the collection of WW-local morphisms coincides with WW.

Proposition

Up to isomorphism, the WW-local objects are precisely the objects of FF in EE

Proof

First assume that aFa \in F. We need to show that aa is WW-local.

Notice that the existence of the required isomorphism Hom F(d,a)Hom F(c,a) Hom_F(d,a) \simeq Hom_F(c,a) is equivalent to the statement that for every diagram

c d h a \array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} \\ a }

there is a unique extension

c d h a. \array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} & \swarrow \\ a } \,.

To see the existence of this extension, hit the original diagram with f *f^* to get

c¯ d¯ h¯ a¯a. \array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} \\ \bar a \simeq a } \,.

By the assumption that cdc \to d is in WW the morphism c¯d¯\bar c \to \bar d here is an isomorphism. By the assumption that aa is already in FF we have a¯a\bar a \simeq a since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

c¯ d¯ h¯ !k a¯a. \array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} & \swarrow_{\exists ! k} \\ \bar a \simeq a } \,.

By the hom-isomorphism (using full faithfullness of f *f_* to work entirely in EE)

Hom E(d¯,a)Hom E(d,a) Hom_E(\bar d, a) \simeq Hom_E(d,a)

this defines a morphism k:dak : d \to a. Chasing kk through the naturality diagram of the hom-isomorphism

Hom E(d¯,a¯) Hom E(d,a¯) Hom E(c¯,a¯) Hom E(c,a¯). \array{ Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a) \\ \downarrow && \downarrow \\ Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a) } \,.

shows that k:dak : d \to a does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So aFa \in F is WW-local.

Now for the converse, assume that a given aa is WW-local.

By one of the above propositions we know that the unit η a:aa¯\eta_a : a \to \bar a is in WW, so by the WW-locality of aa it follows that

a η a a¯ Id a a \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }

has an extension

a η a a¯ Id a ρ a a. \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.

By the 2-out-of-3 property of WW shown in one of the above propositions, (using that Id aId_a, being an isomorphism, is in WW) it follows that ρ a:a¯a\rho_a : \bar a \to a is in WW.

Since a¯\bar a is in FF and therefore WW-local by the above, it follows that also

a¯ ρ a a Id a¯ a¯ \array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }

has an extension

a¯ ρ a a Id a¯ λ a a¯. \array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.

So η a\eta_a has a left inverse ρ a\rho_a which itself has a left inverse λ a\lambda_a. It follows that ρ a\rho_a is also a right inverse to η a\eta_a, since

ρ aη a =ρ aη aρ aλ a Id =ρ aη aρ a Idλ a =ρ aλ a =Id. \begin{aligned} \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} & = \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} \underbrace{ \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} }_{Id} \\ & = \stackrel{\rho_a}{\to} \underbrace{ \stackrel{\eta_a}{\to} \stackrel{\rho_a}{\to} }_{Id} \stackrel{\lambda_a}{\to} \\ &= \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} \\ &= Id \end{aligned} \,.

So if aa is WW-local we find that η a:aa¯\eta_a : a \to \bar a is an isomorphism, hence that aa is isomorphic to an object of FF.

Lemma

Let F:CDF : C \to D be a functor and i:DDi : D' \hookrightarrow D a subcategory such that for every object cCc \in C the object F(c)F(c) is isomorphic to an object in DD'.

Then FF factors weakly through DD' in that there exists a functor F:CDF' : C \to D' and a natural isomorphism

C F D F ϕ i D. \array{ C &&\stackrel{F}{\to}&& D \\ & {}_{F'}\searrow &\Uparrow^{\simeq}_{\phi}& \nearrow_i \\ D' } \,.
Proof

Since we assumed the axiom of choice in SETSET we may choose for each cCc \in C objects F(c)DF'(c) \in D' and isomorphisms

ϕ c:F(c)F(c). \phi_c : F'(c) \stackrel{\simeq}{\to} F(c) \,.

Then for any morhism f:ccf : c \to c' in CC set

F(f):F(c)ϕ cF(c)F(f)F(c)ϕ c 1F(c). F'(f) : F'(c) \stackrel{\phi_c}{\to} F(c) \stackrel{F(f)}{\to} F(c') \stackrel{\phi_{c'}^{-1}}{\to} F'(c') \,.

This is clearly functorial. Moreover, the ϕ c\phi_c are clearly the components of the desired natural isomorphism.

Corollary

FF is equivalent to the full subcategory E WlocE_{W-loc} of EE on WW-local objects.

Proof

By the above lemma there is a functor FE WlocF \to E_{W-loc} and a natural isomorphism

F E E Wloc. \array{ F &&\hookrightarrow&& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow \\ && E_{W-loc} } \,.

Since FEF \hookrightarrow E and E WlocEE_{W-loc} \hookrightarrow E are full and faithful, so is FE WlocF \to E_{W-loc}. Since by the above it is also essentially surjective, it establishes the equivalence FE WlocF \simeq E_{W-loc}.

The homotopy category

Given a category with weak equivalences WW, there is another construction one wants to consider: the homotopy category C[W 1]C[W^{-1}] of CC with respect to WW. We describe the general concept, which will be useful later on for the description of derived categories and then show that in the case of geometric embeddings it reproduces the embedded category.

Frequently one encounters ordinary categories CC which are known in some way or other to be the 1-categorical truncation of higher categories C^\hat C.

Standard examples are the 1-category Top of topological spaces or the 1-category Ch(Ab)Ch(Ab) of chain complexes. Both are obtained from full (infinity,1)-categories by forgetting higher morphisms.

The most important information that is lost by forgetting the higher morphisms of a higher category is that about which 1-morphisms are, while not isomorphisms, invertible up to higher cells, i.e. equivalences.

To the full (,1)(\infty,1)-category C^\hat C is canonically associated a 1-category Ho(C^)Ho(\hat C) called the homotopy category of an (infinity,1)-category, which is obtained from C^\hat C not by simply forgetting the higher morphisms, but by quotienting them out, i.e. by remembering the equivalence classes of 1-morphisms. In the (,1)(\infty,1)-category Top these higher morphisms are literally the homotopies between 1-morphisms, and more generally one tends to address higher cells in (,1)(\infty,1)-categories as homotopies. Therefore the name homotopy category of an (,1)(\infty,1)-category for Ho(C^)Ho(\hat C). In particular Ho(Top^)Ho(\hat{Top}) is the standard homotopy category originally introduced in topology.

Now, given just the truncated 1-category CC but equipped with the structure of a category with weak equivalences which indicates which morphisms in CC are to be regarded as equivalences in a higher categorical context, there is a universal solution to the problem of finding a cartegory Ho(C)Ho(C) equipped with a functor Q:CHo(C)Q : C \to Ho(C) such that QQ sends all (morphisms labeled as) weak equivalences in CC to isomorphisms in Ho(C)Ho(C).

In good situations, one may also find an (,1)(\infty,1)-category C^\hat C corresponding to CC, and the notions of homotopy category Ho(C)Ho(C) and Ho(C^)Ho(\hat C) coincide.

This is in particular the case when CC is equipped with the structure of a combinatorial simplicial model category and C^\hat C is the (,1)(\infty,1)-category presented by CC with its model structure. (For instance HTT, remark A.3.1.8).

Definition

Given a category with weak equivalences, its homotopy category Ho(C)Ho(C) is – if it exists – the category which is universal with the property that there is a functor

p:CHo(C) p : C \to Ho(C)

that sends every weak equivalence in CC to an isomorphism in Ho(C)Ho(C).

One also writes Ho(C):=W 1CHo(C) := W^{-1}C or C[W 1]C[W^{-1}] and calls it the localization of CC at the collection WW of weak equivalences.

More in detail, the universality of Ho(C)Ho(C) means the following:

  • for any other category AA and functor F:CAF : C \to A such that FF sends all wWw \in W to isomorphisms in AA, there exists a functor F Q:Ho(C)AF_Q : Ho(C) \to A and a natural isomorphism
C F A Q F Q Ho(C) \array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) }
Remarks
  • This definition by itself does not use any properties of WW, in particular it need not be a system of weak equivalences for this definition to make sense. However, with the same logic as before, the maximal set WWW ' \supset W of morphisms which is sent to isos by WW will be a system of weak equivalences.

  • The second condition implies that the functor F QF_Q in the first condition is unique up to isomorphism: for F QF_Q and (F Q)(F_Q)' two such functors there is an isomorphism F QQ(F Q)QF_Q \circ Q \simeq (F_Q)' \circ Q and therefore, by the full and faithfulness of ()Q(-) \circ Q, also an isomorphism F Q(F Q)F_Q \simeq (F_Q)'.

  • If it exists, the homotopy category Ho(C)Ho(C) is unique up to equivalence of categories.

For suppose Ho(C)Ho(C) and Ho(C)Ho'(C) are two solutions. Then by assumption there are horizontal morphisms

C Q Q Ho(C) Ho(C) \array{ && C \\ & {}^Q\swarrow &\Updownarrow^{\simeq}& \searrow^{Q'} \\ Ho(C) &&\stackrel{}{\leftrightarrow} && Ho'(C) }

which are unique up to isomorphism. So in particular this yields a morphism Ho(C)Ho(C)Ho(C)Ho(C) \to Ho'(C) \to Ho(C) and conversely, unique up to iso. But since also Id Ho(C)Id_{Ho(C)} fills the corresponding diagram, there is an isomorphism to Id CId_C. Etc. This shows that we have an equivalence of categories.

We now give an explicit construction of Ho(C)Ho(C) for the case that WW is a left multiplicative system. First one simple definition

Definition

For aCa \in C let (W/a)(C/a)(W/a) \hookrightarrow (C/a) be the full subcategory of the over category (C/a)(C/a) on morphisms in WW.

Explicitly, (W(/a)(W(/a) is the category whose objects are given by morphisms p:aap : a' \to a in WW and whose morphism are given by commuting triangles

a f a p p a \array{ a' &&\stackrel{f}{\to}&& a'' \\ & {}_p\searrow && \swarrow_{p'} \\ && a }

in CC (i.e. ff need not be in WW).

Notice that there is still the obvious forgetful functor

(W/a)C (W/a) \to C

obtained by remembering only the horizontal morphism of these triangles.

Proposition

Given a category CC with a left multiplicative system WW, the homotopy category of CC with respect to WW is given (up to equivalence) by the category C[W 1]C[W^{-1}] defined as follows:

  • objects are those of CC;

  • hom-sets are given by

Hom C[W 1](a,b)=colimapWaHom C(a,b):=colim((W/a) opC opHom C(,b)Set) Hom_{C[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_C(a',b) := colim ( (W/a)^{\mathrm{op}} \to C^{\mathrm{op}} \stackrel{Hom_C(-,b)}{\to} Set )
  • composition is defined on representing spans by using the third property of left multiplicative systems.

The functor Q:CC[W 1]Q : C \to C[W^{-1}] is the one which is the identity on objects and regards a morphism f:abf : a \to b in CC trivially as a span

a f b Id a. \array{ a &\stackrel{f}{\to}& b \\ \downarrow^{Id} \\ a } \,.

Before proving this, let us unwrap what it means.

Recall the explicit description of colimits in Set: the disjoint union over all sets involved modulo the equivalence relation that is generated by the relation which regards two elements in different sets as equivalent if their pullback to a common set coincides. For the colimit over a filtered category the first relation here already is an equivalence relation.

Here this means:

a morphism f:abf : a \to b in colimapWaHom C(a,b)\underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_C(a',b) is represented by a span

a f b p a \array{ a' &\stackrel{f'}{\to}& b \\ \downarrow^{p} \\ a }

with pp in WW, and two such spans (f i)(f'_i) are regarded as equivalent if there is a third one f 3f'_3 fitting into a commuting diagram

a 1 p 1 f 1 a p 3 a 3 f 3 b p 2 f 3 a 2 \array{ && a'_1 \\ & {}^{p_1}\swarrow &\uparrow& \searrow^{f'_1} \\ a &\stackrel{p_3}{\leftarrow}& a'_3 &\stackrel{f'_3}{\to}& b \\ & {}_{p_2}\nwarrow &\downarrow& \nearrow_{f'_3} \\ && a'_2 }

Given two composable morphisms f:abf : a \to b and g:bcg : b \to c represented by spans as above, their composite is represented by a span of the form

q b g c p 2 a f b p 1 a, \array{ q &\to& b' &\stackrel{g'}{\to}& c \\ \downarrow && \downarrow^{p_2} \\ a' &\stackrel{f'}{\to}& b \\ \downarrow^{p_1} \\ a } \,,

where qq and the two morphisms out of it exists by one of the axioms satisfied by WW, which also says that the top left vertical morphism is in WW, so that by the axioms satisfied by WW the total composite vertical morphism is in WW.

One checks that this composition is indeed well defined:

  • let qaq \to a' and qaq' \to a' be two different ways to fill the top left square. By one of the axioms satisfied by WW one finds a further qq'' such that q q q a. \array{ q'' &\to& q' \\ \downarrow && \downarrow \\ q &\to& a' } \,. This may not yet be sufficient, since it doesn’t imply that qqbq'' \to q' \to b' equals qqbq'' \to q \to b'. But it does follow that
(qqbp 2b)=(qqbp 2b). (q'' \to q \to b' \stackrel{p_2}{\to} b) = (q'' \to q' \to b' \stackrel{p_2}{\to} b) \,.

Hence by the other axiom on WW there is one more refinement qqq''' \to q'' such that indeed we have

q a q c q. \array{ && q \\ & {}^{}\swarrow &\uparrow& \searrow^{} \\ a &\stackrel{}{\leftarrow}& q''' &\stackrel{}{\to}& c \\ & {}_{}\nwarrow &\downarrow& \nearrow_{} \\ && q' } \,.
  • similarly one finds that for different choices of representatives of ff and gg one obtains spans that represent the same equivalence class;

  • and yet again similarly one finds that this composition is indeed associative and unital.

Proof

We need to check first of all that QQ sends f:bcf : b \to c in WW to an isomorphism.

To see this, we show that for all aa we have

f *:Hom Ho(C)(a,b)Hom Ho(C)(a,c) f_* : Hom_{Ho(C)}(a,b) \stackrel{\simeq}{\to} Hom_{Ho(C)}(a,c)

is an isomorphism.

The result then follows with the corresponding corollary of the Yoneda lemma.

First, f *f_* is surjective:

for every span

a h c p a \array{ a' &\stackrel{h}{\to}& c \\ \downarrow^{p} \\ a }

we may find

a h c f p q b a \array{ a' && \stackrel{h}{\to} &&& c \\ & \nwarrow &&& \nearrow_{f} \\ \downarrow^p && q &\to& b \\ a }

with qaq \to a' in WW. This realizes hh as in the image of f *f_*.

Next, f *f_* is injective: if the two spans

a bfc p a \array{ a' &\stackrel{\to}{\to}& b \stackrel{f}{\to} c \\ \downarrow^p \\ a }

are equal, then by assumption on WW there is qaq \to a' such that

q a b a \array{ q &\to& a' & \stackrel{\to}{\to} & b \\ & \searrow & \downarrow \\ && a }

are equal, hence represent the same morphism.

This shows that QQ indeed sends morphisms in WW to isomorphisms. Next we need to show that it is universal with respect to that property.

So let

F:CA F : C \to A

be any functor that sends morphisms wWw \in W to isomorphism in AA. We obtain from this a functor F QF_Q by setting

  • on objects: F Q| Obj=F| ObjF_Q|_{Obj} = F|_{Obj}

  • on morphisms: apply FF to any one representative span by

F Q:(a f b p a)(F(a)F(p) 1F(a)F(f)F(b)) F_Q : \left( \array{ a' &\stackrel{f}{\to}& b \\ \downarrow^{p} \\ a } \right) \mapsto (F(a) \stackrel{F(p)^{-1}}{\to} F(a') \stackrel{F(f)}{\to} F(b) )

Once can check again explicitly that this is indeed functorial. Alternatively this follows from realizing that this corresponds to the following morphism

(F Q) a,b:Hom Ho(C)(a,b) colimapWaHom C(a,b) colimapWaF a,bcolimapWaHom A(F(a),F(b)) colimapWaHom A(F(a),F(b)) Hom A(F(a),F(b)) Hom A(F Q(a),F Q(b)). \begin{aligned} (F_Q)_{a,b} : Hom_{Ho(C)}(a,b) & \simeq \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_C(a',b) \\ & \stackrel{\underset{a' \stackrel{p \in W}{\to}a}{colim}F_{a',b}}{\to} \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_A(F(a'), F(b)) \\ & \simeq \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_A(F(a), F(b)) \\ & \simeq Hom_A(F(a), F(b)) & \simeq Hom_A(F_Q(a), F_Q(b)) \,. \end{aligned}

Here we used that due to assumption on FF for p:aap : a'\to a in WW the morphism

p *:Hom A(F(a),F(b))Hom A(F(a),F(b)) p^* : Hom_A(F(a),F(b)) \stackrel{\simeq}{\to} Hom_A(F(a'),F(b))

is an isomorphism, and then that a colimit over a constant functor (Hom A(F(a),F(b))Hom_A(F(a), F(b)) does no longer depend on aa') is just the value of that functor .

In particular, we have this way not just an isomorphism but an equality

F QQ=F. F_Q \circ Q = F \,.

So it remains to check that Q *:Func(Ho(C),A)Func(C,A)Q^* : Func(Ho(C),A) \to Func(C,A) is full and faithful, i.e. that for all functors F,G:Ho(C)AF,G : Ho(C) \to A we have

Q *:Hom(F,G) [Ho(C),A]Hom [C,A](FQ,GQ) Q^* : Hom(F,G)_{[Ho(C),A]} \stackrel{\simeq}{\to} Hom_{[C,A]}(F \circ Q, G \circ Q)

is an isomorphism. Since QQ is the identity on objects, it is clear that this map is injective: for if η 1,η 2:FG\eta_1, \eta_2 : F \to G are two transformations such that η 1Q=η 2Q\eta_1 \circ Q = \eta_2 \circ Q then already η 1=η 2\eta_1 = \eta_2, trivially.

To see surjectivity, notice that if η:FQGQ:CA\eta : F \circ Q \to G \circ Q : C \to A is a transformation, then with the same components this is already a transformation FGF \to G:

one needs to check naturality on an arbitrary morphism f:abf : a \to b in Ho(C)Ho(C) given by a span

a f b a \array{ a' &\stackrel{f'}{\to}& b \\ \downarrow \\ a }

But this can be decomposed as the composition of

a Id a p a \array{ a' &\stackrel{Id}{\to}& a' \\ \downarrow^p \\ a }

with

a f b Id a. \array{ a' &\stackrel{f'}{\to}& b \\ \downarrow^{Id} \\ a' } \,.

The latter is in the image of QQ, hence naturality here is ensured. The former, however, is inverse to

a p a Id a \array{ a' &\stackrel{p}{\to}& a \\ \downarrow^{Id} \\ a' }

which is again in QQ, hence on which naturality of η\eta is again ensured. But the components of a would-be natural transformation satisfy their naturality condition on some morphism if and only if they do on its inverse:

F(a) η a G(a) F(f) G(f) F(b) η b G(b)F(b) η b G(b) F(f) 1 G(f) 1 F(a) η a G(a). \array{ F(a) &\stackrel{\eta_a}{\to}& G(a) \\ \downarrow^{F(f)} && \downarrow^{G(f)} \\ F(b) &\stackrel{\eta_b}{\to}& G(b) } \;\;\; \Leftrightarrow \array{ F(b) &\stackrel{\eta_b}{\to}& G(b) \\ \downarrow^{F(f)^{-1}} && \downarrow^{G(f)^{-1}} \\ F(a) &\stackrel{\eta_a}{\to}& G(a) } \,.

This completes the proof of the universality of QQ.

Proposition

Let f:cdf : c \to d be in WW. The inverse of Q(f)Q(f) in C[W 1]C[W^{-1}] is

Q(f) 1=[c Id c d]. Q(f)^{-1} = \left[ \array{ c &\stackrel{Id}{\to}& c \\ \downarrow \\ d } \right] \,.
Proof

From the commutativity of the square

c Id c Id c Id f f c f d \array{ c &\stackrel{Id}{\to}& c &\stackrel{Id}{\to}& c \\ \downarrow^{Id} &\searrow^{f}& \downarrow^{f} \\ c &\stackrel{f}{\to}& d }

we have

Q(f) 1Q(f)=[c Id c Id c] Q(f)^{-1} \circ Q(f) = \left[ \array{ c &\stackrel{Id}{\to}& c \\ \downarrow^{Id} \\ c } \right]

and

Q(f)Q(f) 1=[c f d f d]=[d Id d Id d]. Q(f) \circ Q(f)^{-1} = \left[ \array{ c &\stackrel{f}{\to}& d \\ \downarrow^f \\ d } \right] = \left[ \array{ d &\stackrel{Id}{\to}& d \\ \downarrow^{Id} \\ d } \right] \,.

Back to our geometric embedding f:FE f: F \hookrightarrow E with WW the preimage under f *:EFf^* : E \to F of the isomorphisms in FF. We find that FF also realizes the homotopy category of EE with respect to WW:

Proposition

FF is equivalent to the localization E[W 1]E[W^{-1}] of EE at WW.

Proof

By one of the above propositions we know that WW is a left multiplicative system.

By the above proposition this implies that the localization Ho(E)E[W 1]Ho(E) \simeq E[W^{-1}] is (equivalent to) the category with the same objects as EE, and with hom-sets given by

Hom E[W 1](a,b)=colimapWaHom E(a,b). Hom_{E[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b) \,.

There is an obvious candidate for a functor FE[W 1]F \to E[W^{-1}], namely

FEQE[W 1]. F \hookrightarrow E \stackrel{Q}{\to} E[W^{-1}] \,.

Recall that this is given on objects by the usual embedding by f *f_* and on morphisms by the map which regards a morphism trivially as a span with left leg the identity

(ab)(a b Id a a). (a \to b) \;\; \mapsto \;\; \left( \array{ a &\to& b \\ \downarrow^{Id_a} \\ a } \right) \,.

For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.

To see essential surjectivity, let aa be any object in EE and let η a:aa¯\eta_a : a \to \bar a be the component of the unit of our adjunction on aa, as above. By one of the above propositons, η a\eta_a is in WW. This means that the span

a Id a a η a a¯ \array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }

represents an element in Hom E[W 1](a¯,a)Hom_{E[W^{-1}]}(\bar a,a), and this element is clearly an isomorphism: the inverse is represented by

a η a a¯ Id a a. \array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.

Since every a¯\bar a is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let a,bFa, b\in F be any two objects. By one of the above propositions this means in particular that bb is a WW-local object. As discussed above, this means that every span

a b w a \array{ a' &\to& b \\ \downarrow^w \\ a }

with wWw \in W has a unique extension

a b w a. \array{ a' &\to& b \\ \downarrow^w & \nearrow \\ a } \,.

But this implies that in the colimit that defines the hom-set of E[W 1]E[W^{-1}] all these spans are identified with spans whose left leg is the identity. And these are clearly in bijection with the morphisms in Hom E(a,b)Hom F(a,b)Hom_E(a,b) \simeq Hom_F(a,b) so that indeed

Hom E[W 1](a,b)Hom F(a,b) Hom_{E[W^{-1}]}(a,b) \simeq Hom_{F}(a,b)

for all a,bFa,b \in F. Hence our functor is also full and faithful and therefore defines an equivalence of categories

FE[W 1]. F \stackrel{\simeq}{\to} E[W^{-1}] \,.
Remark

Here we motivated multiplicative systems from geometric embeddings, but in practice multiplicative systems arise more generally. In particular, the map Q:CHo(C)Q : C \to Ho(C) is not in general left exact left adjoint to a fully faithful embedding Ho(C)CHo(C) \hookrightarrow C.

But what does remain true generally is that the multiplicative system may be characteized by the preimage of QQ on isomorphisms. This is clarified by the following proposition.

Proposition

Let WW be a left multiplicative system on a catgeory CC which is not necessarily a system of weak equivalences (i.e. does not necessarily satisfy 2-out-of-3) and let Q:CHo(C)Q : C \to Ho(C) be the corresponding localization functor.

The following two conditions are equivalent

  • WW is not only contained in but even coincides with the collection of morphisms sent by QQ to isomorphisms;

  • WW is a system of weak equivalences in that it does satisfy the 2-out-of-6 property.

It frequently happens that one has functors CAC \to A out of categories with weak equivalences which do not send all weak equivalences in CC to isomorphisms, but do so for all weak equivalences in a subcategory ICI \hookrightarrow C with Ho(I)Ho(C)Ho(I) \simeq Ho(C).

Proposition

Let CC be a category, CCC' \hookrightarrow C a full subcategory, WW a right multiplicative system in CC, W| CMor(C)W|_{C'} \subset Mor(C') the restriction.

Assume that

  • for every cCc \in C there exists w:ccw : c' \to c with cc' in CC' and wWw \in W.

Then

  • W CW_{C'} is a left multiplicative system;

  • the corresponding homotopy categories coincide: Ho(C)Ho(C)Ho(C') \simeq Ho(C).

Last revised on June 29, 2018 at 06:52:43. See the history of this page for a list of all contributions to it.