Schreiber geometric embedding

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• last time we had seen that every continuous map $f : X \to Y$ of topological spaces induces an adjoint pair $f^* \dashv f_*$ of functors of the corresponding presheaf categories $PSh(X) \stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\to}} PSh(Y)$ 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 $L : C \to D$ and $R : D \to C$ are adjoint in that there is a natural isomorphism

$$\Phi : Hom_D(L(-),-) \stackrel{\simeq}{\to} Hom_E(-, R(-)) \,.$$

Lemma

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

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

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

such that for all objects $c \in C$ and $d \in D$

• $(L(c) \stackrel{L(\eta_c)}{\to} L\circ R \circ L(c) \stackrel{\epsilon_{L(c)}}{\to} L(c)) = Id_c$

• $(R(d) \stackrel{\eta_{R(d)}}{\to} R \circ L \circ R(d) \stackrel{R(\epsilon_{d})}{\to} R(d)) = Id_d$.

Proof

Given a unit-counit adjunction $(\eta,\epsilon) : L \dashv R$ define a natural transformation $\Phi : Hom_D(L(-),-) \to Hom_C(-,R(-))$ by setting

$$\Phi : (L(c) \stackrel{f}{\to} d) \mapsto (c \stackrel{\eta_c}{\to} R \circ L (c) \stackrel{R(f)}{\to} R(d)) \,.$$

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

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

$$\Phi^{-1} : (c \stackrel{g}{\to} R(d)) \mapsto (L(c) \stackrel{L(g)}{\to} L\circ R(d) \stackrel{\epsilon_d}{\to} d) \,.$$

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

Conversely, given the hom-isomorphism $\Phi$, define for all objects $c \in C$ and $d \in D$

$$(c \stackrel{\eta_c}{\to} R \circ L(c)) := \Phi( L(c) \stackrel{Id}{\to} L(c))$$

$$(L \circ R (d) \stackrel{\epsilon_d}{\to} d) := \Phi^{-1}( R(d) \stackrel{Id}{\to} R(d)) \,.$$

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

$$\array{ Hom_D(L(c),d) &\stackrel{\Phi}{\to}& Hom_C(c,R(d)) \\ \uparrow^{f \circ -} && \uparrow^{R(f) \circ -} \\ Hom_D(L(c),L(c)) &\stackrel{\Phi}{\to}& Hom_C(c, R \circ L(c)) }$$

that

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

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

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

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

Proposition

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

• $R$ being a full and faithful functor is equivalent to $\epsilon : L \circ R \to Id_D$ being an isomorphism;

• $L$ being a full and faithful functor is equivalent to $\eta : Id_C \to R \circ L$ being an isomorphism;

Proof

This follows directly once one observes that the following diagrams commute

$$\array{ && Hom_C(R(d), R(d')) \\ & {}^{R_{d,d'}}\nearrow && \searrow^{\Phi^{-1}} \\ Hom_D(d,d') &&\stackrel{Hom_D(\epsilon_d, d')}{\to}&& Hom_D(L \circ R(d), d') }$$

and

$$\array{ Hom_C(c,c') &&\stackrel{Hom_C(c, \eta_{c'})}{\to}&& Hom_C(c, R \circ L(c') \\ & {}_{L_{c,c'}}\searrow && \nearrow_{\Phi} \\ && Hom_D(L(c), L(c')) } \,.$$

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

$$\Phi(L \circ R (d) \stackrel{\epsilon_d}{\to} d \stackrel{f}{\to} d') = (R(c) \stackrel{R(f)}{\to} R(d')) \,.$$

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

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

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

• they have all limits and colimits;

• they are cartesian closed.

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

Definition

A topos $E$ is a category that

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

This already implies in particular that $E$

• is cartesian closed.

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

• MacLane-Moerdijk, Sheaves in Geometry and Logic

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

$$f_* : F \to E$$

$$f^* : E \to F \,,$$

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

Recall that, as a left adjoint, $f^*$ 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 $F \hookrightarrow E$.

Notably the inclusion $Sh(S) \hookrightarrow PSh(S)$ of a category of sheaves into its presheaf topos or more generally the inclusion $Sh_j E \hookrightarrow E$ of sheaves in a topos $E$ into $E$ 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 $F \hookrightarrow E$ is the localizations of $E$ at the class $W$ or morphisms that the left adjoint $E \to F$ sends to isomorphisms in $F$.

Definition

For $F$ and $E$ two topoi, a geometric morphism

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

is a geometric embedding if the following equivalent conditions are satisfied

• the direct image functor $f_*$ is full and faithful (so that $F$ is a full subcategory of $E$);

• the counit $\epsilon : f^* f_* \to Id_{F}$ of the adjunction $(f^* \dashv f_*)$ is an isomorphism

Remark

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

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 $\eta : Id_E \to f_* f^*$ for the unit of the adjunction.

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

Definition

A category with weak equivalences is a category $C$ equipped with a subcategory $W \subset C$

• which contains all isomorphisms of $C$;

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

Definition

Write $W$ for the class of morphism in $E$ that are sent to isomorphism under $f^*$,

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

Proposition

$E$ equipped with the class $W$ is a category with weak equivalences, in that $W$ satisfies 2-out-of-3.

Proof

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

Definition

A left multiplicative system in a category $C$ is a collection $W$ of morphisms in $C$ such that

• all isomorphisms are in $W$;

• $W$ is closed under composition;

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

• given $A \stackrel{\stackrel{g}{\to}}{\stackrel{f}{\to}} B \stackrel{w}{\to} C$ with $w \circ f = w \circ g$ there is $D \stackrel{w'}{\to} A \stackrel{\stackrel{g}{\to}}{\stackrel{f}{\to}} B$ such that $f \circ w' = g \circ w'$.

Proposition

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

Proof

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

In more detail:

We have already seen in the previous proposition that

• every isomorphism is in $W$;

• $W$ is closed under composition.

It remains to check the following points:

Given any

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

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

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

with $w' \in W$.

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

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

is a pullback diagram in $F$ with $\bar w' = \bar h^* \bar w$. But by assumption $\bar w$ is an isomorphism. Therefore $\bar w'$ is an isomorphism, therefore $w'$ is in $W$.

Finally for every

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

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

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

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

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

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

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

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

of two equal morphism, hence an identity. So $w'$ is in $W$.

Proposition

For every object $a \in E$

• the unit $\eta_a : a \to \bar a$ is in $W$;

• if $a$ is already in $F$ then the unit is already an isomorphism.

Proof

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

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

and

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

In components they say that

• for every $a \in E$ we have $(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}$

• for every $a \in F$ we have $(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a) = Id_a$

This implies the claim.

Definition

An object $a \in E$ is a $W$-local object if for every $g : c \to d$ in $W$ the map

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

obtained by precomposition is an isomorphism.

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

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

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

Proposition

Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

Proof

First assume that $a \in F$. We need to show that $a$ is $W$-local.

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

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

there is a unique extension

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

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

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

By the assumption that $c \to d$ is in $W$ the morphism $\bar c \to \bar d$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\bar a \simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

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

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

$$Hom_E(\bar d, a) \simeq Hom_E(d,a)$$

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

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

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

So $a \in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

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

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }$$

has an extension

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.$$

By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that $Id_a$, being an isomorphism, is in $W$) it follows that $\rho_a : \bar a \to a$ is in $W$.

Since $\bar a$ is in $F$ and therefore $W$-local by the above, it follows that also

$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }$$

has an extension

$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.$$

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

$$\begin{aligned} \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} & = \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} \underbrace{ \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} }_{Id} \\ & = \stackrel{\rho_a}{\to} \underbrace{ \stackrel{\eta_a}{\to} \stackrel{\rho_a}{\to} }_{Id} \stackrel{\lambda_a}{\to} \\ &= \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} \\ &= Id \end{aligned} \,.$$

So if $a$ is $W$-local we find that $\eta_a : a \to \bar a$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

Lemma

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

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

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

Proof

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

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

Then for any morhism $f : c \to c'$ in $C$ set

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

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

Corollary

$F$ is equivalent to the full subcategory $E_{W-loc}$ of $E$ on $W$-local objects.

Proof

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

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

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

The homotopy category

Given a category with weak equivalences $W$, there is another construction one wants to consider: the homotopy category $C[W^{-1}]$ of $C$ with respect to $W$. 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 $C$ which are known in some way or other to be the 1-categorical truncation of higher categories $\hat C$.

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

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

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

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

Definition

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

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

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

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

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

• for any other category $A$ and functor $F : C \to A$ such that $F$ sends all $w \in W$ to isomorphisms in $A$, there exists a functor $F_Q : Ho(C) \to A$ and a natural isomorphism

$$\array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ Ho(C) }$$

• the functor $Q^* : Func(Ho(C),A) \to Func(C,A)$ is a full and faithful functor.

Remarks

• This definition by itself does not use any properties of $W$, 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 $W ' \supset W$ of morphisms which is sent to isos by $W$ will be a system of weak equivalences.

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

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

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

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

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

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

Definition

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

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

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

in $C$ (i.e. $f$ need not be in $W$).

Notice that there is still the obvious forgetful functor

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

obtained by remembering only the horizontal morphism of these triangles.

Proposition

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

• objects are those of $C$;

• hom-sets are given by

$$Hom_{C[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_C(a',b) := colim ( (W/a)^{\mathrm{op}} \to C^{\mathrm{op}} \stackrel{Hom_C(-,b)}{\to} Set )$$

• composition is defined on representing spans by using the third property of left multiplicative systems.

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

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

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

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

with $p$ in $W$, and two such spans $(f'_i)$ are regarded as equivalent if there is a third one $f'_3$ fitting into a commuting diagram

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

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

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

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

One checks that this composition is indeed well defined:

• let $q \to a'$ and $q' \to a'$ be two different ways to fill the top left square. By one of the axioms satisfied by $W$ one finds a further $q''$ such that $\array{ q'' &\to& q' \\ \downarrow && \downarrow \\ q &\to& a' } \,.$ This may not yet be sufficient, since it doesn’t imply that $q'' \to q' \to b'$ equals $q'' \to q \to b'$. But it does follow that

$$(q'' \to q \to b' \stackrel{p_2}{\to} b) = (q'' \to q' \to b' \stackrel{p_2}{\to} b) \,.$$

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

$$\array{ && q \\ & {}^{}\swarrow &\uparrow& \searrow^{} \\ a &\stackrel{}{\leftarrow}& q''' &\stackrel{}{\to}& c \\ & {}_{}\nwarrow &\downarrow& \nearrow_{} \\ && q' } \,.$$

• similarly one finds that for different choices of representatives of $f$ and $g$ 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 $Q$ sends $f : b \to c$ in $W$ to an isomorphism.

To see this, we show that for all $a$ we have

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

is an isomorphism.

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

First, $f_*$ is surjective:

for every span

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

we may find

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

with $q \to a'$ in $W$. This realizes $h$ as in the image of $f_*$.

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

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

are equal, then by assumption on $W$ there is $q \to a'$ such that

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

are equal, hence represent the same morphism.

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

So let

$$F : C \to A$$

be any functor that sends morphisms $w \in W$ to isomorphism in $A$. We obtain from this a functor $F_Q$ by setting

• on objects: $F_Q|_{Obj} = F|_{Obj}$

• on morphisms: apply $F$ to any one representative span by

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

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

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

Here we used that due to assumption on $F$ for $p : a'\to a$ in $W$ the morphism

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

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

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

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

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

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

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

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

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

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

But this can be decomposed as the composition of

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

with

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

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

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

which is again in $Q$, hence on which naturality of $\eta$ 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:

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

This completes the proof of the universality of $Q$.

Proposition

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

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

Proof

From the commutativity of the square

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

we have

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

and

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

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

Proposition

$F$ is equivalent to the localization $E[W^{-1}]$ of $E$ at $W$.

Proof

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

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

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

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

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

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

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

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 $a$ be any object in $E$ and let $\eta_a : a \to \bar a$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, $\eta_a$ is in $W$. This means that the span

$$\array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }$$

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

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.$$

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

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

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

with $w \in W$ has a unique extension

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

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

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

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

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

Remark

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

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

Proposition

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

The following two conditions are equivalent

• $W$ is not only contained in but even coincides with the collection of morphisms sent by $Q$ to isomorphisms;

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

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

Proposition

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

Assume that

• for every $c \in C$ there exists $w : c' \to c$ with $c'$ in $C'$ and $w \in W$.

Then

• $W_{C'}$ is a left multiplicative system;

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