Presentations of logoi | Higher Topos Theory notes

4.2. Presentations of logoi

Algebraic objects can often be presented in terms of `generators and relations'; think of presentations of groups or rings. In the case of commutative rings, one first forms the free commutative ring $\Z[S]$ on a given set of generators $$S$$ (the polynomial ring), and then forms the quotient of $\Z[S]$ by an ideal generated by a given set of relations. There is an analogous notion of presentations for logoi:

In Section 4.2.1 we construct the free logos $\An[C]$ on a given category of generators $$C$$ .
In Section 4.2.2, we define the quotient logos $$T/K$$ of a logos $$T$$ by a congruence $$K$$ .
In Section 4.2.3, we introduce logos presentations $(C,\Sigma)$ , consisting of a small category $$C$$ and a small collection of morphisms $\Sigma$ in $\An[C]$ , and define the logos presented by $(C,\Sigma)$ as $\lra{C \mid \Sigma} := \An[C] / \Sigma^c$ , where $\Sigma^c$ is the congruence generated by $\Sigma$ .

4.2.1. Free logoi

We start by constructing the free logos on a small category.

Construction 4.9.

Let $$C$$ be a small category. We define the free logos on $$C$$ as

\An[C] \quad := \quad \PSh(C^{\lex}),

where $C^{\lex} \subseteq \PSh(C\catop)\catop$ is defined as the smallest subcategory containing $$C$$ which is closed under finite limits. The category $\An[C]$ comes equipped with a canonical inclusion

C \hookrightarrow C^{\lex} \hookrightarrow \PSh(C^{\lex}) = \An[C].

Proposition 4.10.

For any small category $$C$$ , the inclusion $C \hookrightarrow \An[C]$ exhibits $\An[C]$ as the free logos generated by $$C$$ : for any other logos $$T$$ , restriction along the inclusion induces an equivalence of categories

\Fun_{\bbLog}(\An[C],T) \quad \simeq \quad \Fun(C,T).

Proof

Recall from [Lurie 2009, Proposition 5.3.6.2] that for a small category

\(C\)

, the subcategory

C^{\mathrm{rex}} \subseteq \PSh(C)

generated by the representables under finite colimits is the free finite cocompletion of

\(C\)

. It follows that

C^{\lex} = ((C\catop)^{\mathrm{rex}})\catop

is the free finite limit completion: for any other category

\(D\)

with finite limits, restriction along

C \hookrightarrow C^{\lex}

induces an equivalence

\Fun^{\lex}(C^{\lex},D) \quad \simeq \quad \Fun(C,D).

By the universal property of the presheaf category, we also see that restriction along

C^{\lex} \hookrightarrow \PSh(C^{\lex})

induces an equivalence

\Fun^{\colim}(\PSh(C^{\lex}),T) \quad \simeq \quad \Fun(C^{\lex},T).

By Proposition 2.43, this functor restricts to the desired equivalence

\Fun_{\bbLog}(\An[C],T) \quad = \quad \Fun^{\colim, \lex}(\PSh(C^{\lex}),T) \quad \simeq \quad \Fun^{\lex}(C^{\lex},T).

Remark 4.11.

We may think of this result as a categorification of the isomorphism $\Hom_{\CRing}(\Z[S],R) \cong \Hom_{\Set}(S,R)$ for every set $$S$$ and every commutative ring $$R$$ .

Example 4.12.

For $C = \emptyset$ , we get $\emptyset^{\lex} = *$ and so $\An[\emptyset] = \PSh(*) = \An$ . In particular, $\An$ is the initial logos:

\Fun_{\bbLog}(\An,T) \quad \simeq \quad \Fun(\emptyset,T) \quad \simeq \quad *.

By dualizing, it follows that $\An \in \Topos$ is the terminal topos.

Example 4.13.

When $C = * = \{X\}$ is the terminal category, we get $C^{\lex} = (\An^{\fin})\catop$ , the opposite of the category of finite animae. In particular, the free logos on one generator is the logos

\An[X] \quad := \quad \An[\{X\}] \quad = \quad \PSh((\An^{\fin})\catop) \quad = \quad \Fun(\An^{\fin},\An).

4.2.2. Congruences and quotients of logoi

In classical algebra, we can form the quotient of a ring $$R$$ by an ideal $$I$$ . As we discuss in this section, we may analogously form the quotient of a logos $$T$$ by a class of morphisms $$K$$ . This produces a new logos $$T/K$$ in which the morphisms of $$K$$ have been inverted. We begin by defining the precise conditions on $$K$$ that ensure such a quotient behaves well.

Definition 4.14.

Let $$T$$ be a logos. A collection of morphisms $$K$$ in $$T$$ is called a congruence of small generation if it satisfies the following conditions:

$$K$$ is strongly saturated of small generation (see Section A.3).
$$K$$ is closed under base change.

Recall from Proposition A.15 that condition (1) ensures that the localization $T \to T[K^{-1}]$ exists and is an accessible Bousfield localization (i.e. the target is presentable and the localization functor admits a fully faithful right adjoint). Condition (2) ensures that the localization is compatible with the logos structure, i.e. that it is left exact:

Lemma 4.15. ({cf. [Lurie 2009, Proposition 6.2.1.1]})

Let $L\colon C \to D$ be a Bousfield localization, with fully faithful right adjoint $R\colon D \hookrightarrow C$ . Assume that $$C$$ admits finite limits. Then the following conditions are equivalent:

The localization functor $$L$$ is left exact.
The class of morphisms inverted by $$L$$ is closed under finite limits in $\Ar(C)$ .
The class of morphisms inverted by $$L$$ is closed under base change.

Proof

The implications (1)

\implies

(2)

\implies

(3) are clear. For

(3) \implies (1)

, let

\(K\)

be the class of morphisms inverted by

\(L\)

, and assume

\(K\)

is closed under base change. Observe that an object of

\(C\)

lies in the essential image of

\(R\)

if and only if it is local with respect to

\(K\)

. Since the terminal object

\(1\)

\(C\)

is clearly

\(K\)

-local, it follows that

L(1) \iso 1

, and so

\(L\)

preserves the terminal object. It remains to show that

\(L\)

also preserves pullbacks, so consider two morphisms

X \to Y \leftarrow Z

. Since

\(K\)

-local objects are closed under limits in

\(C\)

, it follows that

RLX \times_{RLY} RLZ

\(K\)

-local; in particular, it defines the pullback in

\(D\)

. It now remains to show that the canonical map

X \times_Y Z \to RLX \times_{RLY} RLZ

lies in

\(K\)

. We may factor this map as a composite

X \times_Y Z \to X \times_{RLY} Z \to RLX \times_{RLY} Z \to RLX \times_{RLY} RLZ.

The second and third maps are base changes of morphisms in

\(K\)

, hence lie in

\(K\)

again by assumption. The first map is a base change of the diagonal of

Y \to RLY

, which is therefore a section of the projection

\pr_1\colon Y \times_{RLY} Y \to Y

. Since

\(K\)

is closed under 2-out-of-3, it suffices to show that

\pr_1

is in

\(K\)

, which is true as it is a base change of

Y \to RLY

We now combine these results with the properties of saturated classes discussed in the appendix to obtain the following characterization. Recall from Definition A.13 that the kernel of a cocontinuous functor $\phi$ between presentable categories is defined as the class of morphisms inverted by $\phi$ .

Corollary 4.16.

Let $$T$$ be a logos and $$K$$ a class of morphisms in $$T$$ . The following are equivalent:

$$K$$ is a congruence of small generation.
$$K$$ is strongly saturated of small generation and is closed under finite limits in $\Ar(T)$ .
The localization functor $L\colon T \to T[K^{-1}]$ is left exact and admits a fully faithful right adjoint (i.e. it is a Bousfield localization).
$$K$$ is the kernel of some morphism of logoi $\phi\colon T \to S$ .

Proof

The equivalence

(1) \iff (3)

follows immediately from Definition 4.14 and Lemma 4.15, combined with Proposition A.15.The implication

(3) \implies (4)

follows since

\(K\)

is the kernel of

L\colon T \to T[K^{-1}]

, see Proposition A.15. For

(4) \implies (2)

, if

\(K\)

is the kernel of a morphism of logoi

\phi

, then

\(K\)

is closed under finite limits because

\phi

is left exact. Furthermore, since

\phi

is a morphism of logoi, it preserves colimits, and thus its kernel is strongly saturated of small generation by [Lurie 2009, Proposition 5.5.4.16] (see Proposition A.14).Finally, for

(2) \implies (1)

, it suffices to note that a class closed under finite limits is in particular closed under base change.

We can now define the quotient of a logos.

Definition 4.17.

Let $$K$$ be a congruence of small generation in a logos $$T$$ . We define the quotient of $$T$$ by $$K$$ as the localization

T/K \quad := \quad T[K^{-1}].

By Corollary 4.16, $$T/K$$ is a logos and the localization map $T \to T/K$ is a morphism of logoi.

This construction satisfies the expected universal property:

Proposition 4.18.

For every logos $$S$$ , restriction along $T \to T/K$ induces a fully faithful functor

\Fun_{\bbLog}(T/K, S) \quad \hookrightarrow \quad \Fun_{\bbLog}(T,S).

Its essential image consists of those morphisms of logoi $\phi\colon T \to S$ that invert $$K$$ (i.e. $K \subseteq \ker(\phi)$ ).

Proof

By definition of localization, restriction along

T \to T/K

induces a fully faithful functor

\Fun(T/K,S) \hookrightarrow \Fun(T,S),

with essential image those functors

T \to S

that invert

\(K\)

. It remains to show that a functor

T/K \to S

preserves colimits and finite limits if and only if the composite

T \to T/K \to S

does.The `only if' direction is clear. For the `if' direction, note that every diagram in

\(T/K\)

may be regarded as a diagram in

\(T\)

via the inclusion. Since

T \to T/K

preserves colimits and finite limits, colimits and finite limits in

\(T/K\)

may be computed by forming them first in

\(T\)

and then localizing. Thus if

T \to S

is a logos morphism, then so is

T/K \to S

Definition 4.19.

A morphism of logoi $\phi\colon T \to S$ is called a quotient map if it is a Bousfield localization, i.e. if its right adjoint is fully faithful.

Note that any quotient map $\phi\colon T \to S$ induces an equivalence of logoi $T/\ker(\phi) \simeq S$ .

Lemma 4.20.

Any quotient map of logoi is an epimorphism in the category of logoi.

Proof

Let

\phi^*\colon T \to S

be a quotient map and put

K=\ker(\phi^*)

. The induced morphism

T/K\to S

is an equivalence of logoi. By the universal property of the quotient from Proposition 4.18, for every logos

\(T'\)

the restriction functor

(\phi^*)^*\colon \Hom_{\Logos}(S,T') = \Fun^{\lex,\colim}(S,T')^{\simeq} \hookrightarrow \Fun^{\lex,\colim}(T,T')^{\simeq} = \Hom_{\Logos}(T,T').

is fully faithful. This is precisely the assertion that

\phi^*

is an epimorphism in

\Logos

Remark 4.21.

The previous lemma implies that any quotient map of logoi is a monomorphism when regarded as a morphism in $\Topos$ . This is analogous to the fact that an epimorphism $R \to S$ of commutative rings induces a monomorphism $\Spec(S) \hookrightarrow \Spec(R)$ of affine schemes. Recall that there are two common constructions for commutative rings that give rise to epimorphisms:

Given a commutative ring $$R$$ and an ideal $$I$$ , we may form the quotient ring $R \to R/I$ by killing off all elements in $$I$$ .
Given a commutative ring $$R$$ and a set $$S$$ of elements of $$R$$ , we may form the localization $R \to R[S^{-1}]$ by formally inverting all elements in $$S$$ .

The quotient map $R \to R/I$ corresponds to a closed immersion of affine schemes. A localization map $R \to R[S^{-1}]$ induces a flat monomorphism $\Spec(R[S^{-1}])\to\Spec(R)$ ; it is an open immersion when $$S$$ is generated by finitely many elements, equivalently when the localization may be written as $R\to R[f^{-1}]$ for a single $f\in R$ . We will discuss the analogous notions of open and closed immersions of topoi in Section 5.4 below.

We conclude with two fundamental examples of congruences.

Example 4.22.

Let $\Sigma$ be a small set of morphisms in a logos $$T$$ . The congruence generated by $\Sigma$ , denoted $\Sigma^c$ , is defined as the intersection of all strongly saturated classes containing $\Sigma$ that are closed under base change. It is a non-trivial fact that $\Sigma^c$ is of small generation (and thus a congruence in the sense of Definition 4.14). This is proved by Lurie (2009, Proposition 6.2.1.2). We will give an independent proof below, see Remark 5.42.

Example 4.23.

Let $\phi\colon S \to T$ be a morphism of logoi. Given a congruence of small generation $$K$$ in $$T$$ , we define its preimage as

\phi^{-1}(K) \quad := \quad \{g\in \Ar(S) \mid \phi(g) \in K\}.

This is again a congruence of small generation. Indeed, consider the composite morphism of logoi

S \xrightarrow{\phi} T \xrightarrow{L} T/K.

As the kernel of $$L$$ is $$K$$ , the kernel of this composite is precisely $\phi^{-1}(K)$ . By Corollary 4.16, the kernel of any morphism of logoi is a congruence of small generation, which proves the claim.

4.2.3. Presentations of logoi

We now combine free logoi with quotients of logoi to arrive at the following definition:

Definition 4.24.

A logos presentation is a pair $(C,\Sigma)$ consisting of a small category $$C$$ together with a small class of morphisms $\Sigma$ in $\An[C]$ . We define the logos presented by $(C,\Sigma)$ as the quotient

\lra{C \mid \Sigma} \quad := \quad \An[C] \, / \, \Sigma^c

of the free logos on $$C$$ by congruence $\Sigma^c$ from Example 4.22. In particular, $\lra{C \mid \Sigma}$ is a logos and the quotient map $\An[C] \to \lra{C \mid \Sigma}$ is a logos morphism.

We wish to think of $$C$$ as the `generators' of $\lra{C \mid \Sigma}$ , and of $\Sigma$ as the `relations'.

More explicitly, $\lra{C \mid \Sigma}$ may be identified with the full subcategory of $\PSh(C^{\lex})$ spanned by the $\Sigma^c$ -local presheaves.

Lemma 4.25.

Let $(C,\Sigma)$ be a logos presentation. Then the quotient functor $\An[C] \to \lra{C \mid \Sigma}$ is universal among logos morphisms $\An[C] \to T$ that invert $\Sigma$ .

Proof

By Proposition 4.18,

\An[C] \to \lra{C \mid \Sigma}

is universal among logos morphisms

\phi\colon \An[C] \to T

satisfying

\Sigma^c \subseteq \ker(\phi)

. But since

\ker(\phi)

is a congruence, this is equivalent to the condition that

\Sigma \subseteq \ker(\phi)

, i.e. that

\phi

inverts

\Sigma

We now show that every logos admits a presentation.

Proposition 4.26.

Let $$T$$ be a logos. Then there exists a logos presentation $(C,\Sigma)$ and an equivalence of logoi

\lra{C \mid \Sigma} \simeq T.

Proof

We saw in the proof of Theorem 2.42 that there is a small category

\(C\)

with finite limits and an accessible left exact localization

L\colon \PSh(C)\longrightarrow T.

Let

i\colon C\hookrightarrow C^{\lex}

be the canonical functor, and let

F\colon C^{\lex}\to C

be the left exact extension of

\id_C

. The universal property of

C^{\lex}

gives

\Hom_{C^{\lex}}(iX,Y)\simeq\Hom_C(X,FY)

for

X\in C

and

Y\in C^{\lex}

. Thus

i\dashv F

, and the unit

\id_C\to Fi

is an isomorphism.The left Kan extension along

\(F\)

is therefore restriction along

\(i\)

F_!\simeq i^*\colon \PSh(C^{\lex})\longrightarrow\PSh(C).

In particular,

\(F_!\)

is left exact. Its right adjoint is the restriction functor

F^*\colon\PSh(C)\to\PSh(C^{\lex})

, which is fully faithful because

Fi\simeq\id_C

. Hence

\(F_!\)

is a left exact Bousfield localization. It follows that the composite

Q\colon \An[C]=\PSh(C^{\lex})\xrightarrow{F_!}\PSh(C)\xrightarrow{L}T

is again a left exact Bousfield localization.Its kernel

K:=\ker(Q)

is a congruence of small generation by Corollary 4.16. Choose a small class

\Sigma\subseteq K

which generates

\(K\)

as a strongly saturated class. Since

\(K\)

is a congruence,

\Sigma^c\subseteq K

; conversely,

\Sigma^c

is strongly saturated and contains

\Sigma

, so

K\subseteq\Sigma^c

. Hence

K=\Sigma^c

. The universal property of the quotient now identifies

\lra{C\mid\Sigma}=\An[C]/\Sigma^c\simeq T,

which is the desired presentation.

Example 4.27.

The following are some examples of presentations of logoi:

As discussed, the initial logos (no generators and no relations) is $\An$ :
$\lra{\emptyset \mid \emptyset} \quad = \quad \An.$
The free logos on a single generator $$X$$ is $\An[X]$ :
$\lra{\{X\} \mid \emptyset} \quad = \quad \An[X] \quad = \quad \Fun(\An^{\fin}, \An).$
More generally, the logos presented by $(C,\emptyset)$ is simply the free logos generated by $$C$$ :
$\lra{C \mid \emptyset} \quad = \quad \An[C] \quad = \quad \PSh(C^{\lex}).$
The free logos generated by some initial object $$X$$ is simply $\An$ :
$\lra{\{X\} \mid \emptyset \to X} \quad \simeq \quad \An.$
Indeed, logos morphisms $F\colon \lra{\{X\} \mid \emptyset \to X} \to T$ correspond to logos morphisms $\An[X] = \lra{\{X\} \mid \emptyset} \to T$ sending the generator $$X$$ to the initial object of $$T$$ , but by definition of $\An[X]$ there is a unique such morphism.
The free logos in which the initial object is terminal is the one-object logos:
$\lra{\{X\} \mid \emptyset \to *} \quad \simeq \quad *.$
Indeed, any logos in which the initial and terminal objects agree is trivial: for all $$Y$$ we have $\emptyset \simeq \emptyset \times Y \simeq * \times Y \simeq Y$ .
The free logos generated by an $$n$$ -truncated object is
$\lra{\{X\} \mid X \to X^{S^{n+1}}} \quad \simeq \quad \Fun((\An_{\leq n})^{\fin},\An),$
where $(\An_{\leq n})^{\fin} \subseteq \An_{\leq n}$ is the subcategory generated under finite colimits by the point. The left-hand side classifies the assignment $T \mapsto T_{\leq n}$ of $$n$$ -truncated objects. Indeed, an object of $$T$$ corresponds to a left exact functor $(\An^{\fin})\catop\to T$ . This object is $$n$$ -truncated precisely when the corresponding functor factors through the opposite of the truncation functor
$\tau_n\colon \An^{\fin}\longrightarrow (\An_{\leq n})^{\fin}.$
The free logos generated by an $$n$$ -connected object $$X$$ is
$\lra{\{X\} \mid \tau_n(X) \to *}.$
It arises as a localization of $\An[X] = \lra{* \mid \emptyset}$ .
The pointed object classifier is the logos
$\lra{[1] \mid \mathrm{source} = *} \quad \simeq \quad \PSh((\An_*^{\fin})\catop).$
Note that a logos morphism $\lra{[1] \mid \mathrm{source} = *} \to T$ is just a morphism in $$T$$ whose source is terminal, i.e. a pointed object. But the right-hand side also classifies pointed objects: we have
$\Fun_{\bbLog}(\PSh((\An_*^{\fin})\catop),T) \, \simeq \, \Fun^{\lex}((\An_*^{\fin})\catop, T) \, \simeq \, \Fun^{\lex}((\An_*^{\fin})\catop, T_*) \, \simeq \, T_*.$
Given a Lawvere theory $$L$$ , there is the $$L$$ -algebra classifier $\lra{L \mid \text{finite products}}$ : for every logos $$T$$ we have
$\Fun_{\bbLog}(\lra{L \mid \text{finite products}}, T) \quad \simeq \quad \Alg_L(T) \quad := \quad \Fun^{\times}(L,T).$
This logos may be explicitly described as
$\lra{L \mid \text{finite products}} \quad \simeq \quad \PSh((\PSh_{\Sigma}(L)^{\fin})\catop).$
Here $\PSh_{\Sigma}(L)\subseteq\Fun(L\catop,\An)$ denotes the full subcategory of finite-product-preserving presheaves.

References

Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.