Congruences, modalities, and saturated classes

5.3. Congruences, modalities, and saturated classes

Four closure notions meet in this section. Their definitions, and the smallness hypotheses needed to turn them into categorical constructions, are summarized as follows: \begin{center} \small \begin{tabularx}{\textwidth}{@{}lXX@{}} \toprule Class & Closure properties & Small-generation consequence \\ \midrule Saturated & Isomorphisms, composition, colimits & Left class of a factorization system. \\ Acyclic & Saturated and stable under base change & Left class of a modality. \\ Strongly saturated & Saturated and 2-out-of-3 & Kernel of an accessible localization. \\ Congruence & Strongly saturated and stable under base change & Kernel of a logos morphism. \\ \bottomrule \end{tabularx} \end{center} In particular, neither “acyclic class” nor “congruence” includes a smallness condition. Small generation is stated separately whenever it is needed to produce a modality or a quotient logos.

We will establish three relations among these notions. First, the acyclic class $\Sigma^m$ generated by a set $\Sigma$ is of small generation and hence defines a modality. Second, congruences are precisely the acyclic classes closed under diagonals. Third, closing a set under iterated diagonals before taking its acyclic closure gives the generated congruence:

\Sigma^c=(\Sigma^{\Delta})^m.

These results will be our main tool for checking that localizations of topoi are again topoi. As an application, they give a short proof that the category of sheaves on a Grothendieck site is a topos; we spell this out in Section 6.1. This application seems to be a new observation by Marc Hoyois which, to the note-taker's knowledge, does not appear in the literature.

Notation 5.30.

We denote by

\Sat(T), \qquad \SSat(T), \qquad \Acyc(T), \qquad \Mdl(T), \qquad\text{and}\qquad \Cong(T)

the partially ordered sets of saturated classes, strongly saturated classes, acyclic classes, modalities, and congruences in $$T$$ , with partial order given by inclusion. We regard a modality as its left class, giving inclusions $\Mdl(T)\subseteq\Acyc(T)\subseteq\Sat(T)$ , while $\Cong(T)\subseteq\Acyc(T)\cap\SSat(T)$ .

Notation 5.31.

Given a class of morphisms $\Sigma \subseteq \Ar(T)$ , we define the following closures:

$\Sigma^s$ : the saturation (closure under composition, identities, and colimits).
$\Sigma^{ss}$ : the strong saturation (closure under composition, colimits, and 2-out-of-3).
$\Sigma^m$ : the smallest acyclic class containing $\Sigma$ .
$\Sigma^c$ : the smallest strongly saturated class closed under base change containing $\Sigma$ .

5.3.1. Generated acyclic classes and modalities

The notation $\Sigma^m$ refers a priori only to an acyclic class. We now show that if $\Sigma$ is a set, this acyclic class is of small generation and therefore is the left class of a modality.

Proposition 5.32. ([Anel et al. 2022, Lemma 3.2.13, Proposition 3.2.18, Corollary 3.2.19])

Let $$T$$ be a topos, let $\Sigma$ be a set of morphisms, and let $\Gg$ be a set of generators of $$T$$ . Let $\Sigma^{bc}$ denote the set of base changes of morphisms in $\Sigma$ to objects in $\Gg$ . Then we have

\Sigma^m \quad = \quad (\Sigma^{bc})^s.

In particular, $\Sigma^m$ is a modality of small generation.

Proof

Write

\Sigma' := (\Sigma^{bc})^s

, the saturation of

\Sigma^{bc}

. It is clear that

\Sigma' \subseteq \Sigma^m

, so it remains to show that also

\Sigma^m \subseteq \Sigma'

. To this end, consider the following class of morphisms

\Sigma'' := \{\,u \in \Sigma' \mid \textup{every base change of $u$ lies in $\Sigma'$}\,\}.

We have

\Sigma'' \subseteq \Sigma'

, so it remains to show that in fact

\Sigma^m \subseteq \Sigma''

. We will do this by showing that

\Sigma''

is a saturated class closed under base change which contains

\Sigma

.It is clear that

\Sigma''

is closed under base change. We claim that it is also saturated. It is clear that

\Sigma''

is closed under composition and contains all isomorphisms, so it remains to show that it is closed under colimits. Consider a diagram of maps

X_{\bullet} \to Y_{\bullet}

\Sigma''

, and let

Z \to \colim_i Y_i

be a map. We need to show that the base change

Z \times_{\colim_i Y_i} \colim_i X_i \to Z

lies in

\Sigma'

. By descent, we may write

\(Z\)

\colim_i Z_i

, and hence this base change map may be written as the colimit of the maps

Z_i \times_{Y_i} X_i \to Z_i

. Since each of these base changes lies in

\Sigma'

by definition of

\Sigma''

, the claim follows from the fact that

\Sigma'

is closed under colimits.It remains to show that

\Sigma''

contains

\Sigma

. In other words, given a morphism

u\colon A \to B

\Sigma

and an arbitrary map

Z \to B

, we need to show that the base change

Z \times_B A \to Z

lies in

\Sigma'

. To this end, let

\Ee \subseteq T_{/B}

be the full subcategory spanned by those

Z \to B

for which this condition holds. By definition of

\Sigma'

, we have

\Sigma^{bc} \subseteq \Sigma'

, and in particular

\Ee

contains all maps

Z \to B

with

Z \in \Gg

. Since

\Sigma'

is closed under colimits, it follows from descent that

\Ee \subseteq T_{/B}

is closed under colimits. But since

\Gg

is a set of generators, it follows that

\Ee = T_{/B}

, showing the claim.

Remark 5.33.

On classes of small generation, the assignments $\Sigma \mapsto \Sigma^m$ and $\Sigma \mapsto \Sigma^c$ define the respective left adjoints to the inclusions of modalities and congruences into saturated classes. Without a smallness restriction, the same closure operations are left adjoint to the inclusions $\Acyc(T)\hookrightarrow\Sat(T)$ and $\Cong(T)\hookrightarrow\Sat(T)$ at the level of partially ordered classes.

5.3.2. Characterizations of congruences

The definition of $\Sigma^c$ involves closure under 2-out-of-3, an operation over which we have little explicit control. The following characterizations replace it by finite-limit or diagonal closure.

Lemma 5.34.

Let $\Ee$ be a category with finite limits and let $$K$$ be a class of morphisms in $\Ee$ which contains all identities and is closed under finite limits. Then $$K$$ satisfies the left cancellation property. Dually, a class containing all identities and closed under finite colimits satisfies the right cancellation property.

Proof

It suffices to prove the first claim, as the second one is dual. Given morphisms

f\colon X \to Y

and

g\colon Y \to Z

, the following commutative square in

\Ar(T)

exhibits

\(f\)

as a limit in

\Ar(T)

of the morphisms

\id_Y

\(gf\)

and

\(g\)

Since

\(K\)

contains identities and is closed under finite limits, if

g, gf \in K

, then

f \in K

Proposition 5.35.

The following conditions are equivalent for a class of morphisms $$K$$ :

$$K$$ is a congruence;
$$K$$ is strongly saturated and stable under base change;
$$K$$ is strongly saturated and closed under finite limits;
$$K$$ is saturated and closed under finite limits;
$$K$$ is an acyclic class and is closed under diagonals: if $f\colon X \to Y$ is in $$K$$ , then so is $\Delta_f\colon X \to X \times_Y X$ .

Proof

The equivalence between (1) and (2) is the definition. A congruence is closed under finite limits in

\Ar(T)

, see [Anel et al. 2022, Proposition 4.2.3], which gives (2)

\Rightarrow

(3). Clearly (3) implies (4). Conversely, finite-limit closure implies stability under base change, since a base-change square is a pullback in

\Ar(T)

. Moreover, Lemma 5.34 gives left cancellation. Saturation already gives composition and right cancellation, so

\(K\)

satisfies 2-out-of-3. This proves (4)

\Rightarrow

(2).Condition (2) implies (5): the projection

X\times_YX\to X

is a base change of

\(f\)

, and its section

\Delta_f

belongs to

\(K\)

by 2-out-of-3. Conversely, assume (5). Since

\(K\)

is saturated, it is closed under composition and has right cancellation. It remains to check left cancellation. Given

X \xrightarrow{f} Y \xrightarrow{g} Z

with

g, gf \in K

, factor

\(f\)

X \xrightarrow{(\id, f)} X \times_Z Y \xrightarrow{\pr_Y} Y.

The second map is a base change of

gf \in K

, hence in

\(K\)

. The first is a base change of

\Delta_g

, which is in

\(K\)

by assumption. Thus

f \in K

5.3.3. Décalage and generated congruences

We now extract the largest congruence contained in an acyclic class. Iterating this construction will turn diagonal closure into the explicit formula for a generated congruence.

Definition 5.36. ([Anel et al. 2024, Section 2.2.7])

Let $$L$$ be an acyclic class in $$T$$ . We define the décalage of $$L$$ by

D(L) := \{\,u \in L \mid \Delta_u \in L\,\} \subseteq L.

We inductively define $$D^n(L)$$ by $$D^0(L) := L$$ and $D^{n+1}(L) := D(D^n(L))$ . Finally, we set $D^{\infty}(L) := \cap_n D^n(L)$ .

Proposition 5.37. ([Anel et al. 2024, Section 2.2.7])

If $$L$$ is an acyclic class, then $$D(L)$$ is again an acyclic class. If $$L$$ is of small generation, then so is $$D(L)$$ ; in this case both classes are left classes of modalities.

Proof

Stability under base change and the inclusion of all isomorphisms are immediate. For composition, let

A \xrightarrow{f} B \xrightarrow{g} C

lie in

\(D(L)\)

. Then

gf\in L

, and

\Delta_{gf}

factors as

A \xrightarrow{ \Delta_f } A \times_B A \longrightarrow A \times_C A ,

where the second map is a base change of

\Delta_g

. Since

\Delta_f,\Delta_g\in L

and

\(L\)

is an acyclic class,

\Delta_{gf}\in L

.For colimits, let

f_{\bullet}\colon X_{\bullet}\to Y_{\bullet}

be a diagram in

\Ar(T)

with every

f_i\in D(L)

, and write

f\colon X\to Y

for its colimit. We already know

f\in L

. The acyclic-descent argument of [Anel et al. 2024, Section 2.2.7] says that colimits of

\(L\)

-cartesian transformations remain

\(L\)

-cartesian. We spell out its application here, since it also fixes the variance of the maps involved.For a morphism

i\to j

in the indexing category, the gap map of the naturality square is

X_i\longrightarrow Y_i\times_{Y_j}X_j.

It factors as

X_i\longrightarrow X_i\times_{Y_j}X_j \longrightarrow Y_i\times_{Y_j}X_j.

The first map is a base change of

\Delta_{f_j}

, and the second is a base change of

\(f_i\)

. Both lie in

\(L\)

, so the transformation

f_{\bullet}

\(L\)

-cartesian. Acyclic descent now shows that for every

\(i\)

the square

\(L\)

-cartesian. Equivalently, the gap map

h_i\colon X_i\to Y_i\times_YX

lies in

\(L\)

.By universality of colimits, the diagonal

\Delta_f

is the colimit of the maps

X_i\to X_i\times_YX

. Each of these factors as

X_i\xrightarrow{\Delta_{f_i}}X_i\times_{Y_i}X_i \longrightarrow X_i\times_YX,

where the second map is a base change of

\(h_i\)

. Thus both maps lie in

\(L\)

. Closure under colimits gives

\Delta_f\in L

, proving that

\(D(L)\)

is saturated. The small-generation assertion is the accessibility part of the cited décalage result.

Corollary 5.38.

For an acyclic class $$L$$ in $$T$$ , the class $D^{\infty}(L)$ is a congruence. Moreover, it is the largest congruence contained in $$L$$ . Thus $L\mapsto D^{\infty}(L)$ defines a right adjoint $\Acyc(T)\to\Cong(T)$ to the inclusion $\Cong(T)\hookrightarrow\Acyc(T)$ .

Proof

By Proposition 5.37, each

\(D^n(L)\)

is an acyclic class. Intersections of acyclic classes are acyclic, so

D^{\infty}(L)=\bigcap_{n\geq0}D^n(L)

is acyclic. It is closed under diagonals by construction and hence is a congruence by Proposition 5.35.If

K\subseteq L

is any congruence, then

\(K\)

is closed under diagonals. Inductively, every

f\in K

belongs to

\(D^n(L)\)

for all

\(n\)

, so

K\subseteq D^{\infty}(L)

. This proves maximality and the adjunction.

We now state and prove the main result of this section. Given a class of morphisms $\Sigma$ , write $\Sigma^{\Delta}$ for the closure of $\Sigma$ under diagonals $f \mapsto \Delta_f$ . Thus $\Sigma^{\Delta}$ consists of the maps in $\Sigma$ and all their iterated diagonals.

Theorem 5.39. (Anel–Biedermann–Finster–Joyal, [Anel et al. 2022, Proposition 4.2.12])

Let $$T$$ be a topos and let $\Sigma \subseteq \Ar(T)$ be a set of maps. Then the congruence $\Sigma^c$ is given by

\Sigma^c = (\Sigma^{\Delta})^m,

the acyclic class generated by the iterated diagonals of $\Sigma$ . This class is of small generation and hence is the left class of a modality.

Proof

Set

L:=(\Sigma^{\Delta})^m

. Since

\Sigma^{\Delta}

is again a set, Proposition 5.32 shows that

\(L\)

is of small generation. Let

K:=D^{\infty}(L)

, the largest congruence contained in

\(L\)

.For

f\in\Sigma

, every iterated diagonal

\Delta^nf

belongs to

\Sigma^{\Delta}\subseteq L

. Hence

f\in D^n(L)

for every

\(n\)

, and therefore

f\in K

. Thus

\(K\)

is a congruence containing

\Sigma

, which gives

\Sigma^c\subseteq K

. Conversely,

\Sigma^c

is an acyclic class containing

\Sigma^{\Delta}

, since it is closed under diagonals by Proposition 5.35. It therefore contains the acyclic closure

\(L\)

. We obtain

\Sigma^c\subseteq K\subseteq L\subseteq\Sigma^c,

so all three classes agree.

Remark 5.40.

The same formula holds for an arbitrary class $\Sigma$ , see [Anel et al. 2022, Proposition 4.2.12]. One writes $\Sigma$ as the union of a filtered system of sets closed under taking the diagonals that have already appeared and then uses that both acyclic and congruence closure preserve these filtered unions. The set case above is the form needed in these notes, and it has the additional advantage that $(\Sigma^{\Delta})^m$ automatically defines a modality.

Warning 5.41.

The formula $\Sigma^m = (\Sigma^{bc})^s$ for the acyclic class generated by a set $\Sigma$ does not have the naive congruence analogue: we generally have $\Sigma^c \neq (\Sigma^{bc})^{ss}$ . Indeed, the right-hand side generally has no reason to be closed under base change.

Remark 5.42.

For a set $\Sigma$ , Theorem 5.39 and Proposition 5.32 show that the congruence $\Sigma^c=(\Sigma^{\Delta})^m$ is of small generation. Thus the quotient by the generated congruence exists as a quotient logos.

Corollary 5.43.

If $\Sigma \subseteq \Ar(T)$ consists of monomorphisms, then $\Sigma^c = \Sigma^m$ .

Proof

\(f\)

is a monomorphism,

\Delta_f

is an isomorphism, hence automatically contained in

\Sigma^m

. The class version of the ABFJ formula from Remark 5.40 therefore gives

\Sigma^c = (\Sigma^{\Delta})^m = \Sigma^m

References

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. I: Higher sheaves. Adv. Math., 400, 64. 2022.
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. II: Grothendieck topologies. J. Pure Appl. Algebra, 228 (3), 63. 2024.