Effective epimorphisms

2.2. Effective epimorphisms

We will now introduce an important class of morphisms \(X \twoheadrightarrow Y\) in a topos, known as the effective epimorphisms. There are various ways of thinking about such morphisms:

1. They are higher categorical analogues of the effective epimorphisms from 1-category theory, i.e. those morphisms that exhibit \(Y\) as the quotient of \(X\) at the equivalence relation \(X \times_Y X \subseteq X \times X\);

2. They are those morphisms for which the pullback functor \(T_{/Y} \to T_{/X}\) is conservative, allowing us to check various conditions of maps in \(T_{/X}\) after pulling back to \(Y\);

3. They are the topos-theoretic analogues of the covers in a Grothendieck site.

Following Lurie, we take perspective (1) as our definition. To make this precise, we first recall the notion of groupoid objects.

2.2.1. Groupoid objects

Recall that a (classical) groupoid is a classical category \(\Gg\) in which all morphisms are invertible. Concretely, it consists of a set \(\Gg_0\) of objects, a set \(\Gg_1\) of morphisms, source/target maps \(s,t\colon \Gg_1 \to \Gg_0\), unit and inverse maps \(e\colon \Gg_0 \to \Gg_1\) and \(i\colon \Gg_1 \to \Gg_1\), and a composition map \(m\colon \Gg_1 \times_{s,\Gg_0,t} \Gg_1 \to \Gg_1\), satisfying the expected relations.

The higher categorical analogue is the following:

Definition 2.18.

Let \(C\) be a category. A simplicial object \(\Gg\colon\simp\catop \to C\) is called a groupoid object if for every \([n] \in \simp\) and every (not necessarily order-preserving) partition \begin{align*} [n] \simeq \{i_0,\dots ,i_k\} \sqcup_{\{i_k\}} \{i_k,\dots,i_n\}, \end{align*} the induced diagram

is a pullback square in \(C\). We let \(\Grpd(C) \subseteq \Fun(\simp\catop,C)\) denote the full subcategory spanned by the groupoid objects in \(C\).

For the purposes of topos theory, we may think of groupoid objects as higher categorical analogues of equivalence relations. Given a groupoid \(\Gg\) in the category of sets, the image of the map \((s,t)\colon \Gg_1 \to \Gg_0 \times \Gg_0\) is an equivalence relation on \(\Gg_0\). Indeed, reflexivity follows from the identity maps, transitivity follows from composition of morphisms, and symmetry follows by inverting morphisms. Conversely, equivalence relations \(R \subseteq X \times X\) on a set \(X\) correspond bijectively to groupoids in sets \(\Gg\) such that \(\Gg_0 = X\) and the map \((s,t)\colon \Gg_1 \hookrightarrow \Gg_0 \times \Gg_0\) is injective. Under this correspondence, the colimit \(\abs{\Gg}\) of the groupoid corresponds to the quotient \(X / R\).

Given a groupoid object \(\Gg_{\bullet}\) in \(C\) with \(X = \Gg_0\), we similarly think of the colimit \(\abs{\Gg_{\bullet}}\) as a “quotient of \(X\) by the relations contained in \(\Gg\)”. Unlike an equivalence relation, a groupoid may have several morphisms between two objects, as well as nontrivial automorphisms. This additional structure contributes nontrivially to the quotient object.

The definitions of universal/effective colimits specialize to groupoids as follows:

2.2.2. Effective epimorphisms and Čech nerves

In 1-category theory, a morphism \(f\colon U \to X\) is called an effective epimorphism if it exhibits its target as the quotient of its source at the equivalence relation \(U \times_X U\) determined by \(f\). Equivalently, \(X\) is the coequalizer of the two projection maps \(U \times_X U \rightrightarrows U\).

The definition in the higher categorical setting is similar. We replace the equivalence relation \(U \times_X U\) by a certain groupoid object \(\check{C}_{\bullet}(f)\) called the Čech nerve of \(f\). In degrees \(\leq 1\), this groupoid is given by the two projection maps \(U \times_X U \rightrightarrows U\).

Proof

(1) Assume first that \(f\) is an effective epimorphism. By Lemma 2.26, \(X \simeq \colim_n \check{C}_n(f)\) is a groupoid colimit. By our assumption on \(C\), the functor \(C_{/X} \hookrightarrow \lim_n C_{/\check{C}_n(f)}\) is fully faithful, hence conservative. In particular, the functors \(C_{/X} \to C_{/\check{C}_n(f)}\) for \([n] \in \simp\catop\) are jointly conservative. Since each of these functors factors through \(C_{/U}\), we conclude that \(f^*\) itself must be conservative.

Conversely, assume that \(f^*\) is conservative. We must show that the map \(\colim_n \check{C}_n(f) \to X\) is an isomorphism in \(C\). Since \(f^*\) is conservative and preserves groupoid colimits, it suffices to show that \(\colim_n \check{C}_n(f) \times_X U \to U\) is an isomorphism. This follows from the fact that the simplicial diagram \([n] \mapsto U^{\times^{(n+1)}_{/X}} \times_X U \iso U^{\times^{(n+2)}_{/X}}\) admits an extra degeneracy.

(2) Consider a pullback square

such that \(f\) is an effective epimorphism. Since the functor \(g^*\colon C_{/X} \to C_{/X'}\) preserves pullbacks, it sends \(\check{C}_{\bullet}(f)\) to \(\check{C}_{\bullet}(f')\). By assumption on \(C\), it also preserves groupoid colimits, hence we see that the map \(\colim_{n} \check{C}_n(f') \simeq (\colim_n \check{C}_n(f)) \times_X X' \to X \times_X X' = X'\) is an equivalence, implying that also \(f'\) is an effective epimorphism.

Corollary 2.30.

Let \(C\) be a category with pullbacks such that groupoid colimits are universal in \(C\) (e.g. \(C\) is a topos). Consider a commutative diagram in \(C\) as follows:

Assume that the map \(Y'' \to Y'\) is an effective epimorphism. If both the left-hand square and the outer rectangle are pullback squares, then so is the right-hand square.

Proof

We need to show that the map \(X'\to X \times_Y Y'\) is an isomorphism. By the previous lemma, this may be tested after pullback along the effective epimorphism \(Y'' \twoheadrightarrow Y'\). The claim now follows by applying the 2-out-of-3 property to the following commutative diagram:

We now show that the category of effective epimorphisms in a topos is equivalent to the category of groupoids, via passage to the Čech nerve. We start with the following simple observation:

We may use this adjunction to reformulate the condition of effectivity of groupoids from Definition 2.21:

Proof

The equivalence between (1) and (2) is Lemma 2.22.

From effectivity to the Čech nerve condition. Assume (2), and consider a groupoid object \(\Gg\). By setting \(\Gg^{+}_{-1} := \abs{\Gg} = \colim_{[n] \in \simp\catop} \Gg_n\), we extend \(\Gg\) to an augmented simplicial object \(\Gg^+_{\bullet}\). More precisely, this is the left Kan extension of \(\Gg\) along \(\simp\catop \hookrightarrow \simp\catop_+\). The first face maps define a transformation \(\Gg^+_{\bullet+1} \to \Gg^+_{\bullet}\) whose restriction to \(\simp\catop\) is cartesian by the groupoid condition. Effectivity therefore implies that the extended transformation is cartesian at the cone point. In particular, we obtain a pullback square

The top right corner is isomorphic to \(\Gg_0\), since the augmented simplicial object \([n] \mapsto \Gg_{n+1}\) over \(\Gg_0\) admits extra degeneracies. By Lemma 2.20, the augmented object \(\Gg^+_{\bullet}\) is therefore the Čech nerve of \(\Gg_0 \to \abs{\Gg}\), proving (3).

From the Čech nerve condition to effectivity. Assume (3). The map \(\Gg_0 \to \abs{\Gg}\) is then an effective epimorphism for every groupoid object \(\Gg\), since its Čech nerve is \(\Gg\) and its geometric realization is \(\abs{\Gg}\).

Consider a cartesian transformation \(X_{\bullet} \to Y_{\bullet}\) of groupoid objects, and set \(A:=\abs{X_{\bullet}}\) and \(B:=\abs{Y_{\bullet}}\). We must prove that the canonical map

\[u\colon X_0 \longrightarrow A \times_B Y_0\]

is an isomorphism. The map \(X_0 \to A\) is an effective epimorphism, so pullback along it is conservative by Lemma 2.29. After this base change, the map \(u\) becomes

\[X_0 \times_A X_0 \longrightarrow X_0 \times_B Y_0.\]

Condition (3) identifies the source with \(X_1\) and identifies \(Y_1\) with \(Y_0 \times_B Y_0\). Cartesianness of \(X_{\bullet} \to Y_{\bullet}\) then gives

\[X_1 \iso X_0 \times_{Y_0} Y_1 \iso X_0 \times_B Y_0.\]

Thus the base change of \(u\) is an isomorphism, and conservativity implies that \(u\) is an isomorphism. This proves effectivity of groupoid colimits.

Proof

Consider the adjunction \(\Fun(\simp\catop,C) \rightleftarrows \Ar(C)\) from Lemma 2.31. By Lemma 2.32, the unit transformation is an isomorphism on groupoid objects. Consequently, the restriction of the realization functor to \(\Grpd(C)\) is fully faithful.

Its essential image consists precisely of those morphisms \(f\colon U \to X\) for which the counit of the adjunction is an isomorphism. This counit takes the form of a commutative square

and by definition this is an isomorphism if and only if \(f\) is an effective epimorphism in \(C\).

Proof

The equivalence \(\Grpd(T) \simeq \mathrm{EffEpi}(T)\) from Lemma 2.33 sits in a commutative triangle

Taking fibers over \(* \in T\) gives the claim.

2.2.3. The epi-mono factorization system

In a topos, every morphism can be factored into an effective epimorphism followed by a monomorphism. This factorization is in fact unique. We formulate this precisely using factorization systems, which are recalled in Chapter A.

Proof

Existence of factorizations. Consider an arbitrary morphism \(f\colon U \to X\). Define \(V := \colim_{[n] \in \simp\catop} \check{C}_n(f)\). The augmentation of the Čech nerve gives a factorization

\[U \xrightarrow{e} V \xrightarrow{m} X.\]

By Lemma 2.32, the groupoid object \(\check{C}_{\bullet}(f)\) is the Čech nerve of its realization map \(e\). Hence \(e\) is an effective epimorphism and \(\check{C}_{\bullet}(e)\iso\check{C}_{\bullet}(f)\).

The second map is a monomorphism. It suffices to show that the first projection \(p_1\colon V \times_X V \to V\) is an isomorphism, since the diagonal is a section of \(p_1\). Pullback along the effective epimorphism \(e\colon U \to V\) is conservative by Lemma 2.29(1). We may therefore test \(p_1\) after this pullback. Universality of groupoid colimits gives

\[V \times_X U \iso \colim_{[n] \in \simp\catop} \check{C}_{n+1}(f).\]

The augmented simplicial object \([n] \mapsto \check{C}_{n+1}(f)\) over \(U\) admits extra degeneracies, starting with the diagonal \(U \to U \times_X U = \check{C}_1(f)\). Its realization is therefore \(U\), so \(V \times_X U \to U\) is an isomorphism. Conservativity now implies that \(p_1\) is an isomorphism and hence that \(m\) is a monomorphism.

Orthogonality. Consider a solid commutative diagram

where \(f\) is an effective epimorphism and \(i\colon A \hookrightarrow B\) is a monomorphism. Because \(i\) is a monomorphism, the anima of fillers is \((-1)\)-truncated. It remains only to prove that it is nonempty.

Work in the slice \(C_{/B}\). Composition with \(i\) defines a fully faithful functor \(C_{/A}\hookrightarrow C_{/B}\), and this functor preserves colimits. The given map \(U\to A\) shows that every term \(\check{C}_n(f)=U^{\times^{n+1}_X}\) of the Čech nerve belongs to its essential image. Since \(X\) is the colimit of this Čech nerve in \(C_{/B}\), the object \(X\to B\) also belongs to the essential image. Thus there exists a map \(X\to A\) giving a filler. The anima of fillers is consequently contractible, proving orthogonality.

2.2.4. Properties of effective epimorphisms

We now prove some basic properties of effective epimorphisms. Throughout this subsection, we fix a category \(C\) in which groupoids are effective and universal.

Lemma 2.40.

Given a commutative triangle

if \(gf\) is an effective epimorphism then so is \(g\).