Structure theory for topoi | Higher Topos Theory notes

5.5. Structure theory for topoi

The epi–mono factorization of individual morphisms induces a parallel structure theory for acyclic classes and morphisms of logoi. We first decompose an acyclic class into its monogenic and epigenic parts. Applied to the kernel of a morphism of logoi, this produces a quotient triple factorization into a monogenic quotient, an epigenic quotient, and a conservative morphism. We then compare monogenic congruences with hypercomplete congruences and refine the conservative factor through the image logos.

More explicitly, a morphism of logoi $\varphi\colon T \to S$ admits a factorization

T \xrightarrow{\,\varphi^{\mono}\,} T/K^{\mathrm{mono}} \xrightarrow{\,\varphi^{\epi}\,} T/K \xrightarrow{\varphi^{\cons}} S.

The morphism $\varphi^{\cons}$ in fact factors further as $T/K \to \lra{\varphi(T)} \hookrightarrow S$ , where the second morphism is fully faithful.

5.5.1. Monogenic and epigenic acyclic classes

Definition 5.50.

For an acyclic class $$L$$ , we define

L^{\mathrm{epi}} := L \cap \EffEpi, \qquad \qquadtext{ and } \qquad L^{\mathrm{mono}} := (L \cap \Mono)^{m} = (L \cap \Mono)^{c},

where the last equality holds by Corollary 5.43. We say $$L$$ is monogenic if $L = L^{\mathrm{mono}}$ (i.e. generated by its monomorphisms), and epigenic if $L = L^{\mathrm{epi}}$ (i.e. it is contained in the effective epimorphisms).

Remark 5.51.

Monogenic/epigenic congruences are also called topological/cotopological in the literature. We will avoid that terminology, as it is less descriptive.

Both $L^{\epi}$ and $L^{\mono}$ are again acyclic classes. The following lemma explains their relation to epi–mono factorizations and will also determine when they define modalities.

Lemma 5.52.

Let $$L$$ be an acyclic class, and let $f\colon A \to B$ be a morphism with epi–mono factorization

A \overset{\coim(f)}{\twoheadrightarrow} \Im(f) \xhookrightarrow{\im(f)} B.

Then $$f$$ is contained in $$L$$ if and only if both $\im(f)$ and $\coim(f)$ are contained in $$L$$ .

Proof

If both

\coim(f)

and

\im(f)

lie in

\(L\)

, then so does their composite

\(f\)

. Conversely, assume

f\in L

. Since

\(L\)

is local by Lemma 5.4 and

\coim(f)

is an effective epimorphism, we may test whether

\coim(f)

lies in

\(L\)

after base change along itself. The resulting projection

A\times_{\Im(f)}A\to A

is isomorphic to

A\times_BA\to A

, because

\Im(f)\to B

is a monomorphism. This projection is a base change of

\(f\)

and hence lies in

\(L\)

. Thus

\coim(f)\in L

, and right cancellation applied to

f=\im(f)\coim(f)

gives

\im(f)\in L

Corollary 5.53.

Given a modality $$L$$ , also $L^{\epi}$ is a modality.

Proof

Let

R := L^{\perp}

and define

R^{\epi} := (L^{\epi})^{\perp}

. We need to show that the pair

(L^{\epi},R^{\epi})

is a factorization system. Orthogonality holds by construction, so it remains to check that any morphism

f\colon A \to B

admits an

(L^{\epi},R^{\epi})

-factorization. Since

L^{\epi} \subset L

, we get

R \subseteq R^{\epi}

. Similarly, from

L^{\epi} \subseteq \EffEpi

it follows that

\Mono \subseteq R^{\epi}

. Consider the

\((L,R)\)

-factorization

A \xrightarrow{l} C \xrightarrow{r} B

\(f\)

, and consider the epi-mono factorization

A \overset{\coim(l)}{\twoheadrightarrow} \Im(l) \xhookrightarrow{\im(l)} C

\(l\)

. By Lemma 5.52 we have that

\coim(l) \in L \cap \EffEpi = L^{\epi}

, while

r \circ \im(l) \in R \circ \Mono \subseteq R^{\epi}

because the right class

R^{\epi}

is closed under composition.

The corollary uses the factorization system and is therefore genuinely a statement about modalities. For $L^{\mono}$ , small generation will provide the required factorization system.

Notation 5.54.

Given a morphism $f \colon A \to B$ in $$T$$ , we denote by $A \overset{\coim(f)}{\twoheadrightarrow} \Im(f) \xhookrightarrow{\im(f)} B$ its epi-mono factorization. If $\Sigma$ is a class of morphisms in $$T$$ , we define

\im(\Sigma) := \{\im(f) \mid f \in \Sigma\} \qquadtext{ and } \coim(\Sigma) := \{\coim(f) \mid f \in \Sigma\}.

Remark 5.55.

If $$L$$ is an acyclic class, it follows from Lemma 5.52 that $\im(L) = L \cap \Mono$ and $\coim(L) = L \cap \EffEpi$ .

Lemma 5.56. ([Anel et al. 2024, Proposition 4.1.14])

For a set of morphisms $\Sigma$ in $$T$$ , we have

(\Sigma^m)^{\mono} = (\im(\Sigma))^m \qquadtext{ and } (\Sigma^c)^{\mono} = (\im(\Sigma^{\Delta}))^m = (\im(\Sigma^{\Delta}))^c.

In particular, if $$L$$ is an acyclic class of small generation, then $L^{\mono}$ is an acyclic class of small generation and hence defines a modality. If $$L$$ is a congruence of small generation, then so is $L^{\mono}$ .

Proof

The second relation follows from the first, since

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

by Theorem 5.39. The last two claims immediately follow by writing

L = \Sigma^m

and noting that

\im(\Sigma)

and

\im(\Sigma^{\Delta})

are small. We need to show the first relation.Let

L := \Sigma^m

. By Lemma 5.52, we have

\im(\Sigma) \subseteq L^{\mono}

. Since

L^{\mono}

is an acyclic class, it follows that

\im(\Sigma)^m \subseteq L^{\mono}

. For the converse, consider the class

\(L'\)

of morphisms

f\colon A \to B

such that

\im(f) \in \im(\Sigma)^{m}

. We are done if we can show that

L \subseteq L'

, since then

L \cap \Mono = \im(L) \subseteq \im(\Sigma)^{m}

and thus also

L^{\mono} = (L \cap \Mono)^m \subseteq \im(\Sigma)^{m}

. By construction we have

\Sigma \subseteq L'

, so it remains to show that

\(L'\)

is an acyclic class.To this end, note that

\im(\Sigma)^m = \im(\Sigma)^c

by Corollary 5.43, so in particular

\im(\Sigma)^m

is a congruence of small generation. Consider now the quotient map

T \to T/\im(\Sigma)^m

. This localization preserves effective epimorphisms because it preserves colimits, and it preserves monomorphisms because it preserves finite limits. It therefore preserves epi–mono factorizations. It follows that

f \in L'

if and only if the morphism

\(f\)

is sent to an effective epimorphism under this localization functor. Since the effective epimorphisms in

T/\im(\Sigma)^c

form an acyclic class, the same follows for

\(L'\)

, finishing the proof.

Corollary 5.57.

Let $\Sigma$ be a small class of monomorphisms in $$T$$ . Then the congruence $\Sigma^c$ generated by $\Sigma$ is monogenic.

Proof

For a monomorphism, every positive iterated diagonal is an isomorphism. Thus

\im(\Sigma^{\Delta})

consists of the maps in

\Sigma

together with isomorphisms, which do not change the acyclic class they generate. The claim is therefore immediate from the lemma.

Remark 5.58.

By definition, a monogenic congruence $$K$$ is completely determined by the class of monomorphisms $K \cap \Mono$ . The classes of monomorphisms in $$T$$ arising this way are precisely the Grothendieck topologies on $$T$$ , see Proposition 6.9.

Corollary 5.59.

For every acyclic class $$L$$ , we have

L \;=\; L^{\mathrm{epi}} \,\vee\, L^{\mathrm{mono}},

i.e. $$L$$ is the smallest acyclic class containing both $L^{\mathrm{epi}}$ and $L^{\mathrm{mono}}$ .

Proof

It is clear that

\(L\)

contains both

L^{\epi}

and

L^{\mono}

. Conversely, let

\(L'\)

be an acyclic class containing both parts. For every

f\in L

, Lemma 5.52 places

\coim(f)

L^{\epi}

and

\im(f)

L^{\mono}

. Hence

f=\im(f)\coim(f)

lies in

\(L'\)

, proving

L\subseteq L'

Lemma 5.60.

If $$L$$ is a monogenic acyclic class, then $$L$$ is a congruence. In particular, $L^{\mono}$ is a congruence for every acyclic class $$L$$ .

Proof

By assumption we have

L = L^{\mono} = (L \cap \Mono)^m

. Since

L \cap \Mono

consists of monomorphisms, it follows from Corollary 5.43 that

(L \cap \Mono)^m

agrees with

(L \cap \Mono)^c

, hence is a congruence.

Lemma 5.61.

A congruence $$K$$ is epigenic if and only if it is contained in the class of $\infty$ -connected maps.

Proof

\(K\)

is contained in

\Conn_{\infty}

, then it is in particular contained in

\EffEpi

, so that

\(K\)

is epigenic. Conversely, assume that

K \subseteq \EffEpi

. For any morphism

\(f\)

\(K\)

, the iterated diagonals

\Delta^n_f

\(f\)

also lie in

\(K\)

, hence are effective epimorphisms by assumption. By Theorem 3.22 it follows inductively that

\(f\)

\(n\)

-connected for all

n\geq 0

, hence

\infty

-connected.

5.5.2. The quotient triple factorization

We now discuss the promised triple factorization of morphisms of logoi. The terminology and perspective is due to [Anel et al. 2025].

Definition 5.62.

Let $\varphi\colon T \to S$ be a morphism of logoi.

We call $\varphi$ a monogenic quotient if it is a quotient map (Definition 4.19) whose kernel is a monogenic congruence.
We call $\varphi$ an epigenic quotient if it is a quotient map whose kernel is an epigenic congruence.
We call $\varphi$ conservative if it only inverts isomorphisms.
We call $\varphi$ weakly conservative if it only inverts effective epimorphisms.

Notation 5.63.

Let $\varphi\colon T \to S$ be a morphism of logoi with kernel $K := \ker(\varphi)$ . Then we have an inclusion $K^{\mono} \subseteq K$ , giving rise to three morphisms of logoi as follows:

T \xrightarrow{\,\varphi^{\mono}\,} T/K^{\mathrm{mono}} \xrightarrow{\,\varphi^{\epi}\,} T/K \xrightarrow{\varphi^{\cons}} S.

We refer to this as the quotient triple factorization of $\varphi$ . We further write

\varphi^{\quottext} := \varphi^{\epi} \circ \varphi^{\mono} \qquadtext{ and } \varphi^{\wcons} := \varphi^{\cons} \circ \varphi^{\epi}.

Proposition 5.64.

Let $\varphi\colon T \to S$ be a morphism of logoi.

The morphisms $\varphi^{\mono}$ , $\varphi^{\epi}$ , $\varphi^{\cons}$ , $\varphi^{\quottext}$ and $\varphi^{\wcons}$ are, respectively, a monogenic quotient, an epigenic quotient, conservative, a quotient map, and weakly conservative;
The decomposition $\varphi = \varphi^{\cons} \circ \varphi^{\quottext}$ is the unique decomposition of $\varphi$ into a quotient map followed by a conservative map;
The decomposition $\varphi = \varphi^{\wcons} \circ \varphi^{\mono}$ is the unique decomposition of $\varphi$ into a monogenic quotient followed by a weakly conservative map;
The decomposition $\varphi = \varphi^{\cons} \circ \varphi^{\epi} \circ \varphi^{\mono}$ is the unique decomposition of $\varphi$ into a monogenic quotient followed by an epigenic quotient followed by a conservative map.

Proof

(1) It is clear that

\varphi^{\mono}

is a monogenic quotient, that

\varphi^{\quottext}

is a quotient map, and that

\varphi^{\cons}

is conservative. Let

q\colon T\to T/K^{\mono}

denote the first quotient. The kernel of the induced quotient

\varphi^{\epi}\colon T/K^{\mono}\to T/K

is the congruence generated by the image

\(q(K)\)

. By [Anel et al. 2024, Proposition 4.2.5], this image congruence is contained in the

\infty

-connected maps. It is therefore epigenic by Lemma 5.61. Consequently,

\varphi^{\wcons}

is weakly conservative.(2) Given a decomposition

T \xrightarrow{\psi} T' \to S

into a quotient map and a conservative map, we see that

\ker(\psi) = \ker(\varphi) = K

. Since

\psi

is a quotient map, it follows that

T' \iso T/K

.(3) Given a decomposition

T \xrightarrow{\psi} T' \to S

into a monogenic quotient map and a weakly conservative map, the fact that the second map only inverts effective epimorphisms implies

\ker(\psi)\cap\Mono=\ker(\varphi)\cap\Mono=K\cap\Mono.

Since

\psi

is a monogenic quotient, we have

\ker(\psi)=\ker(\psi)^{\mono}=K^{\mono}

, and thus

T'\iso T/K^{\mono}

.(4) This follows by first factoring

\varphi

as in (2) and then applying (3) to the quotient map

T\to T/K

. These identifications also determine the comparison maps between any two such factorizations, proving uniqueness.

Remark 5.65.

In [Anel et al. 2025, Remark 2.1.34], the authors make the following analogy with ring theory. Given a morphism $\varphi\colon A\to B$ of commutative rings with kernel $$I$$ , its usual image factorization is

A\longrightarrow A/I\lhook\joinrel\longrightarrow B.

The injectivity of the second map is analogous to the conservativity of $\varphi^{\cons}\colon T/K\to S$ . Let $W\subseteq A$ be the multiplicative subset of elements whose images in $$A/I$$ are units. Then the quotient map $A\to A/I$ factors further as

A\longrightarrow A[W^{-1}]\longrightarrow A/I.

Altogether, this gives the triple factorization

A\longrightarrow A[W^{-1}]\longrightarrow A/I\lhook\joinrel\longrightarrow B,

which the authors compare with the factorization of $T\to S$ through $T/K^{\mono}$ and $$T/K$$ .

5.5.3. Hypercomplete congruences

We now discuss the hypercompletion of a congruence.

Definition 5.66.

Given a congruence $$K$$ of small generation, we define its hypercompletion as the kernel of the composite:

K^{\hyp} \quad := \quad \ker \bigl( T \to T/K \to (T/K)^{\hyp} \bigr),

where $(-)^{\hyp}$ denotes the hypercompletion of a topos (localization at $\infty$ -connected maps). Note that we always have $K \subseteq K^{\hyp}$ . We say a congruence $$K$$ is hypercomplete if $K = K^{\hyp}$ .

Lemma 5.67.

Let $$K$$ be a congruence of small generation. Hypercompletion does not affect its monogenic part:

K^{\hyp} \cap \Mono \;=\; K \cap \Mono.

Furthermore, the hypercompletion only depends on the monogenic part of a congruence: we have

(K^{\mono})^{\hyp} = K^{\hyp}.

Proof

For the first part, consider the sequence of localizations

T \to T/K \to (T/K)^{\hyp}

. Let

\(f\)

be a monomorphism in

\(T\)

contained in

K^{\hyp}

; we need to show that

\(f\)

is in fact contained in

\(K\)

. Since

T \to T/K

is left exact, the image of

\(f\)

\(T/K\)

is a monomorphism that becomes an equivalence in the hypercompletion

(T/K)^{\hyp}

. Thus, in

\(T/K\)

, the map

\(f\)

\infty

-connected. But a map that is both a monomorphism and

\infty

-connected is an equivalence. Therefore,

\(f\)

is already inverted in

\(T/K\)

, so

f \in K

.For the second part, we always have

(K^{\mono})^{\hyp} \subseteq K^{\hyp}

. For the converse, we may consider the monogenic-epigenic factorization of the quotient

T \to T/K

T \to T/K^{\mono} \to T/K.

The second map is an epigenic quotient by Proposition 5.64, hence its kernel is contained in the

\infty

-connected maps. An epigenic quotient induces an equivalence on hypercompletions, see [Anel et al. 2025, Lemma 2.1.33]. We therefore obtain

(T/K^{\mono})^{\hyp}\iso(T/K)^{\hyp}.

It follows that

(K^{\mono})^{\hyp} = K^{\hyp}

, as desired.

Let $\MonoCong(T)$ and $\HypCong(T)$ denote the posets of monogenic and hypercomplete congruences of small generation, respectively. It follows from the lemma that hypercompletion defines an equivalence

(-)^{\hyp}\colon\MonoCong(T)\iso\HypCong(T),

with inverse $K\mapsto K^{\mono}$ . The inclusion $\MonoCong(T)\hookrightarrow\Cong(T)$ has right adjoint $K\mapsto K^{\mono}$ , while the inclusion of the epigenic congruences has right adjoint $K\mapsto K^{\epi}$ . Under the preceding equivalence, hypercompletion is the fully faithful further right adjoint to $K\mapsto K^{\mono}$ , as proved in [Anel et al. 2025, Theorem 2.1.26]. Thus the three constructions record, respectively, the largest monogenic congruence below $$K$$ , the largest epigenic congruence below $$K$$ , and the largest hypercomplete congruence with the same monomorphisms as $$K$$ .

5.5.4. Images

In Remark 5.65, we compared the factorization $T \to T/K \to S$ of a logos morphism $\varphi\colon T \to S$ with the image factorization of a ring morphism. The analogy is not yet exact, since $T/K\to S$ is merely conservative and need not be a monomorphism. We now factor it further through an actual sublogos $\lra{\varphi(T)}\subseteq S$ .

Construction 5.68.

Let $\varphi\colon T \to S$ be a morphism of logoi. We define its image $\lra{\varphi(T)}$ as the smallest full subcategory of $$S$$ that contains the objects $\varphi(X)$ for all $X \in T$ and is closed under colimits and finite limits.

Proposition 5.69.

Let $\varphi\colon T \to S$ be a morphism of logoi. Then $\lra{\varphi(T)}$ is a logos, and the inclusion $\lra{\varphi(T)} \hookrightarrow S$ is a morphism of logoi. In particular, $\varphi$ admits a factorization in $\Logos$ as

T \longrightarrow \lra{\varphi(T)} \lhook\joinrel\longrightarrow S,

where the second map is a monomorphism in $\Logos$ .

Proof

Presentability is the substantive point and is proved in [Lurie 2009, Proposition 6.3.6.2]. By construction, the inclusion

\lra{\varphi(T)}\hookrightarrow S

preserves all colimits and finite limits. Consequently, colimits in the image are universal, coproducts are disjoint, and groupoid objects are effective because the corresponding statements hold in

\(S\)

and the inclusion preserves and reflects the diagrams involved. Thus the Giraud conditions of Theorem 2.42 hold in

\lra{\varphi(T)}

.The inclusion is therefore a morphism of logoi. Since it is fully faithful, it is a monomorphism in

\Logos

Remark 5.70.

Let $\varphi\colon T \to S$ be a morphism of logoi with kernel $K = \ker(\varphi)$ . Combining the quotient triple factorization with the image factorization gives

T \xrightarrow{(1)} T/ K^{\mono} \xrightarrow{(2)} T/K \xrightarrow{(3)} \lra{\varphi(T)} \xhookrightarrow{(4)} S .

The four factors isolate the following properties:

\begin{array}{c|c|c} \text{factor} & \text{property of the logos morphism} & \text{kernel or image condition} \\ \hline (1) & \text{monogenic quotient} & \ker=K^{\mono} \\ (2) & \text{epigenic quotient} & \ker\subseteq\Conn_\infty \\ (3) & \text{conservative and algebraic} & \text{its image generates the target} \\ (4) & \text{fully faithful} & \text{inclusion of the image logos}. \end{array}

The composite $$(1)(2)$$ is the quotient part of $\varphi$ , while $$(3)(4)$$ is its conservative part. The composite $$(1)(2)(3)$$ is algebraic in Lurie's terminology. Further terminology for composites, especially variants of “surjective”, depends on whether one works with classical or higher topoi, so we will use the explicit properties above.

References

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. II: Grothendieck topologies. J. Pure Appl. Algebra, 228 (3), 63. 2024.
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.
Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.