# Kerodon

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### 10.2.5 Regular $\infty$-Categories

We now formulate an $\infty$-categorical counterpart of Definition 10.2.0.6.

Definition 10.2.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. We say that $\operatorname{\mathcal{C}}$ is regular if it satisfies the following conditions:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits finite limits.

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}$ has images. That is, every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ can be extended to a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y, }$

where $q$ is a quotient morphism and $i$ is a monomorphism.

$(3)$

The collection of quotient morphisms in $\operatorname{\mathcal{C}}$ is closed under pullback. That is, for every pullback diagram

$\xymatrix@C =50pt@R=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y }$

of $\operatorname{\mathcal{C}}$, if $f$ is a quotient morphism, then $f'$ is also a quotient morphism.

Example 10.2.5.2. Let $\operatorname{\mathcal{C}}$ be a category. Then $\operatorname{\mathcal{C}}$ is regular (in the sense of Definition 10.2.0.6) if and only if $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a regular $\infty$-category (in the sense of Definition 10.2.5.1). See Corollary 10.2.2.7.

Definition 10.2.5.1 admits a number of reformulations.

Proposition 10.2.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which has images and admits finite limits. The following conditions are equivalent:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is regular: that is, the collection of quotient morphisms in $\operatorname{\mathcal{C}}$ is closed under pullbacks.

$(2)$

Every quotient morphism in $\operatorname{\mathcal{C}}$ is a universal quotient morphism.

$(3)$

Every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ can be realized as the composition of a universal quotient morphism $q: X \twoheadrightarrow Y_0$ and a monomorphism $i: Y_0 \hookrightarrow Y$.

$(4)$

For every pullback diagram

10.27
$$\begin{gathered}\label{equation:regular-via-universal-quotient} \xymatrix@C =50pt@R=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d]^{u} \\ X \ar [r]^-{f} & Y, } \end{gathered}$$

the image $\operatorname{im}(f')$ coincides with $u^{-1}( \operatorname{im}(f) )$ (as an element of the set $\operatorname{Sub}(Y')$).

Proof. The equivalence of $(1) \Leftrightarrow (2)$ is a special case of Corollary 10.2.4.8, and the implication $(2) \Rightarrow (3)$ is immediate from the definitions. We next show that $(3)$ implies $(4)$. Fix a diagram of the form (10.27). If condition $(3)$ is satisfied, then we can choose a diagram

$\xymatrix@C =50pt@R=50pt{ & & Y' \ar [d]^{u} \\ X \ar [r]^-{q} & Y_0 \ar [r]^-{i} & Y }$

in the $\infty$-category $\operatorname{\mathcal{C}}$ where $q$ is a universal quotient morphism, $i$ is a monomorphism, and the lower vertical composition coincides with $f$. Since $\operatorname{\mathcal{C}}$ admits finite limits, this diagram admits a right Kan extension

10.28
$$\begin{gathered}\label{equation:regular-via-universal-quotient2} \xymatrix@C =50pt@R=50pt{ Y' \times _{Y} X \ar [r]^-{q'} \ar [d] & Y' \times _{Y} Y_0 \ar [r]^-{i'} \ar [d] & Y' \ar [d]^{u} \\ X \ar [r]^-{q} & Y_0 \ar [r]^-{i} & Y, } \end{gathered}$$

so that the right square and outer rectangle are pullback diagrams. By construction, the inverse image $u^{-1}( \operatorname{im}(f) )$ is the isomorphism class of the fiber product $Y' \times _{Y} Y_0$ (regarded as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{/Y'}$ via the morphism $i'$). On the other hand, the uniqueness of limits guarantees that $Y' \times _{Y} X$ is isomorphic to $X'$ as an object of $\operatorname{\mathcal{C}}_{/Y'}$, so the image of $f'$ coincides with the image of the composite morphism $i' \circ q'$. To prove that this image coincides with $[Y' \times _{Y} Y_0]$, it suffices to show that $q'$ is a quotient morphism in $\operatorname{\mathcal{C}}$. This follows from Corollary 10.2.4.7, since the left half of (10.28) is also a pullback square (Proposition 7.6.3.25).

We now complete the proof by showing that $(4)$ implies $(1)$. Suppose we are given a pullback square (10.27), where $f$ is a quotient morphism; we wish to show that $f'$ is also a quotient morphism. By virtue of Proposition 10.2.3.6, the assumption that $f$ is a quotient morphism guarantees that $\operatorname{im}(f) = [Y]$ is the largest element of $\operatorname{Sub}(Y)$, and we wish to show that $\operatorname{im}(f') = [Y']$ is the largest element of $\operatorname{Sub}(Y')$. This follows immediately from $(4)$, since the inverse image construction $u^{-1}: \operatorname{Sub}(Y) \rightarrow \operatorname{Sub}(Y')$ preserves largest elements (see Construction 9.2.4.31). $\square$

Remark 10.2.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which has images and admits finite limits. Then, for every pullback diagram

$\xymatrix@C =50pt@R=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d]^{u} \\ X \ar [r]^-{f} & Y, }$

we always have a containment $\operatorname{im}(f') \subseteq u^{-1}( \operatorname{im}(f) )$ in $\operatorname{Sub}(Y')$; this follows from the characterization of $\operatorname{im}(f)$ supplied by Proposition 10.2.3.11.

Corollary 10.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits finite limits. Then $\operatorname{\mathcal{C}}$ is regular if and only if every morphism $f: X \rightarrow Y$ can be obtained by composing a universal quotient morphism $q: X \twoheadrightarrow Y_0$ with a monomorphism $i: Y_0 \hookrightarrow Y$.

Corollary 10.2.5.6. Let $\operatorname{\mathcal{S}}$ denote the $\infty$-category of spaces. Then $\operatorname{\mathcal{S}}$ is a regular $\infty$-category.

Proof. Corollary 7.4.5.6 guarantees that $\operatorname{\mathcal{S}}$ admits finite limits. By virtue of Corollary 10.2.5.5, it will suffice to show that every map of Kan complexes $f: X \rightarrow Y$ factors as a composition $i \circ q$, where $q: X \twoheadrightarrow Y_0$ is a universal quotient morphism in $\operatorname{\mathcal{S}}$ and $i: Y_0 \hookrightarrow Y$ is a monomorphism in $\operatorname{\mathcal{S}}$. For this, we can take $i: Y_0 \hookrightarrow Y$ to be the inclusion of the essential image of $f$ (which is a monomorphism by Example 9.2.4.10), and $q: X \rightarrow Y_0$ to be the restriction of $f$ (which is a universal quotient morphism by Proposition 10.2.4.17. $\square$

Remark 10.2.5.7. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $Y_0 \subseteq Y$ be its essential image. The proof of Corollary 10.2.5.6 shows that $Y_0$ is an image of $f$ in the $\infty$-category $\operatorname{\mathcal{S}}$.

Warning 10.2.5.8. Let $\operatorname{\mathcal{QC}}$ denote the $\infty$-category of (small) $\infty$-categories. Then $\operatorname{\mathcal{QC}}$ contains quotient morphisms which are not universal quotient morphisms (see Variant 10.2.2.17). In particular, $\operatorname{\mathcal{QC}}$ is not regular.

Proposition 10.2.5.9. Let $\operatorname{\mathcal{C}}$ be a regular $\infty$-category. Then, for every object $Z \in \operatorname{\mathcal{C}}$, the slice $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$ is regular.

Proof. It follows from Remark 7.1.2.11 that the $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$ admits finite limits. By virtue of Corollary 10.2.5.5, it will suffice to show that every morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ can be realized as the composition of a universal quotient morphism $\widetilde{X} \twoheadrightarrow \widetilde{Y}_0$ with a monomorphism $\widetilde{Y}_0 \hookrightarrow \widetilde{Y}$. Let $f: X \rightarrow Y$ denote the image of $\widetilde{f}$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is regular, we can choose a $2$-simplex $\sigma :$

$\xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y }$

of $\operatorname{\mathcal{C}}$, where $q$ is a universal quotient morphism and $i$ is a monomorphism. The inclusion map $\operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$ is right anodyne (Lemma 4.3.7.8), so we can lift $\sigma$ to a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ & \widetilde{Y}_0 \ar [dr]^{ \widetilde{i} } & \\ \widetilde{X} \ar [ur]^{ \widetilde{q} } \ar [rr]^{ \widetilde{f} } & & \widetilde{Y} }$

in the $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$. We conclude by observing that $\widetilde{i}$ is a monomorphism (Remark 9.2.4.23) and $\widetilde{q}$ is a universal quotient morphism (Proposition 10.2.4.15). $\square$

We now study functors between regular $\infty$-categories.

Definition 10.2.5.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be regular $\infty$-categories. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is regular if it preserves finite limits and carries quotient morphisms of $\operatorname{\mathcal{C}}$ to quotient morphisms of $\operatorname{\mathcal{D}}$.

Remark 10.2.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which preserves pullbacks. Then $F$ carries monomorphisms in $\operatorname{\mathcal{C}}$ to monomorphisms in $\operatorname{\mathcal{D}}$ (Proposition 9.2.4.20). In particular, for every object $Y \in \operatorname{\mathcal{C}}$, the functor $F$ carries subobjects of $Y$ to subobjects of $F(Y)$, and therefore induces a map of partially ordered sets $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$.

Proposition 10.2.5.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories which have images, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves pullback squares. The following conditions are equivalent:

$(1)$

The functor $F$ carries quotient morphisms in $\operatorname{\mathcal{C}}$ to quotient morphisms in $\operatorname{\mathcal{D}}$.

$(2)$

For every $2$-simplex $\sigma :$

$\xymatrix@C =50pt@R=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y }$

in the $\infty$-category $\operatorname{\mathcal{C}}$ which exhibit $Y_0$ as an image of $f$, the $2$-simplex $F(\sigma )$ exhibits $F(Y_0)$ as an image of $F(f)$ in the $\infty$-category $\operatorname{\mathcal{D}}$.

$(3)$

For every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$ of Remark 10.2.5.11 carries $\operatorname{im}(f)$ to $\operatorname{im}( F(f) )$.

Proof. The implication $(1) \Rightarrow (2)$ follows from the observation that $F$ preserves monomorphisms (Proposition 9.2.4.20), and the implication $(2) \Rightarrow (3)$ is immediate from the definitions. We will complete the proof by showing that $(3)$ implies $(1)$. Let $f: X \twoheadrightarrow Y$ be a quotient morphism in $\operatorname{\mathcal{C}}$; we wish to show that $F(f)$ is a quotient morphism in $\operatorname{\mathcal{D}}$. By virtue of Proposition 10.2.3.6, our hypothesis can be reformulated as an equality $\operatorname{im}(f) = [Y]$ in the partially ordered set $\operatorname{Sub}(Y)$, and we wish to prove an equality $\operatorname{im}( F(f) ) = [ F(Y) ]$ in the partially ordered set $\operatorname{Sub}( F(Y) )$. This is clear, since the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$ preserves largest elements (see Remark 10.2.5.11). $\square$

Remark 10.2.5.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories with images, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves pullback diagrams. For any morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, we always have an inclusion $F( \operatorname{im}(f) ) \subseteq \operatorname{im}( F(f) )$ in the partially ordered set $\operatorname{Sub}( F(Y) )$.

Corollary 10.2.5.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be regular $\infty$-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves finite limits. Then $F$ is regular if and only if, for every morphism $f: X \rightarrow Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$, the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}(F(Y))$ of Remark 10.2.5.11 carries $\operatorname{im}(f)$ to $\operatorname{im}( F(f) )$.

Example 10.2.5.15. Let $\operatorname{\mathcal{C}}$ be a regular $\infty$-category. Then, for every object $X \in \operatorname{\mathcal{C}}$, the slice $\infty$-category $\operatorname{\mathcal{C}}_{/X}$ is also regular (Proposition 10.2.5.9). Moreover, for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, Proposition 7.6.3.16 guarantees that there exists a functor

$f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X} \quad \quad Z \mapsto X \times _{Y} Z$

given by pullback along $f$. The functor $f^{\ast }$ is regular: it preserves finite limits since it right adjoint to the postcomposition functor $(f \circ \bullet ): \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$, and preserves quotient morphisms by virtue of Proposition 10.2.5.3.

Proposition 10.2.5.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories which admit finite limits and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves finite limits and geometric realizations of simplicial objects. Then $F$ carries quotient morphisms of $\operatorname{\mathcal{C}}$ to quotient morphisms of $\operatorname{\mathcal{D}}$. In particular, if the $\infty$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are regular, then the functor $F$ is regular.

Proof. Let $f: X \twoheadrightarrow Y$ be a quotient morphism in $\operatorname{\mathcal{C}}$; we wish to that $F(f)$ is quotient morphism in $\operatorname{\mathcal{D}}$. Let $\operatorname{\check{C}}_{\bullet }(X/Y): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ be a Čech nerve of $f$ (Notation 10.1.5.5. Since $f$ is a quotient morphism (Remark 10.2.4.2), $\operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.2.2.4). Our assumption that $F$ preserves geometric realizations guarantees that $F \circ \operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$. Since $F$ preserves finite limits, $F \circ \operatorname{\check{C}}_{\bullet }(X/Y)$ is a Čech nerve of the morphism $F(f): F(X) \rightarrow F(Y)$. Applying Proposition 10.2.2.4 again, we deduce that $F(f)$ is a quotient morphism in $\operatorname{\mathcal{D}}$. $\square$

We now record some closure properties for the collection of regular $\infty$-categories.

Proposition 10.2.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits pullbacks, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory which is closed under the formation of pullbacks, and let $q: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}_0$. If $q$ is a quotient morphism in $\operatorname{\mathcal{C}}$, then it is also a quotient morphism in $\operatorname{\mathcal{C}}_0$. If $q$ is a universal quotient morphism in $\operatorname{\mathcal{C}}$, then it is also a universal quotient morphism in $\operatorname{\mathcal{C}}_0$.

Proof. Assume that $q$ is a quotient morphism in $\operatorname{\mathcal{C}}$; we will show that it is also a quotient morphism in $\operatorname{\mathcal{C}}_0$ (the analogous assertion for universal quotient morphisms then follows from the criterion of Corollary 10.2.4.7). Since $\operatorname{\mathcal{C}}$ admits pullbacks and $\operatorname{\mathcal{C}}_0$ is stable under the formation of pullbacks, it follows that $\operatorname{\mathcal{C}}_0$ also admits pullbacks. Applying Proposition 10.1.5.6, we deduce that $q$ admits a Čech nerve $\operatorname{\check{C}}_{\bullet }(X/Y): \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}_0$, which is also a Čech nerve of $q$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $q$ is a quotient morphism, $\operatorname{\check{C}}_{\bullet }(Z/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.2.2.4). It follows that $\operatorname{\check{C}}_{\bullet }(X/Y)$ is also a colimit diagram in $\operatorname{\mathcal{C}}_0$, so that $q$ is a quotient morphism in $\operatorname{\mathcal{C}}_0$. $\square$

Corollary 10.2.5.18. Let $\operatorname{\mathcal{C}}$ be a regular $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory which is closed under finite limits. Assume that, for every morphism $f$ of $\operatorname{\mathcal{C}}_0$, the image $\operatorname{im}(f)$ (formed in the $\infty$-category $\operatorname{\mathcal{C}}$) can be chosen to belong to $\operatorname{\mathcal{C}}_0$. Then $\operatorname{\mathcal{C}}_0$ is also regular.

Proof. Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is regular, $f$ can be factored as the composition of a universal quotient morphism $q: X \twoheadrightarrow Y_0$ with a monomorphism $i: Y_0 \hookrightarrow Y$. If $X$ and $Y$ belong to $\operatorname{\mathcal{C}}_0$, then our assumption guarantees that we can arrange that $Y_0$ is also contained in $\operatorname{\mathcal{C}}_0$. In this case, $i$ is also a monomorphism in the $\infty$-category $\operatorname{\mathcal{C}}_0$, and Proposition 10.2.5.17 guarantees that $q$ is a universal quotient morphism in the subcategory $\operatorname{\mathcal{C}}_0$. Allowing $f$ to vary and invoking Corollary 10.2.5.5, we conclude that $\operatorname{\mathcal{C}}_0$ is regular. $\square$

Proposition 10.2.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Suppose that the $\infty$-category $\operatorname{\mathcal{C}}$ is regular and that $F$ preserves finite limits. Then the $\infty$-category $\operatorname{\mathcal{D}}$ is also regular, and $F$ is a regular functor.

Proof. Since the functor $F$ has a right adjoint, it preserves geometric realizations of simplicial objects (Corollary 7.1.3.21). Applying Proposition 10.2.5.16, we deduce that the functor $F$ carries quotient morphisms in $\operatorname{\mathcal{C}}$ to quotient morphisms in $\operatorname{\mathcal{D}}$. It will therefore suffice to show that $\operatorname{\mathcal{D}}$ is regular.

It follows from Corollary 7.1.3.27 (together with Corollary 6.2.2.17) that the $\infty$-category $\operatorname{\mathcal{D}}$ admits finite limits. We next show that every morphism $v: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$ has an image. Let $\epsilon : (F \circ G) \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the counit of an adjunction between $F$ and $G$. Since $G$ is fully faithful, the natural transformation $\epsilon$ is an isomorphism. We can therefore replace $v$ by the morphism $(F \circ G)(v)$, and thereby reduce to the case where $v = F(u)$ for some morphism $u: C \rightarrow C'$ in $\operatorname{\mathcal{C}}$. In this case, our assumption that $\operatorname{\mathcal{C}}$ has images guarantees that we can factor $u$ as a composition $C \stackrel{q}{\twoheadrightarrow } C'_0 \stackrel{i}{\hookrightarrow } C'$, where $q$ is a quotient morphism and $i$ is a monomorphism (in the $\infty$-category $\operatorname{\mathcal{C}}$). It follows that $v$ can be written as the composition of $F(q)$ (which is a quotient morphism in $\operatorname{\mathcal{D}}$, as noted above) with $F(i)$ (which is a monomorphism in $\operatorname{\mathcal{D}}$ by virtue of Proposition 9.2.4.20). In particular, the object $F'(C_0)$ is an image of $v$.

We now complete the proof by showing that if

10.29
$$\begin{gathered}\label{equation:reflective-localization-regular} \xymatrix@C =50pt@R=50pt{ X' \ar [r]^-{f'} \ar [d] & Z' \ar [d]^{g} \\ X \ar [r]^-{f} & Z } \end{gathered}$$

is a pullback diagram in $\operatorname{\mathcal{C}}$ where $f$ is a quotient morphism, then $f'$ is also a quotient morphism. Since $\operatorname{\mathcal{C}}$ is regular, we can choose a $2$-simplex

$\xymatrix@C =50pt@R=50pt{ & Y \ar [dr] & \\ G(X) \ar [ur]^{q} \ar [rr]^{ G(f) } & & G(Z), }$

which exhibits $Y$ as an image of $G(f)$. It follows that $F(\sigma )$ exhibits $F(Y)$ as an image of the morphism $(F \circ G)(f)$ in the $\infty$-category $\operatorname{\mathcal{D}}$. Note that $(F \circ G)(f)$ is isomorphic to $f$ (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$), and is therefore a quotient morphism (Corollary 10.2.2.12). Applying Proposition 10.2.3.6, we conclude that $F(i)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Amalgamating the $2$-simplex $\sigma$ with $G(g)$, we obtain a diagram

$\xymatrix@C =50pt@R=50pt{ & & G(Z') \ar [d]^{ G(g)} \\ G(X) \ar [r]^-{q} & Y \ar [r]^-{i} & G(Z) }$

in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits finite limits, this diagram admits a right Kan extension

10.30
$$\begin{gathered}\label{equation:reflective-localization-regular2} \xymatrix@C =50pt@R=50pt{ G(X) \times _{G(Z)} G(Z') \ar [r]^-{q'} \ar [d] & Y \times _{G(Z)} G(Z') \ar [r]^-{i'} \ar [d] & G(Z') \ar [d]^{g} \\ G(X) \ar [r]^-{q} & Y \ar [r]^-{i} & G(Z), } \end{gathered}$$

so that the square on the right and the outer rectangle are pullback squares. Note that, after applying the functor $F$, the outer rectangle of this diagram is isomorphic to (10.29). We are therefore reduced to showing that the functor $F$ carries the upper horizontal composition in (10.30) to a quotient morphism in $\operatorname{\mathcal{D}}$. Since $F$ preserves pullback squares, $F(i')$ is a pullback of $F(i)$ and is therefore an isomorphism (Corollary 7.6.3.24). Using Corollary 10.2.2.12, we are reduced to showing that $F(q')$ is a quotient morphism in $\operatorname{\mathcal{D}}$. In fact, we claim that $q'$ is a pullback morphism of $\operatorname{\mathcal{C}}$. This follows from our assumption that $\operatorname{\mathcal{C}}$ is regular, since $q$ is a quotient morphism by construction and the left half the diagram (10.30) is a pullback square (Proposition 7.6.3.25). $\square$

Corollary 10.2.5.20. Let $\operatorname{\mathcal{C}}$ be a regular $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, so that the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$. If the functor $L$ preserves finite limits, then $\operatorname{\mathcal{C}}_0$ is a regular $\infty$-category and $L$ is a regular functor.