topos theory

# Contents

## Idea

A geometric embedding is the right notion of embedding or inclusion of topoi $F↪E$, i.e. of subtoposes.

Notably the inclusion $\mathrm{Sh}\left(S\right)↪\mathrm{PSh}\left(S\right)$ of a category of sheaves into its presheaf topos or more generally the inclusion ${\mathrm{Sh}}_{j}E↪E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.

Another perspective is that a geometric embedding $F↪E$ is the localizations of $E$ at the class $W$ or morphisms that the left adjoint $E\to F$ sends to isomorphisms in $F$.

The induced geometric morphism of a topological immersion $X↪Y$ is a geometric embedding. The converse holds if $Y$ is a ${T}_{0}$ space. (Example A4.2.12(c) in (Johnstone))

## Definition

For $F$ and $E$ two topoi, a geometric morphism

$F\stackrel{f}{\to }E\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}F\stackrel{\stackrel{{f}_{*}}{\to }}{\underset{{f}^{*}}{←}}E$F \stackrel{f}{\to} E \;\;\;\; F \stackrel{\stackrel{f_*}{\to}}{\underset{f^*}{\leftarrow}} E

is a geometric embedding if the following equivalent conditions are satisfied

• the direct image functor ${f}_{*}$ is full and faithful (so that $F$ is a full subcategory of $E$);

• the counit $ϵ:{f}^{*}{f}_{*}\to {\mathrm{Id}}_{F}$ of the adjunction $\left({f}^{*}⊣{f}_{*}\right)$ is an isomorphism

• there is a Lawvere-Tierney topology on $E$ and an equivalence of categories $e:F\stackrel{\simeq }{\to }{\mathrm{Sh}}_{j}E$ such that the diagram of geometric morphisms $\begin{array}{ccc}F& \stackrel{{f}_{*}}{\to }& E\\ & {}_{e}{↘}^{\simeq }& {↑}^{i}\\ & & {\mathrm{Sh}}_{j}E\end{array}$ commutes up to natural isomorphism ${e}^{*}{i}^{*}\simeq {f}^{*}$

That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of (MacLaneMoerdijk)

## Properties

### Relation to localization

There is a close relation between geometric embedding and localization.

Let $f:F↪E$ be a geometric embedding and let $W\subset \mathrm{Mor}\left(E\right)$ be the class of morphisms sent by ${f}^{*}$ to isomorphisms in $F$.

###### Theorem

We have:

• $F$ is equivalent to the localization $E\left[W{\right]}^{-1}$;
• $F$ is equivalent to the full subcategory of $E$ on $W$-local objects.

This fact connects for instance the description of sheafification in terms of geometric embedding $\mathrm{Sh}\left(S\right)↪\mathrm{PSh}\left(S\right)$ as described for instance in

with that in terms of localization at local isomorphisms, as described in

Moreover, this is the basis on which sheafification is generalized to (∞,1)-sheafification in

The following gives a detailed proof of the above assertion.

Write $\eta :{\mathrm{Id}}_{E}\to {f}_{*}{f}^{*}$ for the unit of the adjunction.

Since ${f}_{*}$ is fully faithful we will identify objects and morphism of $F$ with their images in $E$. To further trim down the notation write $\overline{\left(-\right)}:={f}^{*}$ for the left adjoint.

###### Definition

Write $W$ for the class of morphism that are sent to isomorphism under ${f}^{*}$,

$W=\left({f}^{*}{\right)}^{-1}\left\{g:c\stackrel{\simeq }{\to }d\in \mathrm{Mor}\left(E\right)\right\}\phantom{\rule{thinmathspace}{0ex}}.$W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\} \,.
###### Proposition

$E$ equipped with the class $W$ is a category with weak equivalences, in that $W$ satisfies 2-out-of-3.

###### Proof

Follows since isomorphisms satisfy 2-out-of-3.

###### Proposition

$W$ is a left multiplicative system.

###### Proof

This follows using the fact that ${f}^{*}$ is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

• every isomorphism is in $W$;

• $W$ is closed under composition.

It remains to check the following points:

Given any

$\begin{array}{ccc}& & a\\ & & {↓}^{w}\\ b& \stackrel{h}{\to }& c\end{array}$\array{ && a \\ && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with $w\in W$, we have to show that there is

$\begin{array}{ccc}d& \to & a\\ {↓}^{w\prime }& & {↓}^{w}\\ b& \stackrel{h}{\to }& c\end{array}$\array{ d &\to& a \\ \downarrow^{w'} && \downarrow^w \\ b &\stackrel{h}{\to}& c }

with $w\prime \in W$.

To get this, take this to be the pullback diagram, $w\prime :={h}^{*}w$. Since ${f}^{*}$ preserves pullbacks, it follows that

$\begin{array}{ccc}\overline{d}& \to & \overline{a}\\ {↓}^{\overline{w}\prime }& & {↓}^{\overline{w}}\\ \overline{b}& \stackrel{\overline{h}}{\to }& \overline{c}\end{array}$\array{ \bar d &\to& \bar a \\ \downarrow^{\bar w'} && \downarrow^{\bar w} \\ \bar b &\stackrel{\bar h}{\to}& \bar c }

is a pullback diagram in $F$ with $\overline{w}\prime ={\overline{h}}^{*}\overline{w}$. But by assumption $\overline{w}$ is an isomorphism. Therefore $\overline{w}\prime$ is an isomorphism, therefore $w\prime$ is in $W$.

Finally for every

$a\stackrel{\stackrel{r}{\to }}{\stackrel{s}{\to }}b\stackrel{w}{\to }c$a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c

with $w\in W$ such that the two composites coincide, we need to find

$d\stackrel{w\prime }{\to }a\stackrel{\stackrel{r}{\to }}{\stackrel{s}{\to }}b$d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b

with $w\prime \in W$ such that the composites again coincide.

To get this, take $w\prime$ to be the equalizer of the two morphisms. Sending everything with ${f}^{*}$ to $F$ we find from

$\overline{a}\stackrel{\stackrel{\overline{r}}{\to }}{\stackrel{\overline{s}}{\to }}b\stackrel{\overline{w}}{\to }c$\bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c

that $\overline{r}=\overline{s}$, since $\overline{w}$ is an isomorphism. This implies that $\overline{w}\prime$ is the equalizer

$\overline{d}\stackrel{\overline{w}\prime }{\to }a\stackrel{\stackrel{\overline{r}}{\to }}{\stackrel{\overline{s}}{\to }}b$\bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b

of two equal morphism, hence an identity. So $w\prime$ is in $W$.

###### Proposition

For every object $a\in E$

• the unit ${\eta }_{a}:a\to \overline{a}$ is in $W$;

• if $a$ is already in $F$ then the unit is already an isomorphism.

###### Proof

This follows from the zig-zag identities of the adjoint functors.

$\begin{array}{cccc}& ↗& {⇓}^{\eta }& {↘}^{{\mathrm{Id}}_{E}}\\ E& \stackrel{\overline{\left(-\right)}}{\to }& F& ↪& E& \stackrel{\overline{\left(-\right)}}{\to }& F\\ & & & ↘& {⇓}^{\simeq }& {↗}_{{\mathrm{Id}}_{F}}\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{cc}& ↗{↘}^{\overline{\left(-\right)}}\\ E& {⇓}^{\mathrm{Id}}& F\\ & ↘{↗}_{\overline{\left(-\right)}}\end{array}$\array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F \\ &&& \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow^{\bar{(-)}} \\ E &\Downarrow^{Id}& F \\ & \searrow \nearrow_{\bar{(-)}} }

and

$\begin{array}{cccccc}& & & ↗& {⇓}^{\eta }& {↘}^{{\mathrm{Id}}_{E}}\\ F& ↪& E& \stackrel{\overline{\left(-\right)}}{\to }& F& ↪& E\\ & ↘& {⇓}^{\simeq }& {↗}_{{\mathrm{Id}}_{F}}\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{cc}& ↗↘\\ F& {⇓}^{\mathrm{Id}}& E\\ & ↘↗\end{array}$\array{ &&& \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow \\ F &\Downarrow^{Id}& E \\ & \searrow \nearrow }

In components they say that

• for every $a\in E$ we have $\left(\overline{a}\stackrel{{\overline{\eta }}_{a}}{\to }\overline{\overline{a}}\stackrel{\simeq }{\to }\overline{a}\right)={\mathrm{Id}}_{\overline{a}}$

• for every $a\in F$ we have $\left(a\stackrel{{\eta }_{a}}{\to }\overline{a}\stackrel{\simeq }{\to }a\right)={\mathrm{Id}}_{a}$

This implies the claim.

###### Definition

An object $a\in E$ is $W$-local object if for every $g:c\to d$ in $W$ the map

${g}^{*}:{\mathrm{Hom}}_{E}\left(d,a\right)\stackrel{\simeq }{\to }{\mathrm{Hom}}_{E}\left(c,a\right)$g^* : Hom_E(d,a) \stackrel{\simeq}{\to} Hom_E(c,a)

obtained by precomposition is an isomorphism.

###### Proposition

Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

###### Proof

First assume that $a\in F$. We need to show that $a$ is $W$-local.

Notice that the existence of the required isomorphism ${\mathrm{Hom}}_{F}\left(d,a\right)\simeq {\mathrm{Hom}}_{F}\left(c,a\right)$ is equivalent to the statement that for every diagram

$\begin{array}{ccc}c& \stackrel{}{\to }& d\\ {↓}^{h}\\ a\end{array}$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} \\ a }

there is a unique extension

$\begin{array}{ccc}c& \stackrel{}{\to }& d\\ {↓}^{h}& ↙\\ a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} & \swarrow \\ a } \,.

To see the existence of this extension, hit the original diagram with ${f}^{*}$ to get

$\begin{array}{ccc}\overline{c}& \stackrel{\simeq }{\to }& \overline{d}\\ {↓}^{\overline{h}}\\ \overline{a}\simeq a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} \\ \bar a \simeq a } \,.

By the assumption that $c\to d$ is in $W$ the morphism $\overline{c}\to \overline{d}$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\overline{a}\simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

$\begin{array}{ccc}\overline{c}& \stackrel{\simeq }{\to }& \overline{d}\\ {↓}^{\overline{h}}& {↙}_{\exists !k}\\ \overline{a}\simeq a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} & \swarrow_{\exists ! k} \\ \bar a \simeq a } \,.

By the hom-isomorphism (using full faithfullness of ${f}_{*}$ to work entirely in $E$)

${\mathrm{Hom}}_{E}\left(\overline{d},a\right)\simeq {\mathrm{Hom}}_{E}\left(d,a\right)$Hom_E(\bar d, a) \simeq Hom_E(d,a)

this defines a morphism $k:d\to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism

$\begin{array}{ccc}{\mathrm{Hom}}_{E}\left(\overline{d},\overline{a}\right)& \stackrel{\simeq }{\to }& {\mathrm{Hom}}_{E}\left(d,\overline{a}\right)\\ ↓& & ↓\\ {\mathrm{Hom}}_{E}\left(\overline{c},\overline{a}\right)& \stackrel{\simeq }{\to }& {\mathrm{Hom}}_{E}\left(c,\overline{a}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a) \\ \downarrow && \downarrow \\ Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a) } \,.

shows that $k:d\to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So $a\in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

By one of the above propositions we know that the unit ${\eta }_{a}:a\to \overline{a}$ is in $W$, so by the $W$-locality of $a$ it follows that

$\begin{array}{ccc}a& \stackrel{{\eta }_{a}}{\to }& \overline{a}\\ {↓}^{{\mathrm{Id}}_{a}}\\ a\end{array}$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }

has an extension

$\begin{array}{ccc}a& \stackrel{{\eta }_{a}}{\to }& \overline{a}\\ {↓}^{{\mathrm{Id}}_{a}}& {↙}_{{\rho }_{a}}\\ a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.

By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that ${\mathrm{Id}}_{a}$, being an isomorphism, is in $W$) it follows that ${\rho }_{a}:\overline{a}\to a$ is in $W$.

Since $\overline{a}$ is in $F$ and therefore $W$-local by the above, it follows that also

$\begin{array}{ccc}\overline{a}& \stackrel{{\rho }_{a}}{\to }& a\\ {↓}^{{\mathrm{Id}}_{\overline{a}}}\\ \overline{a}\end{array}$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }

has an extension

$\begin{array}{ccc}\overline{a}& \stackrel{{\rho }_{a}}{\to }& a\\ {↓}^{{\mathrm{Id}}_{\overline{a}}}& {↙}_{{\lambda }_{a}}\\ \overline{a}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.

So ${\eta }_{a}$ has a left inverse ${\rho }_{a}$ which itself has a left inverse ${\lambda }_{a}$. It follows that ${\rho }_{a}$ is also a right inverse to ${\eta }_{a}$, since

$\begin{array}{rl}\stackrel{{\rho }_{a}}{\to }\stackrel{{\eta }_{a}}{\to }& =\stackrel{{\rho }_{a}}{\to }\stackrel{{\eta }_{a}}{\to }{\underset{⏟}{\stackrel{{\rho }_{a}}{\to }\stackrel{{\lambda }_{a}}{\to }}}_{\mathrm{Id}}\\ & =\stackrel{{\rho }_{a}}{\to }{\underset{⏟}{\stackrel{{\eta }_{a}}{\to }\stackrel{{\rho }_{a}}{\to }}}_{\mathrm{Id}}\stackrel{{\lambda }_{a}}{\to }\\ & =\stackrel{{\rho }_{a}}{\to }\stackrel{{\lambda }_{a}}{\to }\\ & =\mathrm{Id}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} & = \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} \underbrace{ \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} }_{Id} \\ & = \stackrel{\rho_a}{\to} \underbrace{ \stackrel{\eta_a}{\to} \stackrel{\rho_a}{\to} }_{Id} \stackrel{\lambda_a}{\to} \\ &= \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} \\ &= Id \end{aligned} \,.

So if $a$ is $W$-local we find that ${\eta }_{a}:a\to \overline{a}$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

###### Corollary

$F$ is equivalent to the full subcategory ${E}_{W-\mathrm{loc}}$ of $E$ on $W$-local objects.

###### Proof

By standard reasoning (e.g. KS lemma 1.3.11) there is a functor $F\to {E}_{W-\mathrm{loc}}$ and a natural isomorphism

$\begin{array}{ccccc}F& & ↪& & E\\ & ↘& {⇓}^{\simeq }& ↗\\ & & {E}_{W-\mathrm{loc}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ F &&\hookrightarrow&& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow \\ && E_{W-loc} } \,.

Since $F↪E$ and ${E}_{W-\mathrm{loc}}↪E$ are full and faithful, so is $F\to {E}_{W-\mathrm{loc}}$. Since by the above it is also essentially surjective, it establishes the equivalence $F\simeq {E}_{W-\mathrm{loc}}$.

###### Proposition

$F$ is equivalent to the localization $E\left[{W}^{-1}\right]$ of $E$ at $W$.

###### Proof

By one of the above propositions we know that $W$ is a left multiplicative systems.

This implies that the localization $E\left[{W}^{-1}\right]$ is (equivalent to) the category with the same objects as $E$, and with hom-sets given by

${\mathrm{Hom}}_{E\left[{W}^{-1}\right]}\left(a,b\right)=\underset{a\prime \stackrel{p\in W}{\to }a}{\mathrm{colim}}{\mathrm{Hom}}_{E}\left(a\prime ,b\right)\phantom{\rule{thinmathspace}{0ex}}.$Hom_{E[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b) \,.

There is an obvious candidate for a functor

$F\to E\left[{W}^{-1}\right]$F \to E[W^{-1}]

given on objects by the usual embedding by ${f}_{*}$ and on morphism by the map which regards a morphism trivially as a span with left leg the identity

$\left(a\to b\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& \to & b\\ {↓}^{{\mathrm{Id}}_{a}}\\ a\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$(a \to b) \;\; \mapsto \;\; \left( \array{ a &\to& b \\ \downarrow^{Id_a} \\ a } \right) \,.

For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.

To see essential surjectivity, let $a$ be any object in $E$ and let ${\eta }_{a}:a\to \overline{a}$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, ${\eta }_{a}$ is in $W$. This means that the span

$\begin{array}{ccc}a& \stackrel{{\mathrm{Id}}_{a}}{\to }& a\\ {↓}^{{\eta }_{a}}\\ \overline{a}\end{array}$\array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }

represents an element in ${\mathrm{Hom}}_{E\left[{W}^{-1}\right]}\left(\overline{a},a\right)$, and this element is clearly an isomorphism: the inverse is represented by

$\begin{array}{ccc}a& \stackrel{{\eta }_{a}}{\to }& \overline{a}\\ {↓}^{{\mathrm{Id}}_{a}}\\ a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.

Since every $\overline{a}$ is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let $a,b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-local object. As discussed above, this means that every span

$\begin{array}{ccc}a\prime & \to & b\\ {↓}^{w}\\ a\end{array}$\array{ a' &\to& b \\ \downarrow^w \\ a }

with $w\in W$ has a unique extension

$\begin{array}{ccc}a\prime & \to & b\\ {↓}^{w}& ↗\\ a\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ a' &\to& b \\ \downarrow^w & \nearrow \\ a } \,.

But this implies that in the colimit that defines the hom-set of $E\left[{W}^{-1}\right]$ all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in ${\mathrm{Hom}}_{E}\left(a,b\right)\simeq {\mathrm{Hom}}_{F}\left(a,b\right)$ so that indeed

${\mathrm{Hom}}_{E\left[{W}^{-1}\right]}\left(a,b\right)\simeq {\mathrm{Hom}}_{F}\left(a,b\right)$Hom_{E[W^{-1}]}(a,b) \simeq Hom_{F}(a,b)

for all $a,b\in F$. Hence our functor is also full and faithful and therefore define an equivalence of categories

$F\stackrel{\simeq }{\to }E\left[{W}^{-1}\right]\phantom{\rule{thinmathspace}{0ex}}.$F \stackrel{\simeq}{\to} E[W^{-1}] \,.

### Factorizations and images

There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings. The factorization of a geometric morphism can be said to construct its image in the topos-theoretic sense.

In the more general context of (∞,1)-topos theory an $\left(\infty ,1\right)$-geometric embedding is an (∞,1)-geometric morphism

$\left({f}^{*}⊣{f}_{*}\right):𝒳\stackrel{←}{↪}𝒴$(f^* \dashv f_*) : \mathcal{X} \stackrel{\leftarrow}{\hookrightarrow} \mathcal{Y}

such that the right adjoint direct image ${f}_{*}$ is a full and faithful (∞,1)-functor.

See reflective sub-(∞,1)-category for more details.

## References

Section VII, 4 of

and section A4.2 of

Revised on April 4, 2013 14:53:32 by Urs Schreiber (82.169.65.155)