Classifying topos if and only if compactly assembled

Proposition 6.148.

Let $$C$$ be a presentable category. Then a classifying topos for $$C$$ -valued sheaves exists if and only if $$C$$ is compactly assembled.

Proof

It remains to show the “only if” part. Let

\(E\)

be such a classifying topos, and write it as a left exact localization

i_*\colon E \hookrightarrow \PSh(D)

of a presheaf category. This induces a geometric morphism

i_*\colon \Shv_C(E) \hookrightarrow \Shv_C(\PSh(D))

. Let

F_{\univ} \in \Shv_C(E)

be the universal

\(C\)

-valued sheaf on

\(E\)

. By universality, there exists a morphism of topoi

f\colon \PSh(D) \to E

satisfying

i_*(F_{\univ}) \simeq f^*(F_{\univ})

. By full faithfulness of

\(i_*\)

, we have

F_{\univ} \simeq i^*i_*(F_{\univ}) \simeq (f \circ i)^*(F_{\univ})

. Universality again gives

f \circ i \simeq \id_E

, so

\(E\)

is a retract of

\PSh(D)

\Topos

.Applying

\Pt(-)

shows that

C \simeq \Shv_C(\An) \simeq \FunR(\An,E)

is a retract of

\Pt(\PSh(D))

\Cat^{\omega}

. Points of

\PSh(D)

are flat functors

D\to\An

, and their category is the filtered-colimit completion

\Ind(D\catop)

. It is therefore compactly generated. Hence

\(C\)

is a retract of a compactly generated category in

\Cat^\omega

, and is compactly assembled by Proposition 6.142.