Compactly assembled characterization | Higher Topos Theory notes

Proposition 6.142.

Let $$C$$ be a category with filtered colimits. Then the following are equivalent:

There exists a small category $$C_0$$ such that $$C$$ is a retract of $\Ind(C_0)$ in $\Cat^{\omega}$ (the category of categories with filtered colimits).
There exists a small category $$C_0$$ such that there exists an adjunction $C \rightleftarrows \Ind(C_0)$ in $\Cat^{\omega}$ such that $C \hookrightarrow \Ind(C_0)$ is fully faithful.
$$C$$ is accessible, and the “colimit functor” $\colim\colon \Ind(C) \to C$ (defined as the left adjoint to $y\colon C \hookrightarrow \Ind(C)$ ) admits a further left adjoint $y'\colon C \hookrightarrow \Ind(C)$ .

Proof

Note that (2) clearly implies (1), since the unit of the adjunction exhibits the map

\Ind(C_0) \to C

as a retraction of

C \hookrightarrow \Ind(C_0)

.For (3)

\implies

(2), we write

\Ind(C) = \bigcup_{\lambda > \kappa} \Ind(C^{\lambda})

, where

\(C\)

\kappa

-accessible. Since

C^{\kappa}

is small, we have

y'(C^{\kappa}) \subseteq \Ind(C^{\lambda})

for some

\lambda

. Since

\(C\)

is generated under colimits by

C^{\kappa}

and

\(y'\)

preserves colimits, this implies that

y'(C) \subseteq \Ind(C^{\lambda})

. But then the functor

y'\colon C \to \Ind(C^{\lambda})

is a left adjoint to

\colim\colon \Ind(C^{\lambda}) \to C

, giving (2).Finally, we prove that (1)

\implies

(3). The assumption gives us functors

C \xrightarrow{u} \Ind(C_0) \xrightarrow{v} C

satisfying

vu = \id_C

. We may now consider the idempotent

uv\colon \Ind(C_0) \to \Ind(C_0)

. We may then write

C = \lim(\dots \to \Ind(C_0) \xrightarrow{uv} \Ind(C_0) \xrightarrow{uv} \Ind(C_0)),

and in particular

\(C\)

is a limit of accessible categories, hence itself accessible. Consider the following diagram

The middle vertical map admits a left adjoint

y'\colon \Ind(C_0) \hookrightarrow \Ind(\Ind(C_0))

.Now, we show that the left adjoint

y'\colon C \hookrightarrow \Ind(C)

exists. We may do this pointwise, so consider some

X \in C

. The map

\Hom(X,\colim(-))

is a retract of the representable copresheaf

\Hom(\hat{v}y'u(X),-)

\Fun(\Ind(C),\An)

. Since

\Ind(C)

is idempotent complete, this means that

\Hom(X,\colim(-))

is representable. Hence there exists some

y'(X) \in \Ind(C)

, which then provides the desired left adjoint object to

\(X\)