presentable category and ind
1. Proposition
Let 
be a regular cardinal, 
a locally small infinity category which admits small coproducts
TFAE:
- is presentable
- is accessible
- admits kappa filtered colimits and has a set
of kappa compact generator
2. Proof
2.1. 1) 
2)
special case
2.2. 2) 3)
see:
where the kappa filtered colimits exist by assumption
2.3. 3) 1)
2.3.1. cocomplete
2.3.2. object as colimit of
Let 
be the smallest full infinity subcategory containing and 
-small colimits of .
Then is small, as we have an epi on objects given by the functor
We want to show that 
There exists a factorization
since is cocomplete (cf. universal property of the presheaf category)
Then 
is a left adjoint with right adjoint given by restriction 
, i.e.
Here 
preserves limits (cf. mapping space functor preserves limits), hence restricts
(cf. ind object and preserving filtered limits)
This provides an adjunction
Then as consists only of -compact objects (cf. kappa small colimit of kappa compact objects), we may conclude that
preserves -filtered colimits.
Then we may proceed similar to accessible category and Ind TODO