equivalences in/of -categories
The notion of essentially small -category is the generalization of the notion of essentially small category from category theory to (∞,1)-category theory.
A quasi-category is essentially -small for some regular cardinal if
the collection of equivalence classes in is -small;
for every morphism in the homotopy sets of the hom ∞-groupoid at (that is, the sets ) are -small.
is essentially small if the above conditions hold “absolutely,” i.e. with ”-small” replaced by “small.”
This appears as HTT, def. 5.4.1.3, prop. 5.4.1.2.
In the presence of the regular extension axiom (which follows from the axiom of choice), essential smallness is equivalent to being essentially -small for some small regular cardinal .
Let be an (∞,1)-category and an uncountable regular cardinal. The following are equivalent:
is -small.
is a -compact object in (∞,1)Cat.
is equivalently given by a quasi-category whose underlying simplicial set is a -small set.
This is HTT, prop. 5.4.1.2
The analogous statement holds for ∞-groupoids.
For an ∞-groupoid and an uncountable regular cardinal, the following are equivalent
For each object the homotopy sets are -small sets.
is presented by a -small simplicial set/Kan complex.
is a -compact object in ∞Grpd.
This is (HTT, corollary 5.4.1.5).
This is the topic of section 5.4.1 of