Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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).
Notice that this proposition really requires that be uncountable. When it is not true: the -compact objects of ∞Grpd are the homotopy retracts of finite CW-complexes, while the -small ∞-groupoids are just the finite CW-complexes. Not every retract of a finite CW-complex has the homotopy type of a finite CW-complex: there is an obstruction, defined for a retract of a finite CW-complex, which is an element of , is called Wall’s finiteness obstruction, and vanishes if and only if has the homotopy type of a finite CW-complex. See Wall’s paper in the references.
This is the topic of section 5.4.1 of
Wall’s finiteness obstruction was defined in
Last revised on March 16, 2015 at 20:13:44. See the history of this page for a list of all contributions to it.