equivalences in/of $(\infty,1)$-categories
The notion of essentially small $(\infty,1)$-category is the generalization of the notion of essentially small category from category theory to (∞,1)-category theory.
A quasi-category $C$ is essentially $\kappa$-small for some regular cardinal $\kappa$ if
the collection of equivalence classes in $C$ is $\kappa$-small;
for every morphism $f : x \to y$ in $C$ the homotopy sets of the hom ∞-groupoid at $f$ (that is, the sets $\pi_i(Hom^R(x,y),f)$) are $\kappa$-small.
$C$ is essentially small if the above conditions hold “absolutely,” i.e. with “$\kappa$-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 $\kappa$-small for some small regular cardinal $\kappa$.
Let $C$ be an (∞,1)-category and $\kappa$ an uncountable regular cardinal. The following are equivalent:
$C$ is $\kappa$-small.
$C$ is a $\kappa$-compact object in (∞,1)Cat.
$C$ is equivalently given by a quasi-category whose underlying simplicial set is a $\kappa$-small set.
This is HTT, prop. 5.4.1.2
The analogous statement holds for ∞-groupoids.
For $X$ an ∞-groupoid and $\kappa$ an uncountable regular cardinal, the following are equivalent
For each object $x \in C$ the homotopy sets $\pi_n(X,x)$ are $\kappa$-small sets.
$X$ is presented by a $\kappa$-small simplicial set/Kan complex.
$X$ is a $\kappa$-compact object in ∞Grpd.
This is (HTT, corollary 5.4.1.5).
Notice that this proposition really requires that $\kappa$ be uncountable. When $\kappa = \omega$ it is not true: the $\omega$-compact objects of ∞Grpd are the homotopy retracts of finite CW-complexes, while the $\omega$-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 $X$ of a finite CW-complex, which is an element of $\tilde{K}_0(\mathbb{Z}[\pi_1(X)])$, is called Wall’s finiteness obstruction, and vanishes if and only if $X$ 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