nLab essentially small (infinity,1)-category

Idea

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.

Definitions

Definition

A quasi-category $C$ is essentially $\kappa$-small for some regular cardinal $\kappa$ if

1. the collection of equivalence classes in $C$ is $\kappa$-small;

2. 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 $\kappa$.

Properties

Proposition

Let $C$ be an (∞,1)-category and $\kappa$ an uncountable regular cardinal. The following are equivalent:

1. $C$ is $\kappa$-small.

2. $C$ is a $\kappa$-compact object in (∞,1)Cat.

3. $C$ 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.

Proposition

For $X$ an ∞-groupoid and $\kappa$ an uncountable regular cardinal, the following are equivalent

1. For each object $x \in C$ the homotopy sets $\pi_n(X,x)$ are $\kappa$-small sets.

2. $X$ is presented by a $\kappa$-small simplicial set/Kan complex.

3. $X$ 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.

Related concepts

• essentially small category

• compact object in an (∞,1)-category

References

This is the topic of section 5.4.1 of

• Jacob Lurie, Higher Topos Theory .

Wall’s finiteness obstruction was defined in

• C. T. C. Wall, Finiteness conditions for CW-complexes.