nLab
essentially small (infinity,1)-category

Context

Compact objects

(,1)(\infty,1)-Category theory

Contents

Idea

The notion of essentially small (,1)(\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 CC is essentially κ\kappa-small for some regular cardinal κ\kappa if

  1. the collection of equivalence classes in CC is κ\kappa-small;

  2. for every morphism f:xyf : x \to y in CC the homotopy sets of the hom ∞-groupoid at ff (that is, the sets π i(Hom R(x,y),f)\pi_i(Hom^R(x,y),f)) are κ\kappa-small.

CC is essentially small if the above conditions hold “absolutely,” i.e. with ”κ\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 κ\kappa-small for some small regular cardinal κ\kappa.

Properties

Proposition

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

  1. CC is κ\kappa-small.

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

  3. CC is equivalently given by a quasi-category whose underlying simplicial set is a κ\kappa-small set.

This is HTT, prop. 5.4.1.2

The analogous statement holds for ∞-groupoids.

Proposition

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

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

  2. XX is presented by a κ\kappa-small simplicial set/Kan complex.

  3. XX is a κ\kappa-compact object in ∞Grpd.

This is (HTT, corollary 5.4.1.5).

Notice that this proposition really requires that κ\kappa be uncountable. When κ=ω\kappa = \omega it is not true: the ω\omega-compact objects of ∞Grpd are the homotopy retracts of finite CW-complexes, while the ω\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 XX of a finite CW-complex, which is an element of K˜ 0([π 1(X)])\tilde{K}_0(\mathbb{Z}[\pi_1(X)]), is called Wall’s finiteness obstruction, and vanishes if and only if XX has the homotopy type of a finite CW-complex. See Wall’s paper in the references.

References

This is the topic of section 5.4.1 of

Wall’s finiteness obstruction was defined in

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

Revised on July 16, 2013 17:18:57 by Omar Antolín-Camarena? (140.247.0.118)