nLab
Kan fibrant replacement

Context

Homotopy theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

This entry describes special methods for the construction of fibrant resolutions in the standard model structure on simplicial sets.

Contents

Idea

In as far as we may think of simplicial sets having some suitable properties as a simplicial model for weak ω-categories (for instance for quasi-categories) and of a simplicial set that has the property of being a Kan complex as an ∞-groupoid, Kan fibrant replacement of simplicial sets is the operation of \infty-groupoidification in that it sends an \infty-category to the \infty-groupoid obtained by freely inverting all its non-invertible k-morphisms.

Technically, the terminology comes from the fact that with respect to the standard model structure on simplicial sets the Kan complexes are precisely the fibrant objects.

There are several methods to actually construct the Kan fibrant replacement. One convenient one, called the ExEx functor – described below – constructs an \infty-groupoid from (the nerve of) an \infty-category CC by

  • taking its 1-morphisms to be (co)spans in CC;

  • taking its 2-morphisms to be cospan-of-cospan multispans in CC:

  • taking its 3-morphisms to be cospan-of-cospan-of-cospan multispans in CC:

  • etc.

ExEx functor

For Δ[k]\Delta[k] the simplicial kk-simplex let sdΔ[k]sd \Delta[k] be its barycentric subdivision : this is the simplicial set that is the nerve of the poset of non-degenerate sub-simplicies in Δ[k]\Delta[k].

Notice that this simplicial set sdΔ ksd \Delta^k encodes the shape of an nn-fold cospan of cospans.

For instance

sdΔ 1={0(0,1)1} sd \Delta^1 = \{0 \to (0,1) \leftarrow 1\}

is the ordinary cospan.

These multi-cospan simplicial sets define a functor Ex:SSetSSetEx : SSet \to SSet by setting

(ExX) k=Hom SSet(sdΔ[k],X). (Ex X)_k = Hom_{SSet}(sd \Delta[k], X) \,.

So this functor reads in a simplicial set XX and spits out the simplicial set whose 1-cells are cospans in XX.

This comes with a natural map

XExX. X \to Ex X \,.

Iterating this construction indefinitely defines a simplicial set Ex XEx^\infty X to be the colimit over

XExXExExX. X \to Ex X \to Ex Ex X \to \cdots \,.

The 1–cells in Ex XEx^\infty X are zig-zags in XX.

Then

Proposition

  • Ex XEx^\infty X is a Kan complex;

  • XEx XX \to Ex^\infty X is a natural weak equivalence.

References

An original reference is

  • Dan Kan, On c.s.s. complexes, Amer. J. Math. 79 (1957), 449-476.

A standard textbook reference is

A summary of the basics is in

  • Bertrand Guillou, Kan’s Ex Ex^\infty-functor (pdf)

Discussion in the context of simplicial presheaves is section 3 of

Revised on January 9, 2014 11:13:00 by Anonymous Coward (24.5.241.75)