nLab
Kan fibrant replacement

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

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 $\mathrm{\infty }$-groupoidification in that it sends an $\mathrm{\infty }$-category to the $\mathrm{\infty }$-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 $\mathrm{Ex}$ functor – described below – constructs an $\mathrm{\infty }$-groupoid from (the nerve of) an $\mathrm{\infty }$-category $C$ by

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

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

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

• etc.

$\mathrm{Ex}$ functor

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

Notice that this simplicial set $\mathrm{sd}{\Delta }^k$ encodes the shape of an $n$-fold cospan of cospans.

For instance

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

is the ordinary cospan.

These multi-cospan simplicial sets define a functor $\mathrm{Ex}:\mathrm{SSet}\to \mathrm{SSet}$ by setting

$$(\mathrm{Ex}X{)}_n={\mathrm{Hom}}_{\mathrm{SSet}}(\mathrm{sd}\Delta [k],X).$$(Ex X)_n = Hom_{SSet}(sd \Delta[k], X) \,.

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

This comes with a natural map

$$X\to \mathrm{Ex}X.$$X \to Ex X \,.

Iterating this construction indefinitely defines a simplicial set ${\mathrm{Ex}}^{\mathrm{\infty }}X$ to be the colimit over

$$X\to \mathrm{Ex}X\to \mathrm{Ex}\mathrm{Ex}X\to \cdots .$$X \to Ex X \to Ex Ex X \to \cdots \,.

The 1–cells in ${\mathrm{Ex}}^{\mathrm{\infty }}X$ are zig-zags in $X$.

Then

Proposition

• ${\mathrm{Ex}}^{\mathrm{\infty }}X$ is a Kan complex;

• $X\to {\mathrm{Ex}}^{\mathrm{\infty }}X$ 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

• Paul Goerss Rick Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, 1999.

A summary of the basics is in

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

Discussion in the context of simplicial presheaves is section 3 of

• Rick Jardine, Simplicial presheaves (pdf)