Kan fibrant replacement
Paths and cylinders
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
This entry describes special methods for the construction of fibrant resolutions in the standard model structure on simplicial sets.
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 -groupoidification in that it sends an -category to the -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 functor – described below – constructs an -groupoid from (the nerve of) an -category by
taking its 1-morphisms to be (co)spans in ;
taking its 2-morphisms to be cospan-of-cospan multispans in :
taking its 3-morphisms to be cospan-of-cospan-of-cospan multispans in :
For the simplicial -simplex let be its barycentric subdivision : this is the simplicial set that is the nerve of the poset of non-degenerate sub-simplicies in .
Notice that this simplicial set encodes the shape of an -fold cospan of cospans.
is the ordinary cospan.
These multi-cospan simplicial sets define a functor by setting
So this functor reads in a simplicial set and spits out the simplicial set whose 1-cells are cospans in .
This comes with a natural map
Iterating this construction indefinitely defines a simplicial set to be the colimit over
The 1–cells in are zig-zags in .
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 -functor (pdf)
Discussion in the context of simplicial presheaves is section 3 of