Paths and cylinders
Categories of simplices
For a simplicial set its category of simplices is the category whose objects are the simplices in and whose morphisms are maps between these, as simplices in .
In particular the subcategory on the non-degenerate simplices has a useful interpretation: it is the poset of subsimplex inclusions whose nerve is the barycentric subdivision of , at least if every non-degenerate simplex in comes from a monomorphism , as for a simplicial complex.
Let sSet be a simplicial set.
The category of simplices of is equivalently (in increasing order of explicitness)
An -simplex is said to be nondegenerate if it is not in the image of any degeneracy map.
for the full subcategory on the nondegenerate simplices, with monomorphisms between them.
This is called the category of non-degenerate simplices.
See at barycentric subdivision – Relation to the category of simplices.
If has the property that every face of every non-degenerate simplex is again non-degenerate, then the inclusion of the category of non-generate simplices has a left adjoint and is hence a reflective subcategory.
Write for the canonical functor that sends to .
The colimit over the functor is itself
In the textbook literature this appears for instance as (Hovey, lemma 3.1.3).
A colimit-preserving functor is uniquely determined by its action on the standard simplices:
Important colimit-preserving functors out of sSet include
The nerve and subdivision
Let Cat sSet denote the simplicial nerve functor on categories.
An -simplex of is determined by a string of composable morphisms
along with a map , i.e. an element of Thus, each the functor from is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in are levelwise, the statement follows.
Therefore, the simplicial set itself can be computed as a colimit over the category of the simplicial sets .
A basic disussion is for instance in section 3.1 of
- Mark Hovey, Model categories, Mathematical surveys and monographs volume 63, American Mathematical Society
Homotopy finality of the non-degenerate simplices is discussed in section 4.1 of
For more on barycentric subdivision see also section 2 of
- Rick Jardine, Simplicial approximation, Theory and Applications of Categories, Vol. 12, 2004, No. 2, pp 34-72. (web)