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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)
the category of elements of the presheaf ;
the comma category , where denotes the Yoneda embedding .
the category whose objects are homomorphisms of simplicial sets from a standard simplicial simplex to , and whose morphisms are morphisms in the simplex category such that the diagram
An -simplex is said to be non-degenerate if it is not in the image of any degeneracy map.
Write
for the (non-full) subcategory on the non-degenerate simplices. Notice that a morphism of with source a non-degenerate simplex of is necessary a monomorphism.
This is called the category of non-degenerate simplices.
If every non-degenerate simplex in comes from a monomorphism , then the nerve is also called the barycentric subdivision of .
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-degenerate simplices has a left adjoint and is hence a reflective subcategory.
The category of simplices is a Reedy category.
Write for the canonical functor that sends to .
By the co-Yoneda lemma.
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
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 .
The terminology “category of simplices” is attributed by Hovey 1999 to:
Textbook account:
Homotopy finality of the non-degenerate simplices:
More on barycentric subdivision:
Last revised on August 19, 2022 at 21:48:43. See the history of this page for a list of all contributions to it.