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For $X$ a simplicial set its category of simplices is the category whose objects are the simplices in $X$ and whose morphisms are maps between these, as simplices in $X$.
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 $X$, at least if every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, as for a simplicial complex.
Let $X \in$ sSet be a simplicial set.
The category of simplices of $X$ is equivalently (in increasing order of explicitness)
the category of elements of the presheaf $X_\bullet : \Delta^{op} \to Set$;
the comma category $(\Delta\downarrow X)$, where $\Delta$ denotes the Yoneda embedding $[n] \mapsto \Delta^n$.
the category whose objects are homomorphisms of simplicial sets $c : \Delta^n \to X$ from a standard simplicial simplex $\Delta^n$ to $X$, and whose morphisms $c \to c'$ are morphisms $f : \Delta^n \to \Delta^{n'}$ in the simplex category $\Delta$ such that the diagram
An $n$-simplex $x\in X_n$ 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 $\Delta\downarrow X$ with source a non-degenerate simplex of $X$ is necessary a monomorphism.
This is called the category of non-degenerate simplices.
If every non-degenerate simplex in $X$ comes from a monomorphism $\Delta^n \to X$, then the nerve $N((\Delta \downarrow X)_{nondeg})$ is also called the barycentric subdivision of $X$.
See at barycentric subdivision – Relation to the category of simplices.
If $X$ 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 $(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X)$ has a left adjoint and is hence a reflective subcategory.
The category of simplices is a Reedy category.
Write $(\Delta \downarrow X) \to sSet$ for the canonical functor that sends $(\Delta^n \to X)$ to $\Delta^n$.
The colimit over the functor $(\Delta \downarrow X) \to sSet$ is $X$ itself
By the co-Yoneda lemma.
In the textbook literature this appears for instance as (Hovey, lemma 3.1.3).
A colimit-preserving functor $F\colon sSet \to C$ is uniquely determined by its action on the standard simplices:
Important colimit-preserving functors out of sSet include
Let $N\colon$ Cat $\to$ sSet denote the simplicial nerve functor on categories.
The functor $sSet \to sSet$ that assigns barycentric subdivision, def. ,
preserves colimits.
An $n$-simplex of $N(\Delta\downarrow X)$ is determined by a string of $n+1$ composable morphisms
along with a map $\Delta^{k_0} \to X$, i.e. an element of $X_{k_0}$ Thus, each the functor $X\mapsto N(\Delta\downarrow X)_n$ from $SSet \to Set$ is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in $SSet$ are levelwise, the statement follows.
Therefore, the simplicial set $N(\Delta\downarrow X)$ itself can be computed as a colimit over the category $(\Delta\downarrow X)$ of the simplicial sets $N(\Delta\downarrow \Delta^n)$.
A basic disussion is for instance in section 3.1 of
Homotopy finality of the non-degenerate simplices is discussed in section 4.1 of
For more on barycentric subdivision see also section 2 of
Last revised on April 17, 2019 at 07:50:54. See the history of this page for a list of all contributions to it.