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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 nondegenerate if it is not in the image of any degeneracy map.
Write
for the full subcategory on the nondegenerate simplices, with monomorphisms between them.
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-generate 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. 2,
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 September 21, 2015 at 06:52:29. See the history of this page for a list of all contributions to it.