# The realization of the simplicial object ${\bf A}[A|X_\bullet|B]$

Let ${\bf A}$ be a category. Let $\Delta[n]$ be the representable presheaf $\Delta(-,[n])$.

What is the colimit $$\int^{n\in\Delta}\Delta[n]\times \coprod_{X_0,\dots X_n} {\bf A}(A,X_0)\times {\bf A}(X_0, X_1)\times\cdots \times {\bf A}(X_n,B)$$ What if ${\bf A}$ is simplicial and each ${\bf A}(X,Y)$ is a simplicial set, and the operation above lives in $\bf sSet$? Are there explicit descriptions of this object?

• If ${\mathbf A}$ is not simplicial, this is the nerve of the category $A \downarrow {\mathbf A} \downarrow B$ consisting of diagrams $A \to X \to B$. You can make a similar statement if you allow yourself to discuss category objects in $\mathbf{sSet}$ where the object set is nonconstant. (Whether ${\mathbf A}$ is simplicial or not, this nerve is weakly contractible because "$A \to A \to B$" is an initial object and "$A \to B \to B$" is a terminal object.) – Tyler Lawson Nov 13 '16 at 23:14