# Simplicial space and its simplicial replacement?

Following the notation from Dugger: "Primer on homotopy colimits":

For a small category $\mathcal{J}$ the nerve of the opposite category $\textrm{N}{\mathcal{J}^{\mathrm{op}}}$ is isomorphic to the simplicial set $\tilde{\textrm{N}}{\mathcal{J}}$ that in dimension $n \geqslant 0$ contains all chains $[i_{n} \rightarrow i_{n-1} \rightarrow \ldots \rightarrow i_{0}] = [i_{0} \leftarrow i_{1} \leftarrow \ldots \leftarrow i_{n}]$ of $n$ composable morphisms in $\mathcal{J}$ with $\alpha_{j}: i_{j} \rightarrow i_{j-1}$ for all $1 \leq j \leq n$. Again $s_{j}^{\ast}: (\tilde{\textrm{N}}{\mathcal{J}})_{n} \rightarrow (\tilde{\textrm{N}}{\mathcal{J}})_{n+1}$ for $0 \leq j \leq n$ inserts the identity morphism on $i_{j}$ and $d_{k}^{\ast}: (\tilde{\textrm{N}}{\mathcal{J}})_{n} \rightarrow (\tilde{\textrm{N}}{\mathcal{J}})_{n-1}$ for $0 \leq k \leq n$ covers up $i_{k}$. ATTENTION: $\tilde{\textrm{N}}{\mathcal{J}}$ is very similar to $\textrm{N}{\mathcal{J}}$, but the order of face and degeneracy maps has been reversed!

Let $\mathcal{J}$ be a small category and $D: \mathcal{J} \rightarrow \mathcal{T}op$ be a small diagram. The simplicial replacement of $D$ is defined as follows:
$$(\textrm{srep}_{\mathcal{J}}{D})_n := \coprod_{[i_0 \leftarrow i_1 \leftarrow \ldots \leftarrow i_n] \in (\tilde{\mathrm{N}}{\mathcal{J}})_{n}}D(i_n)$$ Let $\sigma = [i_0 \leftarrow i_1 \leftarrow \ldots \leftarrow i_n ]$ with $\alpha_{j}: i_{j} \rightarrow i_{j-1}$ for $1 \leq j \leq n$ be an element of $(\tilde{\textrm{N}}{\mathcal{J}})_{n}$. The degeneracy maps $s_j^{\ast}: (\textrm{srep}_{\mathcal{J}}{D})_n \rightarrow (\textrm{srep}_{\mathcal{J}}{D})_{n+1}$ for $0 \leq j \leq n$ map the summand $D(i_n)$ corresponding to $\sigma$ with the identity morphism to the same summand $D(i_n)$ indexed by $\tilde{\textrm{N}}(s_{j})(\sigma) \in (\tilde{\textrm{N}}{\mathcal{J}})_{n+1}$. The degeneracy maps $d_{j}^{\ast}: (\textrm{srep}_{\mathcal{J}}{D})_{n} \rightarrow (\textrm{srep}_{\mathcal{J}}{D})_{n-1}$ send $D(i_{n})$ to the identical copy in $(\textrm{srep}_{\mathcal{J}}{D})_{n-1}$ for $j \leq n-1$ or to $D(i_{n-1})$ using $D(\alpha_{n}): D(i_{n}) \rightarrow D(i_{n-1})$ for $j = n$ indexed by $\tilde{\textrm{N}}(d_{j})(\sigma) \in (\tilde{\textrm{N}}{\mathcal{J}})_{n-1}$.

My questions are: 1) Are there any natural maps between a simplicial space and its simplicial replacement? 2) Why is the homotopy colimit of a simplicial space weakly equivalent to its geometric realization?

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Robin Chapman, thank you very much for solving my problem with MathOverflow! – user6499 Jun 6 '10 at 16:09
In question 1), are you asking about what happens when $\mathcal{J}$ is the simplical indexing category $\Delta$? – Charles Rezk Jun 15 '10 at 18:46