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I'm trying to understand the object $\text{Fun}(\Delta^{\text{op}},\textbf{Grpd})$ as mentioned in 2.9 of http://arxiv.org/pdf/1512.07573v2.pdf.

So far I have tried to use Higher Topos Theory by Lurie in understanding this object. The problem I run in to is that it seems there is an awful lot of data floating around, and I was hoping there is perhaps a more succinct way of understanding this, preferably all in terms of quasi-categories, i.e. simplicial sets satisfying the weak Kan condition. Also I would like to verify my understanding so far.

$\text{Fun}(\Delta^{\text{op}},\textbf{Grpd})$ is the functor $\infty$-category, so it should be a quasi-category itself. It is given by \begin{align*} n \mapsto \text{Set}^{\Delta^{\text{op}}}( \Delta^{\text{op}} \times \Delta^n, \textbf{Grpd}) \end{align*} Hence the vertices are simplicial maps $X:N \Delta^{\text{op}} \to \textbf{Grpd}$. Here $N\Delta^{\text{op}}$ is the ordinary nerve, which is a quasi-category, and \begin{align*} \textbf{Grpd} = N(\textbf{Kan}) \end{align*} is the simplicial nerve of the full subcategory $\textbf{Kan}$ of $\text{Set}^{\Delta^{\text{op}}}$, considered as simplicial category, so that \begin{align*} \text{Map}_{\textbf{Kan}}(\mathcal{C},\mathcal{D})_n = [n] \mapsto \text{Set}^{\Delta^{\text{op}}}(\Delta^n \times \mathcal{C},\mathcal{D}). \end{align*} for Kan complexes $\mathcal{C},\mathcal{D}$ and $n\in \mathbb{N}$. This gives us \begin{align*} \textbf{Grpd}_n = \text{Hom}_{\text{Cat}_\Delta}(\mathfrak{C}[\Delta^n],\mathbf{Kan}) \end{align*}

From the above it follows that giving a map of simplicial sets $X: N\Delta^{\text{op}} \to \textbf{Grpd}$ amounts to:

  • In degree zero we have just a function $X^{(0)}:N\Delta^{\text{op}}_0 = \Delta^{\text{op}} \ni [n] \mapsto X_n^{(0)} \in \textbf{Grpd}_0 = \textbf{Kan}$
  • In degree one we have a function $X^{(1)}:\text{Cat}([1],\Delta^{\text{op}}) \to \text{Hom}_{\text{Cat}_\Delta}(\mathfrak{C}[\Delta^1],\textbf{Kan}) $
  • In degree $n$, we have a function $X^{(n)}:\text{Cat}([n],\Delta^{\text{op}}) \to \text{Hom}_{\text{Cat}_\Delta}(\mathfrak{C}[\Delta^n],\textbf{Kan}) $

Now concretely, what is a good way of writing this function $X^{(1)}$, and understanding its relation to $X^{(0)}$? And for higher degrees?

Also, in http://arxiv.org/pdf/1512.07573v2.pdf the authors write $X$ just as $\Delta^{\text{op}} \ni [r] \mapsto X_r \in \textbf{Grpd}$. This suggests one can understand $X$ as a sort of simplicial groupoid, with the caveat that 'the simplicial identities are not strictly commutative squares; rather, they are $\Delta[1] \times \Delta[1]$-diagrams in $\textbf{Grpd}$'. How does this follow from the above data? Should I not have understood $\Delta^{\text{op}}$ as $\infty$-category via its nerve?

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1 Answer 1

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In the mean time, I think I know what is going on here.

Let $X$ be an object of $\text{Fun}(\Delta^{\text{op}},\textbf{Grpd})$. Then $X$ is a simplicial map from $\Delta^{\text{op}}$ (considered as simplicial set via its ordinary nerve) to the simplicial nerve of the category of Kan complexes.

One can write the data as follows. For $n \geq 0$ we write $X_n$ for $X_0(n)$. For $\sigma \in \Delta_d$ we write $X\sigma$ for $X_d(\sigma)$.

Now $X$ induces:

  • For $f: [n] \to [m]$ in $\Delta_1$, a simplicial map $f^*:X_m \to X_n$
  • For $\sigma = [n] \xrightarrow{f} [m] \xrightarrow{g} [k]$ in $\Delta_2$ we have a simplicial map $\sigma^*:\Delta^1 \times X_k \to X_n$, inducing a homotopy $f^* \circ g^* \simeq (gf)^*$
  • In general, for $\theta = [n_0] \xrightarrow{f_1} [n_1] \xrightarrow{f_2} \dots \xrightarrow{f_k} [n_k]$ in $\Delta_k$, we have a functor of simplicial categories \begin{align*} X \theta : \mathfrak{C}[\Delta^k] \to \textbf{Kan} \end{align*} inducing a simplicial map \begin{align*} \theta^*: \mathfrak{C}[\Delta^k](0,k) = (\Delta^1)^{\times (k-1)} \to \text{Map}(X_{n_k},X_{n_0}) \end{align*} witnessing that all the various ways to compose the $f_i$'s are homotopic.

We can summarize as follows. $X$ is like a simplicial set, except that each $X_n$ is a Kan complex, the face- and degeneracy maps are simplicial maps, and all strict simplicial identities become homotopies. Furthermore, two such homotopies witnessing the same identity are again homotopic. And in general, two homotopies between the same homotopy at a lower level are homotopic at a higher level.

A 1-vertex $\alpha:X \to Y$ in $\text{Fun}(\Delta^{\text{op}},\textbf{Grpd})$ is simplicial map $\Delta^{\text{op}} \times \Delta^1 \to \textbf{Grpd}$ which is $X$ on $\Delta^{\text{op}} \times \{0\}$ and $Y$ on $\Delta^{\text{op}} \times \{1\}$. It induces a family of maps $\alpha_k:X_k \to Y_k, k \geq 0$ that behaves like a natural transformation except that every diagram build up from naturality squares becomes a homotopy coherent diagram instead of a commutative one.

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