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From the wikipedia page on simplicial sets, "Δ denotes the simplex category whose objects are finite strings of ordinal numbers..." I think that $\Delta\downarrow X$ is a slight abuse of notation, as the simplex category of a simplicial set $X$ is the same as the comma category $$\Delta^{(-)}\downarrow X$$ where ...

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Let $n \in \mathbb{N}_{\geq 1}$. The simplicial category $\mathfrak{C} [\partial \Delta^n]$ has as objects the set $\{0,\ldots,n\}$, and the simplicial set $Hom_{\mathfrak{C}[ \partial \Delta^n]}(i,j)$, for $0 \leq i < j \leq n$ and $(i,j) \neq (0,n)$, is $N(P_{i,j})$, the nerve of the partially ordered set of subsets $I \subseteq \{ i,\ldots,j \}$ with ...

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The trick is to embed the category of abstract simplicial complexes inside the category of symmetric simplicial sets (= functor $\mathbf{F}^\mathrm{op} \to \mathbf{Set}$, where $\mathbf{F}$ is the category of positive finite cardinals): this can be done by sending an abstract simplicial complex $X$ to the symmetric simplicial set ...

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You can define a "simplex" as having an orientation, thus getting an easier answer. A $k$-simplex is the convex hull of a set of $k+1$ points. But what does it mean for a set to have $k+1$ points? That there is a bijection from $\{1,2,\dots,k+1\}$. So simply define "$k$-simplex" in terms of a map $\{1,2,\dots,k+1\}\to\mathbb R^n$ and you can pick an ...

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