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Here is a simple counterexample with $X = Y = \mathbb{R}$: Send a simplex $\sigma : |\Delta^n| \to \mathbb{R}$ to the affine function $F(\sigma) : |\Delta^n| \to \mathbb{R}$ with the same values at th …

$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in Kapulkin-Lumsdaine) is that "the base of the universal fibration is fibrant". Exp …

Pick pointed topological spaces $A$ and $B$ which admit pointed injective continuous maps $A \to B$ and $B \to A$ for which $A$ is contractible but $B$ has nonvanishing reduced homology. For example, …

$\newcommand{\SSet}{\mathsf{SSet}}\DeclareMathOperator{\Map}{Map}$First, let's record the fact that for any $A$ in $\SSet_{/S}$ and any right fibration $p : X \to S$, the simplicial set $\Map_{\SSet_{ …

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ob …

$\newcommand{\y}{\mathbf{y}} $Take $C = \mathcal{P}(a \stackrel{t}{\to} b) = \mathrm{Set}^{\cdot \leftarrow \cdot}$, so $C$ is freely generated under colimits by a morphism $\y t : \y a \to \y b$. Aga …

The inclusion $\partial \Delta^n \times \Delta^1 \subseteq X(n+1)$ isn't any kind of anodyne extension, though. It's formed by attaching an n-simplex to $\partial \Delta^n \times \Delta^1$ with bound …

I'm not 100% sure, but I think the answer is that you should choose a cellular model for PSh(C) (the category of presheaves of sets on C), which is a set S of monomorphisms in PSh(C) such that every m …

Since the join of simplicial sets is associative and $\Delta^m = \Delta^0 \star \cdots \star \Delta^0$ ($m+1$ times), we should start by trying to understand things like $\Lambda^n_j \star \Delta^0$, …

If I understand correctly, these are constructible sheaves with respect to the stratification of your simplicial complex by its skeleta. I think by a theorem of MacPherson the category of such sheave …

More examples: the functor Δ → Δ sending a totally ordered set S to S ∐ S, where the elements in the left copy are all less than the elements in the right copy. Restriction along this functor is the …

I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×hZ Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets. Assume for con …