# Tagged Questions

For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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### Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
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### Show simplicial complex is Hausdorff

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if ...
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I started another question related to this. Consider $|\Delta^n|:=\{(x_0,..,x_n)\in\mathbb{R}^{n+1}:\sum_{i=0}^n x_i=1, x_i\in[0,1]\;\forall i\}$ the geometric realization of the standard n-simplex $\... 1answer 42 views ### Paths between 0-cells in a classifying space. II Let$\mathcal{C}$be a small category and$X,Y$objects within. If$X$and$Y$(as$0$-cells) lie within the same path-component in$B\mathcal{C}$, can one say anything in general on how$X$and$...
I have such a problem. Let $K,L$ be simplicial complexes, where $K$ is connected, countable and locally finite. In particular we can express $$K=\bigcup_{i=0}^\infty K_i,$$ where $K_0$ is some full ...
Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where \$N(\...