# 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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### Double Nerve Preserving Filtered Colimits

The double nerve (https://ncatlab.org/nlab/show/double+nerve) is a functor $\mathcal{N}_2:2Cat\rightarrow sSSet$ from the 3-category of 2-categories to the category of bisimplicial sets. Does the this ...
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### geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
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### A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
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### Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?

Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample? The only things I can say are that it ...
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### CW complex structure of geometric realization

In Ralph Cohen's notes on the topology of fiber bundles he makes the following claims: on pp.69, he says the geometric realization of a simplicial set is a CW complex on pp.70, he says the geometric ...
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### Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
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### What is a “cubical map” between cubical complexes?

What is a natural definition of a cubical map between cubical complexes? What is its geometric realization? I found some definitions, such as here or here, where a cubical map between cubical ...
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### Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
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### pointed simplicial set as coequalizer

Im studying simplcial sets and homotopy theory. I found this statement that seems quite immediate but for me it is not. Let $X$ be a pointed simplicial set, then $X$ can be realized as the ...
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### Fibrant (Kan complex) geometric meaning

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$: Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any ...
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### Is there any expression to calculate the homology groups of a quotient space?

Let $B \subset A$ where $A$ is a topological space and $A/B$ the space obtained from $A$ via collapsing $B$ to a single point. I was wondering if there is any expression for $H_k(A/B)$ in terms of ...
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### How can I calculate the homology group of an infinite torus using Mayer-Vietoris?

I want to calculate the (simplicial) homology of the following space using Mayer-Vietoris: I have tried to do it by cutting it along the axis and getting two subspaces homeomorphic to something ...
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### Is there a sensible way to form the geometric realization of a “simplicial space up to homotopy”?

I am new to simplicial methods, and have some naive questions. As such, the questions may be malformed and the terminology might not even be right. I assume these are answered in some reference and ...
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### When are the geometric realisations of two simplicial sets homeomorphic?

I'm given two simplicial sets $X,Y : \Delta^{\operatorname{op}} \to Set$. Of, course, if I study their geometric realisations $\lvert X \rvert$ and $\lvert Y \rvert$, I might find a homeomorphism, or ...
could somebody please explain me why the geometric realisation of cyclic maps is $S^1$-equivariant? This is e.g. used in the proof of 7.3.11 in Loday's cyclic homology Thanks