Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [simplicial-stuff]

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-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

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-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

732 questions
7
votes
1 answer

Boundary of a simplicial set in terms of a coequalizer

I am trying to understand why we have a coequalizer $\sqcup_{0 \leq i < j \leq n} |\Delta^{n-2}| \rightrightarrows \sqcup_{0 \leq i \leq n} |\Delta^{n-1}| \rightarrow |\partial \Delta^n|$. What are all the three maps?
Fran
  • 71
6
votes
2 answers

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 preserves limits because it is a right adjoint and…
Ronald Bernard
  • 2,589
5
votes
1 answer

Is the simplicial $n$-sphere a reduced Kan simplicial set?

The simplicial $n$-sphere is the unique simplicial set with one non-degenerate 0-simplex, one non-degenerate $n$-simplex, and no other non-degenerate simplices. The $n$-sphere is obtained from $\Delta^n$ by identifying the boundary to a point. We…
user394412
  • 169
5
votes
1 answer

A Grothendieck topology on $\Delta$

Is there a choice for a Grothendieck topology on $\Delta$ for which most interesting simplicial sets are sheaves (like representables, horns and boundaries, and more generally all categories)? I suspect I can look at the Segal condition as a sheaf…
fosco
  • 11,814
4
votes
1 answer

Completing a delta set to a simplicial set

I am reading "An elementary illustrated introduction to simplicial sets" by Greg Friedman, available online here. He defines Delta sets (or semi-simplicial sets) as a generalisation of a simplicial complex, and then simplicial sets as generalising…
Garnet
  • 986
3
votes
1 answer

The value of the left adjoint of the diagonal functor $\delta^*:\mathsf{bisSet}\to\mathsf{sSet}$ at $\Delta^m\boxtimes \Delta^n$

I'm currently reading the proof of Theorem 5.5.7 in Cisinski's book Higher Categories and Homotopical Algebra. There's one detail I don't understand, and I need someone's help. Let us introduce some notations. We write $\mathsf{sSet}$ and…
Ken
  • 2,544
3
votes
1 answer

Are degeneracy maps always injective?

Are the degeneracy maps of a simplicial set always injective? I wonder, because I'd naively think that it wouldn't make sense to have two $n$-simplices which yield the same degenerate $n+1$-simplex. At least I haven't seen any example of a…
user981667
  • 33
2
votes
1 answer

Understanding pointed simplicial sets

For pointed simplicial sets there are two equivalent definitions of the basepoint. Let $\Delta^0$ be the simplicial set with only one vertex in each degree. Let $X$ be a simplicial set. Then a basepoint in $X$ is either a simplicial map…
Mike
  • 55
1
vote
1 answer

Must the degenerate simplices of a nondegenerate simplex be different?

Let $\sigma$ be a nondegenerate $n$-simplex in a simplicial set. Does it follow that the degenerate simplices $s_0(\sigma)$ and $s_1(\sigma)$ are different? For instance, consider $\Delta^2$, the triangle. Indeed, the nondegenerate 1-simplex $(0,1)$…
user984603
  • 343
1
vote
0 answers

Proof of Homotopy Addition Lemma

Let $X$ be a simplicial set. I'm looking for proof that $[d_0 y] - [d_1 y] + \ldots + (-1)^{n+1}[d_{n+1} y] \in \pi_n(X,x)$ is homotopic to 0, where $y \in X_{n+1}$ and $d_i$ are face maps. Thanks for your help!
Throw Away
  • 87
0
votes
1 answer

An elementary question about $\pi_0$ and two homotopic maps

Let $f\colon A\to B$ be a map of pointed simplicial sets and let $B$ be Kan-fibrant. Let $0\colon A\to B$ denote the map which factorizes over the basepoint of $B$. Is it true that $f$ is homotopic to $0$ if and only if the composition…
howard
  • 137
0
votes
1 answer

Does this specific identity follow from the simplicial identities?

In a simplicial set, if $r
user1005113
  • 478
0
votes
0 answers

Meaning of simplicial homotopies outside of SSet

If $X, Y$ are simplicial sets, and $f, g: X\to Y$ are two simplicial maps between them, we can define a homotopy of simplicial sets $h: X\times I\to Y$ or, equivalently, $h': X\to Y^I$. The latter definition can be broken down into something which…
Patrick Nicodemus
  • 1,245