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In Greg Friedman's article "An elementary illustrated introduction to simplicial sets" (2016), there is the following definition: enter image description here

Given such a Delta set $X$, we can form a chain complex $C_\ast(X)$ by defining $C_n(X)$ to be the free abelian group generated by $X_n$ and $$\partial_n := \sum_{i=0}^{n} (-1)^i\, d_i: C_n(X)\to C_{n-1}(X).$$

Is the corresponding homology isomorphic to the singular homology of the geometric realization $|X|$ of $X$, defined as $$ |X| := \Big( \bigsqcup_{n=0}^\infty (X_n\times\Delta^n) \Big) \Big/ \sim $$ where $\sim$ is the equivalence relation generated by $(x,(t_0,\dots,t_{i-1},0,t_{i+1},\dots,t_{n-1}))\sim (d_i(x),(t_0,\dots,t_{n-1}))$?

I know that this is true if $X$ is a simplicial set. But what about the case of a Delta set? If it is true, does there exist a chain homotopy equivalence between $C_\ast(X)$ as defined above and the singular chain complex $S_\ast(|X|)$ of $|X|$? What about the chain homomorphism $$\theta: C_\ast(X)\to S_\ast(|X|)$$ defined on a generator $x\in X_n$ of $C_n(X)$ by $$\theta(x) := (t\to [x,t])\in S_n(|X|)?$$

Or would it be perhaps easier to use the cellular chain complex?

Any help is much appreciated. Thank you in advance!

Simba
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    You can extend a $\Delta$-set to a simplicial set by adding degeneracies. The chain complex associated with the $\Delta$-set is isomorphic to the normalised chain complex associated with the simplicial set, if I recall correctly. – Zhen Lin Apr 09 '21 at 10:24
  • Thank you! And the normalized chain complex associated to the simplicial set is isomorphic to the cellular chain complex of the geometric realization of the simplicial set, right? But how are the geometric realizations of the Delta set and the corresponding simpicial set related? Are they homotopy equivalent? – Simba Apr 09 '21 at 12:12
  • The geometric realisations are isomorphic. – Zhen Lin Apr 09 '21 at 13:03
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    @ZhenLin I doubt the realisations are homeomorphic, but they should be homotopy equivalent. See also here: https://math.stackexchange.com/questions/636992/fat-geometric-realization-weakly-equivalent-to-the-usual-one – Vincent Boelens Apr 09 '21 at 13:26
  • They are isomorphic. The fat geometric realisation is something else. The “underlying” $\Delta$-set of the simplicial set associated with a $\Delta$-set is not what you started with. – Zhen Lin Apr 09 '21 at 13:39
  • Thank you both! I thought a bit about the relation to the cellular chain complex. For each $n\geq 0$, $C_n(X)$ is the free abelian group generated by the elements of $X_n$, and the cellular chain group of $|X|$ in degree $n$ is the free abelian group generated by the $n$-cells, which are in one-to-one correspondence with the elements of $X_n$, aren't they? So shouldn't it be obvious that both groups are isomorphic? Thus, it suffices to check that this defines a chain homomorphism... Or do I miss something? – Simba Apr 09 '21 at 17:16
  • By the way, I have found a nice reference (from Johannes Ebert and Oscar Randal-Williams) concerning the relation between "thin" and "fat" geometric realizations: link. – Simba Apr 10 '21 at 08:46

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