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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeJan 8th 2012
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexI added the coequalizer description of a [[boundary of a simplex]]. The [[horn]] has such a description, too...

   I added the coequalizer description of a boundary of a simplex. The horn has such a description, too…

2. • CommentRowNumber2.
   • CommentAuthorJosh
   • CommentTimeMay 4th 2023
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   Author: Josh
   Format: TextIs there a known way of making sense of the boundary for more general simplicial sets, like (finite) groupoids?

   Is there a known way of making sense of the boundary for more general simplicial sets, like (finite) groupoids?
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 4th 2023
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   Author: Urs
   Format: MarkdownItexOne can form the *chains* on a simplicial set in the usual sense of singular homology: Formal linear combination of simplices. The boundary operation becomes a linear map of degree -1 on these chains, such that, in particular, the boundary of a chain consisting of a single simplex is the appropriately signed formal sum of its faces in the expected way. (This is described at *[[Moore complex]]*.) Now if the simplicial set is a finite simplicial complex of dimension $n$, then it carries a canonical $n$-chain being the sum of its non-degenerate $n$-simplices that it is made up of. The image under the above boundary map of this chain is what one would want to regard as the boundary of that finite simplicial complex. Beware though that the nerve of a non-trivial groupoid is never a finite simplicial set (not even for the smallest non-trivial groupoid $\mathbf{B} (\mathbb{Z}/2)$) so that it does not have a notion of "volume-filling chain" of which one could take the boundary. However/instead, every (nerve of a) groupoid represents the homotopy type of a 1-truncated topological space (a disjoint union of Eilenberg-MacLane spaces, such as $K(\mathbb{Z}/2, 1)$ in the previous case), and its Moore complex of chains and their boundaries computes the ordinary homology (singular homology) of that space in any degree.

   One can form the chains on a simplicial set in the usual sense of singular homology: Formal linear combination of simplices. The boundary operation becomes a linear map of degree -1 on these chains, such that, in particular, the boundary of a chain consisting of a single simplex is the appropriately signed formal sum of its faces in the expected way.

   (This is described at Moore complex.)

   Now if the simplicial set is a finite simplicial complex of dimension $n$, then it carries a canonical $n$-chain being the sum of its non-degenerate $n$-simplices that it is made up of. The image under the above boundary map of this chain is what one would want to regard as the boundary of that finite simplicial complex.

   Beware though that the nerve of a non-trivial groupoid is never a finite simplicial set (not even for the smallest non-trivial groupoid $\mathbf{B} (\mathbb{Z}/2)$) so that it does not have a notion of “volume-filling chain” of which one could take the boundary. However/instead, every (nerve of a) groupoid represents the homotopy type of a 1-truncated topological space (a disjoint union of Eilenberg-MacLane spaces, such as $K(\mathbb{Z}/2, 1)$ in the previous case), and its Moore complex of chains and their boundaries computes the ordinary homology (singular homology) of that space in any degree.

4. • CommentRowNumber4.
   • CommentAuthorJosh
   • CommentTimeMay 4th 2023
   • (edited May 4th 2023)
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   Author: Josh
   Format: TextThanks for your answer. If a finite simplicial set is a nerve of a category, I think it must be a direct category with finitely many objects?

   Thanks for your answer. If a finite simplicial set is a nerve of a category, I think it must be a direct category with finitely many objects?
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 4th 2023
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   Author: Urs
   Format: MarkdownItex> If a finite simplicial set is a nerve of a category, I think it must be a direct category with finitely many objects? That sounds right. (Incidentally, we ought to have this fact recorded at *[[direct category]]* and at *[[nerve]]*, but currently we don't.)

   If a finite simplicial set is a nerve of a category, I think it must be a direct category with finitely many objects?

   That sounds right. (Incidentally, we ought to have this fact recorded at direct category and at nerve, but currently we don’t.)