A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simplicies in each degree, necessarily finite?
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3$\begingroup$ Just a remark: it is not true that every simplicial set is the colimit of its non-degenerate simplices. For example, let $X$ be obtained from the standard $2$-simplex by collapsing the edge connecting $0$ and $1$. The colimit of non-degenerate simplices of $X$ is the standard $2$-simplex with vertices $0$ and $1$ identified, but the edge connecting them is not collapsed this time. $\endgroup$– Karol SzumiłoNov 17, 2013 at 17:08
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1$\begingroup$ Further to what Karol says, the construction sending a simplicial set $X$ to the colimit of its non-degenerate simplices is not even fully functorial in $X$, if I’m not mistaken. It only respects maps that preserve non-degenerate simplices. $\endgroup$– Peter LeFanu LumsdaineNov 17, 2013 at 17:16
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$\begingroup$ In Lurie's book "higher topos theory" Variant 4.2.3.16 it is mentioned that if $K$ is a finite simplicial set such that every nondegenerate simplex $\Delta^n\to K$ is a monomorphism, then the full subcategory inclusion of the non degenerate simplices of $K$, in the hole simplex category of $K$, is cofinal. I forgot this extra "monomotphism" condition, so thanks. Anyway, the fact that every finite simplicial set has only a finite number of simplicies in each degree is true. $\endgroup$– Ilan BarneaNov 17, 2013 at 20:09
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1$\begingroup$ Hi Ilan, Have you tried to construct an infinite simplicial set with finitely many simplicies in each degree? I think you will find that you can in fact do so (e.g. $\mathbb{CP}^\infty$, and more generally classifying spaces of finite groups). But if you failed to find any, perhaps you can study where the obstructions come from? As is stands, the question looks like you haven't put much thought into finding the answer. $\endgroup$– Theo Johnson-FreydNov 17, 2013 at 23:22
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2 Answers
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Let G be a finite group viewed as a one object category. Then the nerve BG is a simplicial set with finitely many simplices in each dimension but it is not finite.
answered Nov 17, 2013 at 17:07
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Take $X$ to be the “infinite-dimensional dunce’s cap”, with a unique non-degenerate simplex $x_n$ in each dimension, and with every face of $x_n$ equal to $x_{n-1}$.
Explicitly, $X_n = \coprod_{m \leq n} \mathrm{Surj}([n],[m])$. So it’s clear that this has finitely many simplices in each dimension, but infinitely many non-degenerate ones in total.
answered Nov 17, 2013 at 17:21
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2$\begingroup$ Another example in the same spirit is an infinite wedge $\bigvee S^n$ of spheres of different dimension. $\endgroup$ Nov 17, 2013 at 18:11
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$\begingroup$ Simpler: $\Delta[n]\subset \Delta[n+1]$ and then consider the colimit of the tower. $\endgroup$ Nov 18, 2013 at 8:56
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3$\begingroup$ @PhilippeGaucher: that was my first thought too, but it doesn’t work — it ends up having infinitely many n-d simplices in each dimension. $\endgroup$ Nov 18, 2013 at 14:36
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$\begingroup$ @PeterLeFanuLumsdaine yes indeed. $\endgroup$ Nov 18, 2013 at 19:03