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Let $(U_i)_{i\in I}$ be an open covering of a topological space $X$.

At, the nerve of the open covering is defined as follows:

the nerve $N$ is the set of finite subsets of $I$ defined as follows:

  • the empty set belongs to $N$;
  • a finite set $J\subset I$ belongs to $N$ if and only if the intersection of the $U_i$ whose subindices are in $J$ is non-empty.

On the other hand, states:

If $X$ is a topological space with open cover $U_i$, the nerve of the cover is obtained from the above definitions by replacing the cover with the category obtained by regarding the cover as a partially ordered set with relation that of set inclusion.

Here, "the above definitions" refers to the usual construction of the nerve of a category: A vertex for each object, and a $k$-simplex for each $k$-tuple of composable morphisms.

My question is: Does this categorical construction really yield the previously defined nerve of the open covering?

For instance, cover the inverval by two intersecting invervals non of them containing the other one. Then it seems to me that the first construction yields two vertices connected by an edge, while the second construction yields to bare vertices.

What am i missing?

If the second definition is indeed wrong, what is the right way to obtain the nerve of an open covering as a special case of the nerve of a category?

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The second definition needs to be corrected. – Tyler Lawson Mar 24 '10 at 17:35
up vote 5 down vote accepted

I think the second construction is not correct. If you replace the cover with the category whose objects are all intersections of elements of your original cover, then the two notions agree.

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But if I refine by adding all intersections of my covering sets, I obtain the baryzentric subdivision of the nerve of the original covering, don't I? – Rasmus Bentmann Mar 24 '10 at 17:45
You obtain the barycentric subdivision of the first definition of nerve. The problem is that these really are two different definitions of nerve. The first definition of nerve produces a simplicial complex (with simplices determined by their vertices), whereas any category-theoretic nerve produces a simplicial set (where each simplex has an orientation). – Tyler Lawson Mar 24 '10 at 17:58
Tyler: If you'd post this as an answer, I'd accept it. :) – Rasmus Bentmann Mar 24 '10 at 18:07

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