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Stream: theory: algebraic topology

Topic: proper dimension of a simplex

Leopold Schlicht (Oct 18 2021 at 17:23):

Intuitively, degenerate $n$-simplices are $n$-simplices that live in the "wrong" dimension. Is there a standard name for the "proper dimension" of an $n$-simplex? By the "proper dimension" of $\sigma\in X_n$ I mean the smallest $m\leq n$ such that $\sigma$, considered as a map $\Delta^n\to X_\bullet$ (using Yoneda), factors as $\Delta^n\to \Delta^m\to X_\bullet$ (it follows that $\Delta^m\to X_\bullet$ is nondegenerate).

Is this equivalent to the smallest $m$ for which there exist degeneracy maps $d^1, \dots, d^k$ and an $m$-simplex $\tau$ such that $\sigma = (d^k\circ\dots \circ d^1)(\tau)$? (It follows that $\tau$ is nondegenerate, because otherwise $m$ is not minimal.) Here, $d^1$ should carry $m$-simplices to $m+1$-simplices, and so on.

Reid Barton (Oct 18 2021 at 17:58):

I think twice in parentheticals you wrote degenerate when you meant nondegenerate, right?

Leopold Schlicht (Oct 18 2021 at 17:58):

Yes, thanks. :grinning_face_with_smiling_eyes: Now it's correct.

Reid Barton (Oct 18 2021 at 18:01):

Here's an even better statement which I think answers your questions: For any simplex $\sigma : \Delta^n \to X$ there is a unique factorization of $\sigma$ as a degeneracy map $\Delta^n \to \Delta^m$ followed by a nondegenerate simplex $\tau : \Delta^m \to X$.

Reid Barton (Oct 18 2021 at 18:01):

Except for the question about the name of course--I don't think I have a name for this "proper dimension". I'd probably call it the dimension of the nondegenerate simplex through which $\sigma$ factors.

Reid Barton (Oct 18 2021 at 18:03):

A degeneracy map $\Delta^n \to \Delta^m$ is what corresponds to a map $[n] \to [m]$ of the simplex category which is surjective as a map of ordered sets--or if you like, a map $\Delta^n \to \Delta^m$ which is surjective on vertices.

Reid Barton (Oct 18 2021 at 18:38):

or in other words a composition of the generating degeneracies, which are usually denoted by the letter $s$--I think you used $d$ for degeneracy, but unfortunately that letter is traditionally used for the face maps

Ulrik Buchholtz (Oct 18 2021 at 19:10):

Re names: the factorization of a simplex as degeneracy map followed by a nondegenerate simplex is usually called the Eilenberg–Zilber decomposition, and this leads to the notion of an Eilenberg–Zilber category as one where all elements of a presheaf have such a decomposition. I haven't seen a name for the degree of the nondegenerate element in the decomposition, though.

Leopold Schlicht (Oct 19 2021 at 10:32):

Thanks both!

Reid Barton said:

or in other words a composition of the generating degeneracies,

Then I think it's actually equivalent to the characterization I gave. Why do you think your statement is "even better"? :grinning_face_with_smiling_eyes:

which are usually denoted by the letter $s$--I think you used $d$ for degeneracy, but unfortunately that letter is traditionally used for the face maps

Ah, yes, you're right (as always :grinning_face_with_smiling_eyes:). I have to get used to writing $s$ rather than $d$ when I think of the word degenerate, which is really hard.

Spencer Breiner (Oct 19 2021 at 12:29):

How about the "image dimension"?

Leopold Schlicht (Oct 19 2021 at 17:09):

Sounds good, thanks. :grinning_face_with_smiling_eyes: