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Someone anonymous (probably from Gottingen) had changed simplicial homotopy. The old version seems to have had a different convention on the ordeing of the parts which was not ’wrong’ but was slightly different from some sources. I have added a note to the entry pointing out the existence of different conventions, and have cleaned up the use of $\Delta^n$ instead of the more usual $\Delta[n]$ in that entry.
I checked back in the linked entry simplex, there, and found that the entry was a bit confusing as to the simplicial set $\Delta[n]$. I have altered the structure and wording slightly to clarify things (I hope).
The question mentioned is:
Anonymous Coward: With exercise 8.3.5 of Weibel in mind, what is the notion of “cylinder” meant in the assertion “the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any finitely cocomplete category” for a general finitely cocomplete category?
I have answered
I have transferred this question to the nForum where it will be easier for others to reply as well. In the meantime some indication is given in Kamps and Porter as referenced below. I myself do not quite understand your question as it is presently stated, but this may be that I am too near to the subject matter to see the difficulty.
I believe we do not usually care about the directionality of simplicial homotopies as it is only the zig-zags of simplicial homotopies that matters. However, it might still be useful to state that if we follow the convention of simplicial homotopy using cylinders then the one assigned using the combinatorial definition turns out to be naturally in the opposite direction of what we might expect from the numbering.
Mann
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