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I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).
What would be a text that features a graphics which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )
There is some bicategorical generalization and weakening of a cylinder object, under the same name, which does not factorize identity 1-cell but arbitrary 1-cell and have some other weaker properties in more general context.
The left homotopies defined using these cylinders do not compose, but one can consider formal finite sequences of left homotopies to define certain localizations.
I could write few details about it, but I am not sure if this belongs to this entry, despite the name being the same. Should it be here or having a separate entry ?
I’d probably be inclined towards a separate entry, like bicategorical cylinder object perhaps.
formal finite sequences of left homotopies to define certain localizations
I mean considering equivalence classes of sequences.
OK, I will (gradually) write a separate entry and link from cylinder object.
My feeling is that this notion is better thought of as relative to 1-cell. The classical case is an example of rather very special kind.
I did not write the entry yet but I reviewed the paper where the construction appears (without going into the details about the cylinder object itself). The review is in a draft form (very soon to be submitted), so corrections, improvements or suggestions are very welcome.
The review has been improved,
and a variant of it electronically submitted (changes can still be made for few days). Quite an interesting construction.
added pointer to:
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