added pointer to:

- Daniel Quillen, Section I.1, Def. 4, p. 9 (15 of 165) in:,
*Axiomatic homotopy theory*in:*Homotopical Algebra*, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

The review has been improved,

and a variant of it electronically submitted (changes can still be made for few days). Quite an interesting construction.

]]>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.

]]>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’d probably be inclined towards a separate entry, like bicategorical cylinder object perhaps.

]]>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.

- M.E. Descotte E.J., Dubuc M. Szyld,
*A localization of bicategories via homotopies*, Theory and appl. of categories 35:23, abs, arxiv/1805.05248

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 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. )