Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
In the article cocylinder one reads at the bottom:
George Whitehead, Elements of homotopy theory
(This uses the terminology mapping path space.)
(This was added in revision 3 by Mike Shulman.)
However, I was unable to find any occurrence of this terminology in Whitehead’s book.
Indeed, looking at the table on page 141 below Theorem 6.22, we see that Whitehead refers to the dual construction as the mapping cylinder I_f, whereas the original construction is denoted by I^f, but there is no name attached to it.
Furthermore, on page 43 below Theorem 7.31 one reads:
The process of replacing the map f: X→Y by the homotopically equivalent fibration p : I^f→Y
is, in some sense, analogous to that of replacing f by the inclusion map of X into the mapping cylinder of f;
the latter is a cofibration, rather than a fibration.
Pursuing this analogy further, we may consider the fibre T^f of p over a designated point of Y.
We shall call T^f the mapping fibre of f (resisting firmly the temptation to call I^f and T^f the mapping cocylinder and cocone of f!).
I presume you mean the article cocylinder. That addition by me was based on this discussion in which Zoran attributes this terminology to Whitehead. Perhaps it is a different article of Whitehead’s? Or perhaps Zoran’s memory was wrong.
Re #2: Yes, indeed.
The first occurrence of “mapping path space” that I was able to find in the literature is two articles by Peter Hilton from 1968 and 1969. Perhaps the terminology is due to him?
Beats me. There’s no reason we have to attribute it to anyone in particular, only note that it’s an alternative terminology.
I changed the article to indicate that Peter May’s books use this terminology.
1 to 5 of 5