nLab
homotopy

The word homotopy is used for different but closely related notions:

• homotopy (as a transformation) denotes the transformation between two continuous maps between topological spaces, i.e. a 2-morphism in the (∞,1)-category Top – or more generally any 2-morphism in any (∞,1)-category, often modeled as a (left or right) homotopy in a model category or category of fibrant objects;

• homotopy (as an operation) denotes the operation (the functor) of assigning homotopy groups to spaces in Top or, more generally, homotopy groups in an (∞,1)-topos.

The latter operation involves taking equivalence relations with respect to the former transformation.

I think that if we're going to organise the pages like this, then we should go through the links to this page and sort most of them out to point to one of the pages above. Certainly if somebody writes ‘Let $h$ be a homotopy from $f$ to $g$.’, they're going to naturally link here, but now it should link to homotopy (as a transformation). (If most links are like that —I haven't checked yet—, then perhaps that page should come back here.) —Toby

Urs: yes, I know what you mean. it took me some time to decide how to name these entries, once I had decided to separate them and I don’t claim that this is the optimal solution.

I guess currently most links want to go to homotopy (as a transformation) but with David pushing the Eckmann-Hilton duality that may change. I was motivated by his second sentence there.