Exponentiation and nonlinear homology
by
F. William Lawvere
SUNY Buffalo

Hurewicz's homotopy types of spaces are themselves space-like in that they are objects in a suitable cartesian-closed extensive category. The same is true for the types of infinitesimals (if they are properly construed). Yet both of these examples are a dialectical negation of space in that the left and right adjoints to the discrete inclusion coalesce, even though their distinctness is fundamental for spaces. Homology types constitute a third example of abstract space-like objects, and also live in a cartesian closed category (as seems to have motivated Eilenberg and Moore in their work on coalgebras). Much linear algebra is apparently required to compute particular examples. Can the nonlinear content of these constructions be conceptually characterized without involving linear algebra?

Date received: May 31, 2002