### Generalized Homotopy Theory

#### Posted by David Corfield

Over at $n$Lab we’re itching for some discussion as to whether there can be something which is to homotopy as Nonabelian (unstable) cohomology is to cohomology. Can we free things up so we don’t just map spheres into spaces? At the entry homotopy (as an operation) you can read the suggestion of a ‘homotopy with co-coefficients in $B$’, rather than a sphere.

I suppose Moore spaces as domain would be a start as suggested here for spaces of type $(A, 2)$ and suspensions.

A couple of things to check out perhaps:

- Generalized Homotopy Theory by J. Remedios-Gómez and S. Rodríguez-Machín;
- On CW cospectra by M. Hikida.

I have the feeling they’ll be a huge amount out there to learn, e.g., about cofibrant co-grouplike objects.

## Re: Generalized Homotopy Theory

In a 1996 discussion on the cat-list André Joyal writes