symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
An -space is a homotopy type that is equipped with the structure of a monoid up to coherent higher homotopy:
that means it is equipped with
a binary product operation
a choice of associativity homotopy; ;
a choice of pentagon law homotopy between five such s;
and so ever on, as controlled by the associahedra.
In short one may say: an -space is an A-∞ algebra/monoid in an (∞,1)-category in the (∞,1)-category ∞Grpd/Top. See there for more details.
If in the definition of an -space one discards all the higher homotopies and retains only the existence of an associativity-homotopy, then one has the notion of H-monoid?. Put another way, An -space in the (∞,1)-category ∞Grpd/Top becomes an H-monoid? in the homotopy Ho(Top). And lifting an H-monoid? structure to an -space structure means lifting a monoid structure through the projection from the (∞,1)-category ∞Grpd/Top to Ho(Top).
The delooping of an -space is an A-∞ category/(∞,1)-category with a single object. (Beware that in standard literature “-category” is often, but not necessarily, reserved for a stable (∞,1)-category).
There is an equivalence of (∞,1)-categories between pointed connected A-∞ categories/(∞,1)-categories and -spaces.
-spaces were introduced by Jim Stasheff as a refinement of an H-group taking into account higher coherences.
Last revised on January 29, 2024 at 12:21:04. See the history of this page for a list of all contributions to it.