symmetric monoidal (∞,1)-category of spectra
By Alg is usually meant a category with associative algebras as objects and algebra homomorphisms as morphisms.
(This depends on a choice of ground field or even ground ring, which is often left implicit and determined by the context. Over the ground ring of integers, Ring is the category of rings.)
Beware that many other types of algebras exist besides associative algebras (e.g. Lie algebras or generally algebras over an operad or algebras over a monad) and all of them form categories which may in corresponding contexts be denoted “” or similar.
Since associative algebras may be identified with one-object categories enriched in modules over the ground ring), it is sometimes useful to regard as the strict full sub-2-category of the 2-category of Mod-enriched categories. In this case the 2-morphisms between morphisms of algebras come from “intertwiners”: inner endomorphisms of the codomain algebra.
(Analogous statements hold for the category Grp of groups when the latter are regarded as their delooping groupoids.)
With regarded as a strict 2-category this way there is a canonical 2-functor
to the category Bimod of bimodules, which sends algebra homomorphisms to the - bimodule . This exhibits as a framed bicategory in the sense of Shulman.
Last revised on June 9, 2025 at 05:56:19. See the history of this page for a list of all contributions to it.