internalization and categorical algebra
algebra object (associative, Lie, …)
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
monoid theory in algebra:
An idempotent monoid is a monoid, which is idempotent in that “squares to itself” in the evident category-theoretic sense.
An idempotent monoid in a monoidal category is a monoid in whose multiplication morphism is an isomorphism.
Similarly, an idempotent semigroup in (also called a non-unital idempotent monoid in ) is a semigroup in with an isomorphism.
We write for the full subcategory of spanned by the idempotent monoids in .
An idempotent monoid or semigroup is strict if is not only an isomorphism, but in fact the identity morphism of .
A strong monoidal functor (resp. strict monoidal functor) induces a functor
and hence “preserves” idempotent monoids (resp. strict idempotent monoids).
Similarly, strong (strict) semigroupal functors (“non-unital strong monoidal functors”) “preserve” (strict) idempotent semigroups.
(Strict) idempotent monoids in are the same as (strict) strong monoidal functors from the punctual monoidal category .
Similarly, (strict) idempotent semigroups in may be identified with (strict) strong semigroupal functors functors .
(Strict) idempotent semigroups in are also the same as strictly unitary strong monoidal functors (resp. strict monoidal functors) from to , where is the “Boolean monoid”, the initial monoid with an idempotent element.
An idempotent semigroup in for an ordinary monoid (in ) is an idempotent element of , i.e. an element such that .
An idempotent semigroup in with the monoid of endomorphisms of at is an idempotent morphism of , satisfying .
An idempotent monoid in abelian groups is a solid ring.
An idempotent monoid in an endomorphism functor category is an idempotent monad.
An idempotent monoid in the monoidal category of endomorphisms of a bicategory at is an idempotent -morphism of , satisfying up to a coherent -isomorphism.
An idempotent monoid in Spectra is a “solid ring spectrum” as in Gutierrez 2013, Section 4. See also MO #298435.
An idempotent monoid in the category equipped with the Day convolution monoidal structure is a strong monoidal functor. Similarly, strict idempotent monoids in recover strict monoidal functors.
Peter Hines, The categorical theory of self-similarity, 1999. Theory and Applications of Categories. (abstract, pdf, dvi, ps.)
Javier J. Gutiérrez, On solid and rigid monoids in monoidal categories, Applied Categorical Structures 23, no. 2 (2015), 575-589 (arXiv:1303.5265)
Last revised on September 26, 2021 at 01:07:24. See the history of this page for a list of all contributions to it.