Seminar on Triples and Categorical Homology Theory

Quantale-Enriched Multicategories Via Actions

“…Moreover, −⊠ 2 = defines a symmetric closed monoidal structure on Sup with unit 2 ≃ P 2 (1). With respect to this monoidal structure, since for X , Y , P 2 (X ) ⊠ 2 P(Y ) ≃ P 2 (X × Y ), the free functor P 2 : Set → Sup becomes strong monoidal.…”

“…Let (T, m, e) be a monad with T : Set → Set. Then, if we assume the axiom of choice, Set T is cocomplete (see [26,1]). Thus, Eilenberg-Moore categories for strong monads defined on Set always satisfy the hypotesis of Theorem C.5.…”

Compatibility of (co)actions and localizations

“…where the A-action ⊲ T M on M ⊗ k H is given by k-linear extension of the formula a ⊲ T M (m ⊗ h) := ((h (2) ⊲ A a) ⊲ M m)) ⊗ h (1) .…”

“…Throughout the paper k is a ground field, but for most results it can be taken to be a commutative ring. The unadorned tensor symbol means tensoring over k. We assume that the reader is well familiar with adjoint functors and familiar with (co)monads ( [3,12,18]) which some call triples ( [1]). We will speak (co)modules over (co)monads for what many call (co)algebras over (co)monads.…”

Bicategory of entwinings

“…When the preprint version 1 of this article has been posted, G. Böhm has kindly called my attention to the following argument: distributive laws ( [1,2]) are just monads in a 2-category of monads in the sense of formal monad theory ( [10]) of R. Street, and in particular they themselves make a 2-category; the analogue can be easily written out for mixed distributive laws betwen a monad and a comonad; it is not published in detail, but it is widely known among the experts that the formal monad theory can be extended to bicategorical setup, instead of strict 2-categories; finally entwinings are mixed distributive laws in the setup of the bicategory of rings and bimodules; regarding that in a bicategory we can do 2 dualizations (inverting 1-cells and 2-cells) there are thus 4 natural bicategories of entwinings. Our construction is explicit and from scratch and does not use this chain of constructions and translations of data (explicit descriptions are its merit but also its conceptual deficiency).…”