Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and functors. Our first result is that for finite equivalent to the a comonad arising from central Hopf monad. This has several applications: First, we obtain short conceptual proof Ocneanu rigidity. Second, it allows use standard methods theory compute family non-semisimple finite-dimensional algebras generali...

Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn doctrines. Modal interior operators characterised as particular adjoints in Dtn. We show that they can be constructed from comonads well adjunctions it, and two constructions compare. Finally we amount information lost passage comonad, or an adjunction, modal operat...

We discuss Osius’s [22] concept of a recursive coalgebra of a functor from the perspective of programming semantics and give some new sufficient conditions for the recursiveness of a functor-coalgebra that are based on comonads, comonad-coalgebras and distributive laws.

Given an adjunction F ⊣ G connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calcul...

In this paper, we introduce the notion of {bf K}-flat projective fuzzy quantales, and give an elementary characterization in terms of a fuzzy binary relation on the fuzzy quantale. Moreover, we  prove that {bf K}-flat projective fuzzy quantales are precisely the coalgebras for a certain comonad on the category of fuzzy quantales. Finally, we present two special cases of {bf K} as examples.

It is well known that Barr and Beck’s definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induce...

We formulate an elementary condition on an involutive quantaloidQ under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawver...

The paper discusses, in a categorical perspective, some recent works on optimal graph reduction techniques for the-calculus. In particular, we relate the two \brackets" in GAL92a] to the two operations associated with the comonad \!" of Linear Logic. The rewriting rules can be then understood as a \local implementation" of naturality laws, that is as the broadcasting of some information from th...

We give an abstract axiomatic account of timed processes using monoids and their (partial and total) actions. Subsequently, we present categorical formulations thereof, including a novel characterisation of partial monoid actions as coalgebras for an evolution comonad. Adapting the approach of Turi and Plotkin [24], we then exhibit an abstract theory of well-behaved operational rules suitable f...

Comonads are mathematical structures that account naturally for eeects that derive from the context in which a program is executed. This paper reports ongoing work on the interaction between recursion and comonads. Two applications are shown that naturally lead to versions of a comonadic fold operator on the product comonad. Both versions capture functions that require extra arguments for their...