For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C. Two them are Hopf monads, and for pivotal, so remaining two. In that case all Ai isomorphic as monads. We define monadic cointegral to be an Ai-module morphism 1→Ai(D), where D is distinguished invertible object relate cointegrals categorical introduced by Shimizu (2019), and, in braided, integral braid...

Parameterised monads have the same relationship to adjunctions with parameters as monads do to adjunctions. In this paper, we investigate algebras for parameterised monads. We identify the EilenbergMoore category of algebras for parameterised monads and prove a generalisation of Beck’s theorem characterising this category. We demonstrate an application of this theory to the semantics of type an...

The main goal of this publication is to show that the basic constructions in theories fuzzy sets, soft hesitant sets or intuitionistic have a common background, based on theory monads categories. It proven ad hoc defined concepts individual theories, such as power set structures these relations approximation operators by are only special examples applications monad This makes it possible, one h...

We construct a ‘weak’ version EMw(K ) of Lack & Street’s 2-category of monads in a 2-category K , by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EMw(K ) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of ‘wea...

The use of monads to structure functional programs is de scribed Monads provide a convenient framework for simulating e ects found in other languages such as global state exception handling out put or non determinism Three case studies are looked at in detail how monads ease the modi cation of a simple evaluator how monads act as the basis of a datatype of arrays subject to in place update and ...

Iterative algebraic theories were introduced by Calvin C. Elgot in Elgot (1975) as a concept serving the study of computation (on, say, Turing machines) at a level abstracting from the nature of external memory. The main example presented by Elgot is the theory of rational trees, that is, infinite trees that are solutions of systems of finitary iterative equations. Or, equivalently, that posses...

A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law. A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K , respectively. This observa...

For functors L : A→ B and R : B→ A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B, MorB(L(A), B) α // MorA(A,R(B)) β oo . (L,R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonad...

Using Escardó-Flagg approach to injectivity via Kock-Zöberlein monads in T0 topological spaces [3], and Hofmann’s recent study of injectivity for spaces [4], we characterize continuous maps which are injective with respect to special classes of embeddings using convergence: see [1]. In fact, convergence has been shown to be very useful in the characterization of special classes of maps, like ef...