Distributive laws between monads (triples) were defined by Jon Beck in the 1960s; see [1]. They were generalized to monads in 2-categories and noticed to be monads in a 2-category of monads; see [2]. Mixed distributive laws are comonads in the 2-category of monads [3]; if the comonad has a right adjoint monad, the mate of a mixed distributive law is an ordinary distributive law. Particular case...

In this paper, we consider a model of classical linear logic based on coherence spaces endowed with a notion of totality. If we restrict ourselves to total objects, each coherence space can be regarded as a uniform space and each linear map as a uniformly continuous function. The linear exponential comonad then assigns to each uniform space X the finest uniform space !X compatible with X . By a...

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E . In contrast to the wellknown interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H . The canonical map relating H and ΩE both serves to interpret equality and provides a modal...

Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi. We show that such calculi have good computational properties for a variety of intuitionistic modal logics. Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with...

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to M. Escardó which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never pr...

We report progress in two closely related lines of research: the semantic study of sequentiality and parallelism, and the development of a theory of intensional semantics. We generalize Kahn and Plotkin's concrete data structures to obtain a cartesian closed category of generalized concrete data structures and continuous functions. The generalized framework continues to support a de nition of s...

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...

An algebraic characterization of monads which are abstract partial map classi$ers is provided, without the assumption that the categories of total maps possess products. By an abstract partial map classi$er we mean a monad whose Kleisli category is a full subcategory of a partial map category wherein the induced comonad classi$es partial maps in the usual sense. A construction of the correspond...

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to a natural transformation that need not be an isomorphism). These results are proved first in the ...

Every variable-free logic program induces a PfPf -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variable-free logic program with a PfPf -coalgebra on Set and showed that, if C(PfPf ) is the cofree comonad on ...