We seek a unified account of modularity for computational effects. We begin by reformulating Moggi's monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi's exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi's side-effects monad transformer. Finally, we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.
We present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic: in particular we treat the exponentials in detail. We indicate simple applications of the methods and show that they cover familiar examples.
Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature-including of course Joyal's original notion-together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed.
Abstract. We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.
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