A PROP is a way of encoding structure borne by an object of a symmetric monoidal category. We describe a notion of distributive law for PROPs, based on Beck’s distributive laws for monads. A distributive law between PROPs allows them to be composed, and an algebra for the composite PROP consists of a single object with an algebra structure for each of the original PROPs, subject to compatibilit...

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg–Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg–Moore monoids coincide with algebras for the list monad transformer (‘done right’) known from Haskell libraries. From this, we obtai...

Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category V with respect to a specified system of arities j : J ↪→ V . Lawvere’s notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted V -enriched J -cotensor theory, or J -theory for short. For suitable choices of V and J , such...

Strong monads are important for several applications, in particular, the denotational semantics of effectful languages, where strength is needed to sequence computations that have free variables. Strength non-trivial: it can be difficult determine whether a monad has any at all, and strong multiple ways. We therefore review some most known facts about prove new ones. In we present number equiva...

We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as including adjunctions, exponential comonads, multicategories, IL-indexed categories, linearly distributive categories storage, commutative strong monads, CBPV-structures, models polarized calculi, Freyd-categories, skew well ...

Monads have become a fundamental tool for structuring denotational semantics and programs by abstracting a wide variety of computational features such as side-effects, input/output, exceptions, continuations and non-determinism. In this setting, the notion of a monad is equipped with operations that allow programmers to manipulate these computational effects. For example, a monad for side-effec...

The definition of Azumaya algebras over commutative rings R requires the tensor product of modules over R and the twist map for the tensor product of any two R-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category A by considering a monad (F,m, e) on A endowed with a dis...

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoida...

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our ba...

Received Milner's action calculus implements abstraction in monoidal categories, so that familiar-calculi can be subsumed together with the-calculus and the Petri nets. Variables are generalised to names: only a restricted form of substitution is allowed. In the present paper, the well-known categorical semantics of the-calculus is generalised to the action calculus. A suitable functional compl...