We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley’s theorem; by proving that effectful computations (Kle...

Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as Poset, ωCpo, or functor categories such as [Inj, Set] or [Inj, ωCpo]. That involves allowing the arities of Lawvere t...

Generalisation of the foundational basis for many-valued logic programming builds upon generalised terms in form of powersets of terms. A categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions, and compositions thereof. In this paper, substitutions and unifiers appear as constructs in Kleisli categories related...

Collections of objects and morphisms that fail to form categories inas-much as the expected composites of two morphisms need not always be deened have been introduced in 14, 15] under the name composition graphs. In 14, 16], notions of adjunction and weak adjunction for composition graphs have been proposed. Building on these deenitions, we now introduce a concept of monads for composition grap...

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. A part of the so-called Gelfand–Naimark–Segal (GNS) construction is ide...

in this work, we describe an adjunction between the comma category of set-based monads under the v -powerset monad and the category of associative lax extensions of set-based monads to the category of v -relations. in the process, we give a general construction of the kleisli extension of a monad to the category of v-relations.

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 ...

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...

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...

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.