monoid can be defined. Monoidal categories are ubiquitous: cartesian categories (in particular, the category of sets), free word-category over any category, endofunctors category over any category, category of R-modules over a commutative ring R, etc. The abstract monoids definable in the monoidal category incarnate in the usual monoids, triples (or monads), R-algebras, etc. 2 Given a monoidal ...

In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad onMod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ ...

In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomi...

Let SMC denote the 2-category with objects small symmetric monoidal categories, 1cells symmetric monoidal functors and 2-cells monoidal natural transformations. It is shown that the category quotient of SMC by the congruence generated by its 2-cells is symmetric monoidal closed. 1 Summary of results Thomason’s famous result claims that symmetric monoidal categories model all connective spectra ...

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...

Applicative functors define an interface to computation that is more general, and correspondingly weaker, than that of monads. First used in parser libraries, they are now seeing a wide range of applications. This paper sets out to explore the space of non-monadic applicative functors useful in programming. We work with a generalization, lax monoidal functors, and consider several methods of co...

Given a class F of weights, one can consider the construction that takes a small category C to the free cocompletion of C under weighted colimits, for which the weight lies in F . Provided these free F cocompletions are small, this construction generates a 2-monad on Cat, or more generally on V-Cat for monoidal biclosed complete and cocomplete V. We develop the notion of a dense 2-monad on V-Ca...

Notions of guardedness serve to delineate admissible recursive definitions in various settings a compositional manner. In recent work, we have introduced an axiomatic notion symmetric monoidal categories, which serves as unifying framework for examples from program semantics, process algebra, and beyond. the present paper, propose generic metalanguage guarded iteration based on combining this w...

We argue that symmetric (semi)monoidal comonads provide a means to structure context-dependent notions of computation such as notions of dataflow computation (computation on streams) and of tree relabelling as in attribute evaluation. We propose a generic semantics for extensions of simply typed lambda calculus with context-dependent operations analogous to the Moggi-style semantics for effectf...

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...