We discuss conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. In particular, we show that these inherit their categorical properties from monad. further observe various constructions such as prime open filter defined H. Simmons, closed compactification studied Bentley and Herrlich, well separa...

We show, for a monad T, that coalgebra structures on a T-algebra can be described in terms of “braidings”, provided that the monad is equipped with an invertible distributive law satisfying the Yang-Baxter equation.

This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in ε0 play a prominent role. The ...

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...

Motivated by twisted N = 4 supersymmetric Yang-Mills theory in 4 dimensions, a natural extension of the monad (ADHM) construction relevant to D-instantons is considered. We show that a family of Yang-Mills instantons can be constructed from D-instantons. We discuss some possible roles of reciprocity in D-brane physics. We conjecture the existence of universal instantons together with a generali...

The notions of side-effect-freeness and commutativity are typical for probabilistic models, as subclass of quantum models. This paper connects these notions to properties in the theory of monads. A new property of a monad (‘strongly affine’) is introduced. It is shown that for such strongly affine monads predicates are in bijective correspondence with side-effect-free instruments. Also it is sh...

Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad, which encompasses the notion of algebra. The associated notion of morphism of modules (”linear” natural transformations) captures important formal properties about substitution. In this paper, we present basic constructions of modules and we show examples concerning i...

We present some tools for providing situations where the generalised Rota formula of Carlier (Algebr Geometr Topol 20:169–213, 2020) applies. As an example this, we compute Möbius function incidence algebra any directed restriction species, free operad, or more generally monad on a finitary polynomial monad.

A general notion of operad is given, which includes: (1) the operads that arose in algebraic topology in the 1970’s to characterise loop spaces. (2) the higher operads of Michael Batanin [Bat98]. (3) braided and symmetric analogues of Batanin’s operads which are likely to be important in the study of weakly symmetric higher dimensional monoidal categories. The framework of this paper, links tog...

This article demonstrates how a powerful and expressive abstraction from concurrency theory—monads of resumptions—plays a dual rôle as a programming tool for concurrent applications. The article demonstrates how a wide variety of typical OS behaviors may be specified in terms of resumption monads known heretofore exclusively in the literature of programming language semantics. We illustrate the...