It is known that the so-called monadic decomposition, applied to adjunction connecting category of bialgebras vector spaces via tensor and primitive functors , returns usual between (restricted) Lie algebras . Moreover, in this framework, notions augmented monad combinatorial rank play a central role. In order set these results into wider context, we are led substitute decomposition by what cal...

Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic...

Monads can be used to model term rewriting systems by generalising the well-known equivalence between universal algebra and monads on the category Set. In [Lü96], this semantics was used to give a purely categorical proof of the modularity of confluence for the disjoint union of term rewriting systems. This paper provides further support for monadic semantics of rewriting by giving a categorica...

We propose for the Effective Topos an alternative construction: a realisability framework composed of two levels of abstraction. This construction simplifies the proof that the Effective Topos is a topos (equipped with natural numbers), which is the main issue that this paper addresses. In this our work can be compared to Frey’s monadic tripos-to-topos construction. However, no topos theory or ...

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an ObCindexed family of operations of the form αx : (Tx) v −→ (Tx)w provides semantics for algebraic operations on the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy r...

We introduce a simple logic that allows to quantify over the subobjects of a categorical object. We subsequently show that, for the category of graphs, this logic is equally expressive as second-order monadic graph logic (msogl). Furthermore we show that for the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Co...

A formalism for expressing the operational semantics of proof languages used in procedural theorem provers is proposed. It is argued that this formalism provides an elegant way to describe the computational features of proof languages, such as side effects, exception handling, and backtracking. The formalism, called proof monads, finds its roots in category theory, and in particular satisfies t...