A main concern of the paper will be a Curry-Howard interpretation of Intuitionistic Linear Logic. It will be extended with recursion, and the resulting functional programming language will be given operational as well as categorical semantics. The two semantics will be related by soundness and adequacy results. The main features of the categorical semantics are that convergence/divergence behav...

A propositional logic program P may be identified with a PfPf -coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf )coalgebra, where C(PfPf ) is the cofree comonad on PfPf , describes derivations by resolution. That correspondence has been developed to model first-order programs in two ways, with lax semantics and saturated semantics, based on locally ordered ca...

After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We p...

Containers are a neat representation of a wide class of set functors. We have previously [1] introduced directed containers as a concise representation of comonad structures on such functors. Here we examine interpreting the opposite categories of containers and directed containers. We arrive at a new view of a di↵erent (considerably narrower) class of set functors and monads on them, which we ...

Abstract It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power Shkaravska comodels an theory $\mathbb{T}$ – i.e., models in opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ provide a suitable environment for evaluating effects encoded by . As already noted Shkaravska, taking yield...

We introduce the notion of elementary Seely category as a notion of categorical model of Elementary Linear Logic (ELL) inspired from Seely’s definition of models of Linear Logic (LL). In order to deal with additive connectives in ELL, we use the approach of Danos and Joinet [DJ03]. From the categorical point of view, this requires us to go outside the usual interpretation of connectives by func...

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey. A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of c...

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant. Let A be a cofibrantly generated model category. Garner’s version...

Finite triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested data type, i. e., a heterogeneous family of inductive data types, while infinite triangular matrices form an example of a nested coinductive type, which is a heterogeneous family of coinductive data types. Redecoration for infinite triangular matrices is taken up from previous wo...

For an associative ring R, let P be an R-module with S = EndR(P). C. Menini and A. Orsatti posed the question of when the related functor HomR(P, −) (with left adjoint P ⊗S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphi...