In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) over K. The link with existing notions in terms of morphism classes is given via the respective Eilenberg–Moore categories. Dedica...

Abstract. We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, whereas the additive (product and coproduct) and exponential (⊗-comonoid comonad) structures require additional properties and a...

Let K be a comonad on a model category M. We provide conditions under which the associated category MK of K-coalgebras admits a model category structure such that the forgetful functor MK →M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are s...

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of a...

In the theory of coalgebras C over a ring R, the rational functor relates the category C∗M of modules over the algebra C∗ (with convolution product) with the category CM of comodules over C. This is based on the pairing of the algebra C∗ with the coalgebra C provided by the evaluation map ev : C∗ ⊗R C → R. The (rationality) condition under consideration ensures that CM becomes a coreflective fu...

O’Connor [6] made the simple but very useful observation with deep consequences that the (very well-behaved) lenses à la Foster et al. [3] are nothing but coalgebras of the array comonads of Power and Shkaravska [7]. The put operation in these lenses is quite rigid in that a whole new view is merged into the source, there is no flexibility for speaking about small changes to the view. We advoca...

Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. They represent Berry’s bidomains. The representation can proceed in two stages. Bistructures form a categorical model of Girard’s linear logic...

In the coherence space semantics of linear logic, the webs of the spaces interpreting the exponentials may be deened using multi-cliques (multisets whose supports are cliques) instead of cliques. Inspired by the quantitative semantics of Jean-Yves Girard, we give a characterization of the morphisms of the co-Kleisly category of the corresponding comonad (this category is cartesian closed and, t...

We introduce a probabilistic extension of Levy’s Call-By-Push-Value. This extension consists simply in adding a “flipping coin” boolean closed atomic expression. This language can be understood as a major generalization of Scott’s PCF encompassing both call-by-name and call-by-value and featuring recursive (possibly lazy) data types. We interpret the language in the previously introduced denota...

Received We study three comonads derived from the comma construction. The induced coalgebras correspond to the three concepts displayed in the title of the paper. The comonad that yields the-autonomous categories is, in essence, the Chu construction, which has recently awaken much interest in computer science. We describe its couniversal property. It is right adjoint to the inclusion of-autonom...