We show, for a monoidal closed category V = (V0,⊗, I), that the category V -Cat of small V -categories is locally λ-presentable if V0 is so, and that it is locally λ-bounded if the closed category V is so, meaning that V0 is locally λ-bounded and that a side condition involving the monoidal structure is satisfied. Many important properties of a monoidal category V are inherited by the category ...

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to fp-hom-closed definable additive satisfying an exactness criterion. For a fixed finitely accessible category C products and structure appropriate assumptions, we provide bijections between subcategories C, Serre tensor-ideals Cfp-mod closed subsets Ziegler-type topology. prea...

We study a relationship between the Int construction of Joyal et al. and a weakening of biproducts called semibiproducts. We then provide an application of geometry of interaction interpretation for the multiplicative additive linear logic (MALL for short) of Girard. We consider not biproducts but semibiproducts because in general the Int construction does not preserve biproducts. We show that ...

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, which may be used to extend the usual monadic semantics of the computational λ-calculus uniformly...

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...

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...

In here we define the concept of fibered symmetric bimonoidal categories. These are roughly speaking fibered categories Λ : D → C whose fibers are symmetric monoidal categories parametrized by C and such that both D and C have a further structure of a symmetric monoidal category that satisfy certain coherences that we describe. Our goal is to show that we can correspond to a fibered symmetric b...

The structure of a k-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition of operads in a braided category. Further...

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to fp-hom-closed definable additive satisfying an exactness criterion. For a fixed finitely accessible category C products and structure appropriate assumptions, we provide bijections between subcategories C, Serre tensor-ideals Cfp-mod closed subsets Ziegler-type topology. prea...