We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal clos...

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....

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...

A monoidal model category is a model category with a closed monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker, than the results of [SS97]. In part...

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to a natural transformation that need not be an isomorphism). These results are proved first in the ...

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2 completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1 + L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and...

Abrams’ theorem describes a necessary and sufficient condition for the closedness of a linear image of an arbitrary set. Closedness conditions of this type play an important role in the theory of duality in convex programming. In this paper we present generalizations of Abrams’ theorem, as well as Abrams-type theorems characterizing other properties (such as relatively openness or polyhedrality...