In this paper, we classify additive closed symmetric monoidal structures on the category of left R-modules by using Watts’ theorem. An additive closed symmetric monoidal structure is equivalent to an R-module ΛA,B equipped with two commuting right R-module structures represented by the symbols A and B, an R-module K to serve as the unit, and certain isomorphisms. We use this result to look at s...

This paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results together provide a categorical model for the motivic stable category which has an internal symmetric monoidal smash product. The details of the basic construction of the Morel-Voevodsky proper closed simplicial model structure underlying the motivic stable category are requir...

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing obj...

Given any symmetric monoidal (closed) category C and any suitable collections W of objects of C, it is shown how to construct C[W], a polynomial such category, the result of freely adjoining to C a system of monoidal indeterminates for every element of W. It is then shown that all of the categories of “possible worlds” used to treat languages that allow for dynamic creation of “new” variables, ...

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing obj...

I define “symmetric monoids”, and “caterads” in a closed symmetric monoidal category, and I define what it means for a symmetric monoid to be an algebra over a caterad. These notions codify formal structure that appears in motivic cohomology and should be of more general interest.

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product...

Rios and Selinger have recently proposed a categorical model for the quantum programming language Proto-Quipper-M, which is an important fragment of the Quipper language. In this work, we describe an extension to their categorical model with the additional property that it is DCPOenriched, bringing us closer to modeling general recursion in the language. Similar to their model, our model exhibi...

When replacing the non-negative real numbers with their addition by a commutative quantale V, under a metric lens one may then view small V-categories as sets that come with a V-valued distance function. The ensuing category V-Cat is well known to be a concrete topological category that is symmetric monoidal closed. In this paper we show which concrete symmetric monoidalclosed topological categ...

Abstract. We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is reminiscent of Day’s convolution on presheaves. We then make this category into a model for intuitionistic linear logic by defining an additive an...