We develop a type theory and provide a denotational semantics for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic. In our semantics, variables inhabit certain Hilbert bundles, and computations are interpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal...

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W -spaces, orth...

If M is both an abelian category and a symmetric monoidal closed category, then it is natural to ask whether projective objects in M are at, and whether the tensor product of two projective objects is projective. In the most familiar such categories, the answer to these questions is obviously yes. However, the category M G of Mackey functors for a compact Lie group G is a category of this type ...

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoida...

In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of A−A bicomodules. Properties of relations are defi...

We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also o...

In this paper we propose a new categorical formulation of attribute grammars in traced symmetric monoidal categories. The new formulation, called monoidal attribute grammars, concisely captures the essence of the classical attribute grammars. We study monoidal attribute grammars in two categories: Rel and ωCPPO. It turns out that in Rel monoidal attribute grammars correspond to the graphtheoret...

By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a tr...

Assuming only a very rudimentary knowledge of enriched category theory — V-categories, V-functors, and V-natural transformations for a closed, symmetric monoidal, complete and cocomplete category V — we introduce weighted limits and colimits, the appropriate sort of limits for the enriched setting. This “modern” approach was introduced to the author through talks by Mike Shulman at the Category...

Let Sp be the category of symmetric spectra, regarded as having the stable model structure. We prove that the category of small categories enriched over Sp admits a model category structure. The method of proof applies to other categories than symmetric spectra as well, as long as these categories are linked to the category of simplicial sets via a certain strong monoidal Quillen pair.