Ross Street

The geometry of tensor calculus. I

This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals. Mathematics Subject Classifications (1991). 18D10, 52B11, 53A45 , 57M25, 68Q10, 82B23.

Traced monoidal categories

Traced monoidal categories are introduced, a structure theorem is proved for them, and an example is provided where the structure theorem has application.

The algebra of oriented simplexes

Abstract Anm-simplex x in ann-category A consists of the assignment of anr-cell x(u) to each (r + 1)-element subset u of {0, 1,..., m} such that the source and target (r−1)-cells of x(u) are appropriate composites of x(v) for v a proper subset of u. As m increases, the appropriate composites quickly become hard to write down. This paper constructs anm-categoryOm such that anm-functor x:Om →A is precisely an m-simplex in A. This leads to a simplicial set ΔA, called the nerve of A, and provides the basis for cohomology with coefficients in A. Higher order equivalences in A as well as freen-categories are carefully defined. Each Om is free.

Limits indexed by category-valued 2-functors

There is common agreement now on the correct general notion of limit for categories whose horns are enriched in a suitable category 13. The definition involves a v-functor J: A + v which should be thought of as a v-diagram scheme; for simplicity, we shall suppose %q is complete enough so that the category of v-valued %functors on OQ admits its natural enrichment to aC13 -category [ 94;,33 1. A J-indexed limit for a %functor S: d + 3c is then an object lim (J, S) of % together with a c13 -natural isomorphism

Tortile Yang-Baxter operators in tensor categories

Abstract The free tortile tensor category T ∼ on one generating object is shown here, by purely algebraic techniques, to also be the free tensor category containing an object equipped with a tortile Yang-Baxter operator. Shum has a geometric characterization of T ∼ as the category whose arrows are tangled ribbons. This alternative universal property can be used to construct representations of T ∼ .

Variation through enrichment

This paper continues the authors’ various works [3,4,12,14] on categories enriched in bicategories. We treat the elements of the theory again, here from a more algebraic (logical) and less geometric viewpoint. For a bicategory ~1 we first develop V -matrices before passing on to ti -modules, an approach tihich allows a simple proof of the cocompleteness of the 2-category ti -Cat of

Frobenius monads and pseudomonoids

1 -categories. When 5 has precisely one object (and so is a monoidal category) the main results are in works of Bknabou [21p Lawvere [6], and Wolff

Quantum groups : a path to current algebra

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What it means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to “strongly separable” Frobenius algebras and “weak monoidal Morita equivalence.” Wreath products of Frobenius algebras are discussed.

Cauchy characterization of enriched categories

Introduction 1. Revision of basic structures 2. Duality between geometry and algebra 3. The quantum general linear group 4. Modules and tensor products 5. Cauchy modules 6. Algebras 7. Coalgebras and bialgebras 8. Dual coalgebras of algebras 9. Hopf algebras 10. Representations of quantum groups 11. Tensor categories 12. Internal homs and duals 13. Tensor functors and Yang-Baxter operators 14. A tortile Yang-Baxter operator for each finite-dimensional vector space 15. Monoids in tensor categories 16. Tannaka duality 17. Adjoining an antipode to a bialgebra 18. The quantum general linear group again 19. Solutions to exercises References Index.

Flexible limits for 2-categories

SuntoSi caratterizzano le bicategorie di moduli fra categorie arricchite, a meno di biequivalenze. Tale caratterizzazione permette di introdurre la nozione di comsos (non elementare), e di mostrare che risulta chiusa rispetto a numerose costruzioni.SummaryA characterization is given of those bicategories which are biequivalent to categories of modules for some suitable base. These bicategories are the correct (non elementary) notion of cosmos, which is shown to be closed under several basic constructions.