Brian Day

Limits of small functors

Abstract For a small category K enriched over a suitable monoidal category V , the free completion of K under colimits is the presheaf category [ K op , V ] . If K is large, its free completion under colimits is the V -category P K of small presheaves on K , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on P K .

Dualizations and Antipodes

Because an exact pairing between an object and its dual is extraordinarily natural in the object, ideas of R. Street apply to yield a definition of dualization for a pseudomonoid in any autonomous monoidal bicategory as base; this is an improvement on Day and Street, Adv. in Math.129 (1997), Definition 11, p. 114. We analyse the dualization notion in depth. An example is the concept of autonomous (which, usually in the presence of a symmetry, also has been called “rigid” or “compact”) monoidal category. The antipode of a quasi-Hopf algebra H in the sense of Drinfeld is another example obtained using a different base monoidal bicategory. We define right autonomous monoidal functors and their higher-dimensional analogue. Our explanation of why the category Comodf(H) of finite-dimensional representations of H is autonomous is that the Comodf operation is autonomous and so preserves dualization.

Abstract substitution in enriched categories

Abstract Enriched categories equipped with an abstract substitution process are defined in this paper, and called substitudes . They generalize both monoidal enriched categories and operads, and are a little more general than multicategories. They can bear braidings and symmetries. There are two convolution processes with distinctly different properties.

Universal algebra in a closed category

Abstract The development of finitary universal algebra is carried out in a suitable closed category called a π-category. The π-categories are characterized by their completeness and cocompleteness and some product-colimit commutativities. We establish the existence of left adjoints to algebraic functors, completeness and cocompleteness of algebraic categories, a structure-semantics adjunction, a characterization theory for algebraic categories and the existence of the theory generated by a presentation. The conditions on the closed category are sufficiently weak to be satisfied by any (complete and cocomplete) cartesian closed category, semi-additive category, commutatively algebraic category and also the categories of semi-normed spaces, normed spaces and Banach spaces.

Enriched Tannaka reconstruction

Abstract An abstract approach is made to recent theory on “Tannaka” recovery of coalgebras, bialgebras, and Hopf algebras. It is hoped that the methods used shed some light upon the more quantitative aspects of the mathematics involved.

Localisation of locally presentable categories

Abstract The localisations of locally finitely presentable categories are characterised as those categories which admit small colimits, finite limits and a small strong generator, and have filtered colimits commuting with finite limits. Moreover, this is done in the context of enriched categories.

Note on monoidal monads

The representation theory of categories is used to embed each promonoidal monad in a monoidal biclosed monad. The existence of a promonoidal structure on the ordinary Eilenberg- Moore category generated by a promonoidal monad is examined. Several results by previous authors (notably A. Kock and F. E. J. Linton) are reproved and extended.

Localisations of locally presentable categories II

Abstract Localisations of a locally finitely presentable category A are shown to all arise from Grothendieck topologies on the full subcategory C of finitely presentable objects. Twelve further good structural aspects of C are shown to transfer to A .

Gelfand dualities over topological fields

The spectral duality theory of H.-E. Porst and M. B. Wischnewsky is examined in more generality, and examples based on topological fields are described.

On embedding closed categories

This article contains one method of fully embedding a symmetric closed category into a symmetric monoidal closed category. Such an embedding is very useful in the study of coherence problems. Also we give an example of a non-symmetric closed category for which, under the embedding discussed in this article, the resultant monoidal closed structure has associativity not an isomorphism.