Enlargements of Categories

Proponents of category theory long hoped to escape the limits of set theory by founding mathematics on an unlimited category theory in which large categories, such as the category Grp of all groups, the category Top of all topological spaces, and the category Cat of all categories, would be (first-class) entities rather than just classes. Several proposals were put forward by Lawvere, MacLane, and Feferman, but none were successful. Feferman, in 1969 and 2013, proposed three ...

In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In the case of ordinary categories this reduces to something surprising: a category may be enriched in an `fc-multicategory', a very general kind of 2-dimensional structure encompassing monoidal categories, plain multicategories,...

This note has several aims. Firstly, it portrays a non-standard analysis as a functor, namely a functor * that maps any set A to the set *A of its non-standard elements. That functor, from the category of sets to itself, is postulated to be an equivalence on the full subcategory of finite sets onto itself and to preserve finite projective limits (equivalently, to preserve finite products and equalizers). Secondly, "Local" non-standard analysis is introduced as a structure whi...

We give an easy example of a triangulated category, linear over a field $k$, with two different enhancements, linear over $k$, answering a question of Canonaco and Stellari.

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further s...

We introduce the notion of an enriched set, as an abstraction of enriched categories, and a category of enriched sets. The set of enriched sets is itself described as a set enriched over the category of enriched sets. We introduce a method for the construction of sets enriched over the set of enriched sets from a given enriched set with some addition data, and for "functors" from such enriched sets as should thereby arise to the enriched set of enriched sets.

This paper is a sequel of Imamura (2019) (arXiv:1711.01609) where we set up a framework of nonstandard large-scale topology. In the present paper, we apply our framework to various topics in large-scale topology: spaces having with both small-scale and large-scale structures, large-scale structures on nonstandard extensions, size properties of subsets of coarse spaces, and coarse hyperspaces.

We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a "spec datum" is introduced, as a certain relation between categories, of which one has been given a Grothendieck topology. A "geometry" is interpreted as a sub-category of $\Omega$, and a formalism is given by which such a subcategory is to be associated to a spec datum, reflecting the standard construction of the category of schemes from ...

We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by comparing it to other known generalizations of enrichment: namely, those for indexed categories and for generalized multicategories. It turns out that both of these notions are closely related to internal enrichment and, as a corollary, to ea...

In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat detailed discussion of system limits and the exact imbedding of abelian categories. This is followed (in part 3) by a discussion of localization, homological algebra, and generalizations of additive and abelian categories such as triangulated an...