On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which are not immediately apparent in the traditional presentation of the presheaf hyperdoctrine, this reconstruction leads us to an axiomatic treatment of directed equality predicates (modelled by hom presheaves), realizing a vision initially set out by Lawvere (1970). It also leads to a simple calculus of string diagrams (representing presheaves) that is highly reminiscent of C. S. Peirce's existential graphs for predicate logic, refining an earlier interpretation of existential graphs in terms of Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this work extends to a bifibrational setting a number of fundamental ideas of linear logic.Comment: Identical to the final version of the paper as appears in proceedings of LICS 2016, formatted for on-screen readin

Topological semi-abelian algebras

Given an algebraic theory whose category of models is semi-abelian, we study the category of topological models of and generalize to it most classical results on topological groups. In particular, is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite -algebras.http://www.sciencedirect.com/science/article/B6W9F-4CB07X6-1/1/61cf6d089f1d054878b360422bce8da

Internal object actions

summary:We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module

A generalization of Gabriel's Galois covering functors II: 2-categorical Cohen-Montgomery duality

Given a group $G$, we define suitable 2-categorical structures on the class of all small categories with $G$-actions and on the class of all small $G$-graded categories, and prove that 2-categorical extensions of the orbit category construction and of the smash product construction turn out to be 2-equivalences (2-quasi-inverses to each other), which extends the Cohen-Montgomery duality.Comment: 31 pages. I moved the Sec of G-GrCat into Sec 3, and added Lem 5.6. I added more explanations in the proof of Cor 7.6 with (7.5). I added Def 7.7 and Lem 7.8 with the necessary additional assumptions in Props 7.9 and 7.11. I added Lem 8.8 with a short proof, Rmk 8.9 and the proof of Lem 8.10. The final publication is available at Springer via http://dx.doi.org/10.1007/s10485-015-9416-

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember \Forget? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale \pi_1(\spec(R)) in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.Comment: 46 pages + bibliography. Diagrams drawn in Tik

Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Comment: 33 page

State sum construction of two-dimensional open-closed Topological Quantum Field Theories

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma--Hosono--Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.Comment: 33 pages; LaTeX2e with xypic and pstricks macros; v2: typos correcte

Introduction to Categories and Categorical Logic

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 study; a guide to further reading is included. The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra - vector spaces and linear maps, or groups and group homomorphisms - would be helpful.Comment: 96 page