143 research outputs found
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 , we define suitable 2-categorical structures on the class
of all small categories with -actions and on the class of all small
-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 , 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 when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -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 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
- …