990 research outputs found
Crossed modules of racks
We generalize the notion of a crossed module of groups to that of a crossed
module of racks. We investigate the relation to categorified racks, namely
strict 2-racks, and trunk-like objects in the category of racks, generalizing
the relation between crossed modules of groups and strict 2-groups. Then we
explore topological applications. We show that by applying the rack-space
functor, a crossed module of racks gives rise to a covering. Our main result
shows how the fundamental racks associated to links upstairs and downstairs in
a covering fit together to form a crossed module of racks.Comment: 25 pages, 1 figure, accepted in Homology, Homotopy and Application
T-homotopy and refinement of observation (III) : Invariance of the branching and merging homologies
This series explores a new notion of T-homotopy equivalence of flows. The new
definition involves embeddings of finite bounded posets preserving the bottom
and the top elements and the associated cofibrations of flows. In this third
part, it is proved that the generalized T-homotopy equivalences preserve the
branching and merging homology theories of a flow. These homology theories are
of interest in computer science since they detect the non-deterministic
branching and merging areas of execution paths in the time flow of a higher
dimensional automaton. The proof is based on Reedy model category techniques.Comment: 30 pages ; final preprint version before publication ; see
http://nyjm.albany.edu:8000/j/2006/Vol12.ht
Operads within monoidal pseudo algebras
A general notion of operad is given, which includes as instances, the operads
originally conceived to study loop spaces, as well as the higher operads that
arise in the globular approach to higher dimensional algebra. In the framework
of this paper, one can also describe symmetric and braided analogues of higher
operads, likely to be important to the study of weakly symmetric, higher
dimensional monoidal structures
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
- …