Galois theory and a new homotopy double groupoid of a map of spaces

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat^1-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.Comment: 16 A4 pages, xypi

Radicals of rings and pullbacks

AbstractGeneralizing various concrete radicals in associative rings like the nilradical, the Jacobson radical, and so on, A.G. Kurosh and S.A. Amitsur introduced an abstract notion of radical in the early 1950s. The basic notions of their general radical theory can be characterized by properties which are “almost” categorical – in the sense that they can be conveniently defined in the category of rings or even in suitable categories of Ω-groups but not in general categories. Here we are going to characterize radicals of associative rings by means of pullbacks, a notion which is of a purely categorical nature. Throughout the paper we shall work in the category C of associative rings (not necessarily with identity), just calling them “rings”. We hope that our two categorical characterizations of semisimple classes in C can provide natural general frameworks for radical theory, just as localizations do for torsion theories

A simplicial approach to factorization systems and Kurosh–Amitsur radicals

AbstractRegarding categories as simplicial sets via the nerve functor, we extend the notion of a factorization system from morphisms in a category, to 1-simplexes in an arbitrary simplicial set. Applied to what we call the simplicial set of short exact sequences, it gives the notion of Kurosh–Amitsur radical. That is, we present a unified approach to factorization systems and radicals

NON-POINTED EXACTNESS, RADICALS, CLOSURE OPERATORS

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh-Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories

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

Galois theory and commutators

We prove that the relative commutator with respect to a subvariety of a variety of Omega-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Froehlich-Lue and the Janelidze-Kelly notions of central extension. As an example outside the context of Omega-groups we study the reflection of the category of loops to the category of groups where we obtain an interpretation of the associator as a relative commutator.Comment: 14 page

Use of the universal pain assessment tool for evaluating pain associated with temporomandibular disorders in youngsters

Aim: Determine, whether the UPAT could be used as an extra tool to collect data on functional TMJ pain and to assess orofacial pain levels related to temporomandibular disorder(s) (TMD) in youngsters. Methods: Patients were screened at the N1 Dental Clinic of Tbilisi State Medical University. The clinical scores of possible functional jaw pain were collected using the UPAT, to indicate pain severity on a visual scale during different jaw movements (opening, closing and lateral). Statistics: Comparisons of categorised data have been performed by chi-square test and Fisher's Exact test (where expected values were less than 5). The P value less than 0.05 was considered as statistically significant. Results: Two hundred and ninety-one youngsters were screened by calibrated dentists. The majority (59%) of participants were male; age distribution ranged from 8 to 15 years (mean 11.46 +/- 2.11). The results of the UPAT demonstrated the existence of functional TMJ pain in 15.46% (n=45) of the patients without significant prevalence (P > 0.05) in this survey group. Conclusion: According to the results of the present study, the UPAT demonstrated that it could be an additional tool to detect the existence of functional jaw pain possibly associated with TMD and also a valid instrument to score pain intensity associated with TMD in youngster patients