Categorically closed topological groups

Let $\mathcal C$ be a subcategory of the category of topologized semigroups and their partial continuous homomorphisms. An object $X$ of the category ${\mathcal C}$ is called ${\mathcal C}$-closed if for each morphism $f:X\to Y$ of the category ${\mathcal C}$ the image $f(X)$ is closed in $Y$. In the paper we detect topological groups which are $\mathcal C$-closed for the categories $\mathcal C$ whose objects are Hausdorff topological (semi)groups and whose morphisms are isomorphic topological embeddings, injective continuous homomorphisms, continuous homomorphisms, or partial continuous homomorphisms with closed domain.Comment: 26 page

Hereditarily h-complete groups

A topological group G is h-complete if every continuous homomorphic image of G is (Raikov-)complete; we say that G is hereditarily h-complete if every closed subgroup of G is h-complete. In this paper, we establish open-map properties of hereditarily h-complete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of subdirectly represented groups. Numerous applications are presented, among them: 1. Every hereditarily h-complete group with quasi-invariant basis is the projective limit of its metrizable quotients; 2. If every countable discrete hereditarily h-complete group is finite, then every locally compact hereditarily h-complete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.Comment: 12 pages; few changes were made compared to the original submission thanks to the suggestions of the refere

Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC

We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories.The algebraic tools are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context.Comment: To appear in special issue of JMP. Latex documen

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer $n$, the number of elements of order $n$ is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.Comment: 25 pages; To appear in J. of Algebra. Main modifications are the following: (i) Verbose parts of the paper were summarized. (ii) Theorem 6.3 is added. (iii) The relation between Theorem 1.1 and works of Ng and Schauenburg is adde

Categorical models for equivariant classifying spaces

Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.Comment: 29 pages. Revised version, to appear in AGT. Considerable changes of notation and organization and other changes aimed at making the paper more user friendl