Homotopy theory of Moore flows (II)

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

Cross-connection structure of concordant semigroups

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

A LOGICAL AND ALGEBRAIC CHARACTERIZATION OF ADJUNCTIONS BETWEEN GENERALIZED QUASI-VARIETIES

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.

Topological realizations and fundamental groups of higher-rank graphs

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian groups.

MORITA DUALITY AND RING EXTENSIONS

In this paper, we show that there exists a category equivalence between certain categories of A-rings (respectively, ring extensions of A) and B-rings (respectively, ring extensions of B), where A and B are Morita dual rings. In this category equivalence, corresponding two A-ring and B-ring are Morita dual. This is an improvement of a result of Müller, which state that if a ring A has a Morita duality induced by a bimodule BQA and R is a ring extension of A such that RA and Hom A(R, Q)A are linearly compact, then R has a Morita duality induced by the bimodule S End R( Hom A(R, Q))R with S = End R( Hom A(R, Q)). We also investigate relationships between Morita duality and finite ring extensions. Particularly, we show that if A and B are Morita dual rings with B basic, then every finite triangular (respectively, normalizing) extension R of A is Morita dual to a finite triangular (respectively, normalizing) extension S of B, and we give a result about finite centralizing free extensions, which unify a result of Mano about self-duality and a result of Fuller–Haack about semigroup rings.