Categories and General Algebraic Structures with Applications
Volume:3 Issue: 1, Jul 2015

In 2001, S. Bulman-Fleming et al. initiated the study of three flatness properties (weakly kernel flat, principally weakly kernel flat, translation kernel flat) of right acts A S AS over a monoid S S that can be described by means of when the functor A S ⊗− AS⊗− preserves pullbacks. In this paper, we extend these results to S S -posets and present equivalent descriptions of weakly kernel po-flat, principally weakly kernel po-flat and translation kernel po-flat. Moreover, we show that most of flatness properties of S S -posets can be transferred to their coproducts and vice versa.

In the present articleý, ýwe study some categorical properties of the category {ýCpo Sep ýCpoSep -S S } of all {separately strict S S -cpos}; cpo's equipped withý a compatible right action of a separately strict cpo-monoid S S which isý strict continuous in each componentý. ýIn particularý, we show that this category is reflective and coreflective in theý category of S S -cposý, ýfind the free and cofree functorsý, characterize products and coproductsý. ýFurthermoreý, ýepimorphisms andý monomorphisms in {Cpo Sep CpoSep -S S } are studiedý, ýand show thatý {Cpo Sep CpoSep -S S } is not cartesian closedý.

ýýýIn this paperý, ýtheý notion of injectivity with respect to order dense embeddings in ýýthe category of S S -posetsý, ýposets with a monotone action of aý pomonoid S S on themý, ýis studiedý. ýWe give a criterioný, ýlike the Baer condition for injectivity of modulesý, ýor Skornjakov criterion for injectivity of S -setsý, ýfor the order dense injectivityý. ýAlsoý, ýwe consider such injectivity for S itselfý, and its order dense idealsý. ýFurtherý, ýwe define and study some kinds of weak injectivity with respect to order dense embeddingsý, ýconsider their relations with order dense injectivityý. ýAlso investigate if these kinds of injectivity are preserved or reflected by productsý, ýcoproductsý, ýand direct sums ofýýS S -posetsý.

In this article we introduce the notion of \textit{Fractal ω-operad} emerging from a natural ω-operad associated to any coglobular object in the category of higher operads in Batanin's sense, which in fact is a coendomorphism ω-operads. We have in mind coglobular object of higher operads which algebras are kind of higher transformations. It follows that this natural ω-operad acts on the globular object associated to these higher transformations. To construct the natural ω-operad we introduce some general technology and give meaning to saying an ω-operad possesses the \textit{fractal property}. If an ω-operad B0P has this property then one can define a globular object of all higher B0P-transformations and show that the globular object has a B0P -algebra structure.

In this article we discuss examples of fractal ω-operadsý. ýThus we show that there is an ω-operadic approach to explain existence ofý ýthe globular set of globular sets\footnote{Globular sets are also called ω-graphs by the French School.}ý, ýthe reflexive globular set of reflexive globular setsý, ýthe ω-magma of ω-magmasý, ýand also the reflexive ω-magma of reflexive ω-magmasý. ýThusý, ýeven though the existence of theý ýglobular set of globular sets is intuitively evidentý, ýmany other higher structures which \textit{fractality} are less evidentý, ýcould be describedý with the same technologyý, ýusing fractal ω-operadsý. ýWe have in mind the non-trivial question of the existence of theý ýweak ω-category of the weak ω -categories in the globular settingý, ýwhich is described in \cite{kach-ir3} with the same technology up to a contractibilityý ýhypothesisý.

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.