Authors:Jaime Martín Fernández Cestau; Luis Javier Hernández Paricio; María Teresa Rivas Rodríguez Pages: 709 - 745 Abstract: Abstract Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincaré techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9427-1 Issue No:Vol. 25, No. 5 (2017)

Authors:Adriana Balan Pages: 747 - 774 Abstract: Abstract Let U be a strong monoidal functor between monoidal categories. If it has both a left adjoint L and a right adjoint R, we show that the pair (R,L) is a linearly distributive functor and (U,U)⊣(R,L) is a linearly distributive adjunction, if and only if L⊣U is a Hopf adjunction and U⊣R is a coHopf adjunction. We give sufficient conditions for a strong monoidal U which is part of a (left) Hopf adjunction L⊣U, to have as right adjoint a twisted version of the left adjoint L. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if L is precomonadic and \(L\mathbb {1}\) is a Frobenius monoid (where \(\mathbb {1}\) denotes the unit object of the monoidal category), then L⊣U⊣L is an ambidextrous adjunction, and L is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad T on a monoidal category has a right adjoint which is also a Hopf comonad, if the object \(T\mathbb {1}\) is dualizable as a free T-algebra. In particular, if \(T\mathbb {1}\) is a Frobenius monoid in the monoidal category of T-algebras and T is of descent type, then T is a Frobenius monad and a Frobenius monoidal functor. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9428-0 Issue No:Vol. 25, No. 5 (2017)

Authors:Jérémy Dubut; Eric Goubault; Jean Goubault-Larrecq Pages: 775 - 807 Abstract: Abstract In this paper, we define and study a homology theory, that we call “natural homology”, which associates a natural system of abelian groups to every space in a large class of directed spaces and precubical sets. We show that this homology theory enjoys many important properties, as an invariant for directed homotopy. Among its properties, we show that subdivided precubical sets have the same homology type as the original ones ; similarly, the natural homology of a precubical set is of the same type as the natural homology of its geometric realization. By same type we mean equivalent up to some form of bisimulation, that we define using the notion of open map. Last but not least, natural homology, for the class of spaces we consider, exhibits very important properties such as Hurewicz theorems, and most of Eilenberg-Steenrod axioms, in particular the dimension, homotopy, additivity and exactness axioms. This last axiom is studied in a general framework of (generalized) exact sequences. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9438-y Issue No:Vol. 25, No. 5 (2017)

Authors:Carmelo Di Natale Pages: 809 - 861 Abstract: Abstract In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some associative ring R and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9439-x Issue No:Vol. 25, No. 5 (2017)

Authors:Jim Andrianopoulos Pages: 863 - 873 Abstract: Abstract This article shows that the axioms of a skew monoidal category are independent and that its unit is unique up to a unique isomorphism together with an analogue of this result for monoidal functors between skew monoidal categories. It is also noted that these results carry over to skew monoidales before some benefits of certain extra structure on the unit maps of a skew monoidal category are discussed. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9468-5 Issue No:Vol. 25, No. 5 (2017)

Authors:Andrew Salch Pages: 875 - 891 Abstract: Abstract Given an adjoint pair of functors F, G, the composite GF naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad' In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free A-module monad on the category of sets, for A a Dedekind domain. We also prove a criterion, reminiscent of Beck’s monadicity theorem, for when there is essentially (in a precise sense) only a single adjunction that gives rise to a given monad. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9473-8 Issue No:Vol. 25, No. 5 (2017)

Authors:Kamran Sharifi Pages: 893 - 905 Abstract: Abstract K-theory for σ-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips (K-Theory 3, 441–478, 1989). We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital σ-C*-algebra A, represents the functor X↦RK0(C(X,A)) from the category of countably compactly generated spaces to the category of abelian groups. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9474-7 Issue No:Vol. 25, No. 5 (2017)

Authors:Driss Bennis; J. R. García Rozas; Luis Oyonarte Pages: 907 - 915 Abstract: Abstract In this paper we are interested in studying the stability question of subcategories of an abelian category \(\mathcal {A}\) constituted of all objects that admit (proper) coproper resolutions (resp. (coproper) proper coresolutions) with terms in a subcategory \(\mathcal {E}\) of \(\mathcal {A}\) . Using a new approach, we give an affirmative answer to the stability question on these categories under the condition that \(\mathcal {E}\) is closed under finite direct sums. This result generalizes Huang’s affirmative answer to the well-known stability question of Gorenstein categories raised by Sather-Wagstaff, Sharif and White. We end the paper with an example showing that the condition imposed on \(\mathcal {E}\) cannot be dropped. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9478-3 Issue No:Vol. 25, No. 5 (2017)

Authors:Edouard Balzin Pages: 917 - 963 Abstract: Abstract The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal’s Γ-spaces. The formalism of topological operads generalises well to different categories yielding such notions as \(\mathbb E_n\) -algebras in chain complexes, while the Γ-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding “weak” sections of a homotopical Grothendieck fibration. We propose a model for such sections, called derived sections, and study the behaviour of homotopical categories of derived sections under the base change functors. The technology developed for the base-change situation is then applied to a specific class of “resolution” base functors, which are inspired by cellular decompositions of classifying spaces. For resolutions, we prove that the inverse image functor on derived sections is homotopically full and faithful. PubDate: 2017-10-01 DOI: 10.1007/s10485-017-9483-1 Issue No:Vol. 25, No. 5 (2017)

Authors:Victoria Lebed; Friedrich Wagemann Pages: 455 - 488 Abstract: Abstract The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel’d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfel’d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel’d (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators. PubDate: 2017-08-01 DOI: 10.1007/s10485-015-9421-z Issue No:Vol. 25, No. 4 (2017)

Authors:Vasily A. Dolgushev; Christopher L. Rogers Pages: 489 - 503 Abstract: Abstract We construct a symmetric monoidal category \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) whose objects are shifted L ∞ -algebras equipped with a complete descending filtration. Morphisms of this category are “enhanced” infinity morphisms between shifted L ∞ -algebras. We prove that any category enriched over \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L ∞ -algebra point of view. We use this construction in a subsequent paper (Dolgushev et al. Adv. Math. 274, 562–605, 2015) to produce a simplicial model of a (∞,1)-category whose objects are homotopy algebras of a fixed type. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9424-4 Issue No:Vol. 25, No. 4 (2017)

Authors:Jun Pei; Chengming Bai; Li Guo Pages: 505 - 538 Abstract: Abstract This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from the dendriform algebra of Loday splitting the associative operation to the successors splitting binary operads. The separately treated bisuccessor and trisuccessor for binary operads are unified for general operads through the notion of configuration. Applications are provided for various n-algebras, the \(A_{\infty }\) and \(L_{\infty }\) algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra contexts and then generalized to binary operads, is defined for all operads. The well-known connection from Rota-Baxter operators to dendriform algebras and its numerous extensions are expanded as the link from (relative) Rota-Baxter operators on operads to splittings of the operads PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9431-5 Issue No:Vol. 25, No. 4 (2017)

Authors:Xiao-Wu Chen Pages: 539 - 568 Abstract: Abstract For an additive category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. We prove that under certain conditions, the equivarianzation of an additive category with a periodic Serre duality still has a periodic Serre duality. A similar result is proved for fractionally Calabi-Yau triangulated categories. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9432-4 Issue No:Vol. 25, No. 4 (2017)

Authors:Francesco Genovese Pages: 625 - 657 Abstract: Abstract We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable. With this characterisation, we prove an existence result of adjoints, under suitable hypotheses on the dg-categories. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9470-y Issue No:Vol. 25, No. 4 (2017)

Authors:Xabier García-Martínez Pages: 659 - 661 Abstract: Abstract We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden (Pré-Publicações DMUC 16–21, 1–41 2016) which characterises groups amongst monoids as the protomodular objects. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9471-x Issue No:Vol. 25, No. 4 (2017)

Authors:M. Menni Pages: 663 - 707 Abstract: Abstract We extend the work of Schanuel, Lawvere, Blass and Gates in Objective Number Theory by proving that, for any \({L(X) \in \mathbb {N}[X]}\) , the rig \({\mathbb {N}[X]/(X = L(X))}\) is the Burnside rig of a prextensive category. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9475-6 Issue No:Vol. 25, No. 4 (2017)

Authors:Rory B. B. Lucyshyn-Wright Abstract: Abstract We define and study a notion of commutant for \(\mathscr {V}\) -enriched \({\mathscr {J}}\) -algebraic theories for a system of arities \({\mathscr {J}}\) , recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for \(\mathscr {V}\) -monads on a symmetric monoidal closed category \(\mathscr {V}\) . This entails a thorough study of commutation and Kronecker products of operations in \({\mathscr {J}}\) -theories. In view of the equivalence between \({\mathscr {J}}\) -theories and \({\mathscr {J}}\) -ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of \({\mathscr {J}}\) -ary commutant and absolute commutant for \({\mathscr {J}}\) -ary monads, and we show that for finitary monads on \(\text {Set}\) the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere. PubDate: 2017-10-09 DOI: 10.1007/s10485-017-9503-1

Authors:Alexander Campbell Abstract: Abstract This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category \(\mathcal {V}\) , between tensored \(\mathcal {V}\) -categories and hommed \(\mathcal {V}\) -actegories is extended to the skew setting and easily proved by recognising both skew \(\mathcal {V}\) -categories and skew \(\mathcal {V}\) -actegories as equivalent to special kinds of skew \(\mathcal {V}\) -proactegory. PubDate: 2017-10-05 DOI: 10.1007/s10485-017-9504-0

Authors:Scott Balchin Abstract: Abstract We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks. PubDate: 2017-09-20 DOI: 10.1007/s10485-017-9502-2

Authors:Hiroyuki Nakaoka Abstract: Abstract In this article, we introduce the notion of concentric twin cotorsion pair on a triangulated category. This notion contains the notions of t-structure, cluster tilting subcategory, co-t-structure and functorally finite rigid subcategory as examples. Moreover, a recollement of triangulated categories can be regarded as a special case of concentric twin cotorsion pair. To any concentric twin cotorsion pair, we associate a pretriangulated subquotient category. This enables us to give a simultaneous generalization of the Iyama–Yoshino reduction and the recollement of cotorsion pairs. This allows us to give a generalized mutation on cotorsion pairs defined by the concentric twin cotorsion pair. PubDate: 2017-08-07 DOI: 10.1007/s10485-017-9501-3