Authors:Scott Balchin; Richard Garner Pages: 1 - 21 Abstract: We give an account of Bousfield localisation and colocalisation for one-dimensional model categories—ones enriched over the model category of 0-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith’s theorem. PubDate: 2019-02-01 DOI: 10.1007/s10485-018-9537-z Issue No:Vol. 27, No. 1 (2019)

Authors:Clemens Berger; Kruna Ratkovic Pages: 23 - 54 Abstract: We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre sequences up to homotopy and has a weakly invertible strength, then the category of T-algebras is Quillen equivalent to the category of T(I)-modules where I is the monoidal unit. This recovers Schwede’s theorem on connective stable homotopy over a pointed Lawvere theory as special case. PubDate: 2019-02-01 DOI: 10.1007/s10485-018-9539-x Issue No:Vol. 27, No. 1 (2019)

Authors:Fernando Lucatelli Nunes Pages: 55 - 63 Abstract: Given a pseudomonad \(\mathcal {T}\) , we prove that a lax \(\mathcal {T}\) -morphism between pseudoalgebras is a \(\mathcal {T}\) -pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible \(\mathcal {T}\) -transformation. This result encompasses several results on non-canonical isomorphisms, including Lack’s result on normal monoidal functors between braided monoidal categories, since it is applicable in any 2-category of pseudoalgebras, such as the 2-categories of monoidal categories, cocomplete categories, bicategories, pseudofunctors and so on. PubDate: 2019-02-01 DOI: 10.1007/s10485-018-9541-3 Issue No:Vol. 27, No. 1 (2019)

Authors:Jaime Castro Pérez; Mauricio Medina Bárcenas; José Ríos Montes; Angel Zaldívar Corichi Pages: 65 - 84 Abstract: We are concerned with the boolean or more generally with the complemented properties of idioms (complete upper-continuous modular lattices). Simmons (Cantor–Bendixson, socle, and atomicity. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/002-Atom.pdf, 2014) introduces a device which captures in some informal speaking how far the idiom is from being complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative and we give conditions on a nucleus j such that [j, tp] is a complete boolean algebra. We also explore some properties of nuclei j such that \(A_{j}\) is a complemented idiom. PubDate: 2019-02-01 DOI: 10.1007/s10485-018-9543-1 Issue No:Vol. 27, No. 1 (2019)

Authors:Ivan Kobyzev; Ilya Shapiro Pages: 85 - 109 Abstract: We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter–Drinfeld contramodules for Hopf algebras. PubDate: 2019-02-01 DOI: 10.1007/s10485-018-9544-0 Issue No:Vol. 27, No. 1 (2019)

Authors:Ren Wang Abstract: Let \({\mathscr {C}}\) be a finite projective EI category and k be a field. The singularity category of the category algebra \(k{\mathscr {C}}\) is a tensor triangulated category. We compute its spectrum in the sense of Balmer. PubDate: 2019-02-13 DOI: 10.1007/s10485-019-09562-0

Authors:David White; Donald Yau Abstract: This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be \(\Sigma \) -cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter–Shipley about a zig-zag of Quillen equivalences between commutative \(H\mathbb {Q}\) -algebra spectra and commutative differential graded \(\mathbb {Q}\) -algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization. PubDate: 2019-02-12 DOI: 10.1007/s10485-019-09560-2

Authors:Claudia Chaio; Isabel Pratti; María José Souto Salorio Abstract: We consider \(\Lambda \) an artin algebra and \(n \ge 2\) . We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander–Reiten component of \({{\mathbf {C_n}}(\mathrm{proj}\, \Lambda )}\) with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in \({\mathbf {C_n}}(\mathrm{proj}\, \Lambda )\) belong to such a category. For a finite dimensional hereditary algebra H over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which \({\mathbf {C_n}}(\mathrm{proj} \,H)\) is of finite type. PubDate: 2019-02-01 DOI: 10.1007/s10485-019-09557-x

Authors:Anthony Bordg Abstract: In this article, the author endows the functor category \([\mathbf {B}(\mathbb {Z}_2),\mathbf {Gpd}]\) with the structure of a type-theoretic fibration category with a univalent universe, using the so-called injective model structure. This gives a new model of Martin-Löf type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in another work) in the same underlying category and with the same universe, which turns out to be provably not univalent with respect to projective fibrations, provide an example of two Quillen equivalent model categories that host different models of type theory. Thus, we provide a counterexample to the model invariance problem formulated by Michael Shulman. PubDate: 2019-02-01 DOI: 10.1007/s10485-019-09558-w

Abstract: Motivated by certain types of ideals in pointfree functions rings, we define what we call P-sublocales in completely regular frames. They are the closed sublocales that are interior to the zero-sublocales containing them. We call an element of a frame L that induces a P-sublocale a P-element, and denote by \({{\,\mathrm{Pel}\,}}(L)\) the set of all such elements. We show that if L is basically disconnected, then \({{\,\mathrm{Pel}\,}}(L)\) is a frame and, in fact, a dense sublocale of L. Ordered by inclusion, the set \(\mathcal {S}_\mathfrak {p}(L)\) of P-sublocales of L is a complete lattice, and, for basically disconnected L, \(\mathcal {S}_\mathfrak {p}(L)\) is a frame if and only if \({{\,\mathrm{Pel}\,}}(L)\) is the smallest dense sublocale of L. Furthermore, for basically disconnected L, \(\mathcal {S}_\mathfrak {p}(L)\) is a sublocale of the frame \(\mathcal {S}_\mathfrak {c}(L)\) consisting of joins of closed sublocales of L if and only if L is Boolean. For extremally disconnected L, iterating through the ordinals (taking intersections at limit ordinals) yields an ordinal sequence $$\begin{aligned} L\;\supseteq \;{{\,\mathrm{Pel}\,}}(L)\supseteq \;{{\,\mathrm{Pel}\,}}^2(L)\;\supseteq \;\cdots \; \supseteq \;{{\,\mathrm{Pel}\,}}^\alpha (L)\supseteq \;{{\,\mathrm{Pel}\,}}^{\alpha +1}(L)\;\supseteq \cdots \end{aligned}$$ that stabilizes at an extremally disconnected P-frame, that we denote by \({{\,\mathrm{Pel}\,}}^\infty (L)\) . It turns out that \({{\,\mathrm{Pel}\,}}^\infty (L)\) is the reflection to L from extremally disconnected P-frames when morphisms are suitably restricted. PubDate: 2019-01-23 DOI: 10.1007/s10485-019-09559-9

Authors:Florin Panaite; Paul T. Schrader; Mihai D. Staic Abstract: We introduce a new type of categorical object called a hom–tensor category and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of hom-braided category and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the Hom–Yang–Baxter equation fits into this framework and how the category of Yetter–Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom–tensor category (respectively a hom-braided category). PubDate: 2019-01-16 DOI: 10.1007/s10485-019-09556-y

Authors:Tomáš Jakl; Achim Jung; Aleš Pultr Abstract: It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased. PubDate: 2019-01-05 DOI: 10.1007/s10485-018-09553-7

Authors:Sondre Kvamme; René Marczinzik Abstract: We review the theory of Co-Gorenstein algebras, which was introduced in Beligiannis (Commun Algebra 28(10):4547–4596, 2000). We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture. PubDate: 2019-01-04 DOI: 10.1007/s10485-018-09554-6

Authors:Daniel Lin Abstract: Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is equivalent to some partial map category of sheaves. We then use this to show that the Yoneda embedding exhibits the category of join restriction presheaves as the free cocompletion of any small join restriction category. PubDate: 2019-01-04 DOI: 10.1007/s10485-018-09555-5

Authors:Stefan Schröer Pages: 1113 - 1122 Abstract: We show that each sheaf of modules admits a flasque hull, such that homomorphisms into flasque sheaves factor over the flasque hull. On the other hand, we give examples of modules over non-noetherian rings that do not inject into flasque modules. This reveals the impossibility to extend the proof of Serre’s vanishing result for affine schemes with flasque quasicoherent resolutions to the non-noetherian setting. However, we outline how hypercoverings can be used for a reduction to the noetherian case. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9520-8 Issue No:Vol. 26, No. 6 (2018)

Authors:Dikran Dikranjan; Hans-Peter A. Künzi Pages: 1159 - 1184 Abstract: We study cowellpoweredness in the category \(\mathbf{QUnif}\) of quasi-uniform spaces and uniformly continuous maps. A full subcategory \(\mathcal{A}\) of \(\mathbf{QUnif}\) is cowellpowered when the cardinality of the codomains of any class of epimorphisms in \(\mathcal{A}\) , with a fixed common domain, is bounded. We use closure operators in the sense of Dikranjan–Giuli–Tholen which provide a convenient tool for describing the subcategories \(\mathcal{A}\) of \(\mathbf{QUnif}\) and their epimorphisms. Some of the results are obtained by using the knowledge of closure operators, epimorphisms and cowellpoweredness of subcategories of the category \(\mathbf{Top}\) of topological spaces and continuous maps. The transfer is realized by lifting these subcategories along the forgetful functor \(T{:}\mathbf{QUnif}\rightarrow \mathbf{Top}\) and studying when epimorphisms and cowellpoweredness are preserved by the lifting. In other cases closure operators of \(\mathbf{QUnif}\) are used to provide specific results for \(\mathbf{QUnif}\) that have no counterpart in \(\mathbf{Top}\) . This leads to a wealth of cowellpowered categories and a wealth of non-cowellpowered categories of quasi-uniform spaces, in contrast with the current situation in the case of the smaller category \(\mathbf{Unif}\) of uniform spaces, where no example of a non-cowellpowered subcategory is known so far. Finally, we present our main example: a non-cowellpowered full subcategory of \(\mathbf{QUnif}\) which is the intersection of two “symmetric” cowellpowered full subcategories of \(\mathbf{QUnif}\) . PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9523-5 Issue No:Vol. 26, No. 6 (2018)

Authors:Nicholas J. Meadows Pages: 1265 - 1281 Abstract: We develop a model structure on bimplicial presheaves on a small site \({\mathscr {C}}\) , for which the weak equivalences are local (or stalkwise) weak equivalences in the complete Segal model structure. We call this the local Complete Segal model structure. This model structure can be realized as a left Bousfield localization of the Jardine (injective) model structure on the simplicial presheaves on a site \({\mathscr {C}} / {\varDelta }^{op}\) . Furthermore, it is shown that this model structure is Quillen equivalent to the model structure of the author’s paper (Meadows in TAC 31(24):690–711, 2016). This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney (Categories in algebra, geometry and mathematical physics, contemporary mathematics, vol. 431. American Mathematical Society, Providence, pp 277–326, 2007). As an application, we compare the notion of descent in the local Joyal model structure to the notion of descent in the injective model structure. Interestingly, this is a consequence of the Quillen equivalence between the local Joyal and local Complete Segal model structures. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9535-1 Issue No:Vol. 26, No. 6 (2018)

Authors:David Pescod Pages: 1283 - 1304 Abstract: A frieze in the modern sense is a map from the set of objects of a triangulated category \(\mathsf {C}\) to some ring. A frieze X is characterised by the property that if \(\tau x\rightarrow y\rightarrow x\) is an Auslander–Reiten triangle in \(\mathsf {C}\) , then \(X(\tau x)X(x)-X(y)=1\) . The canonical example of a frieze is the (original) Caldero–Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101–124, 2015; Bull Sci Math 140:112–131, 2016), the notion of generalised friezes is introduced. A generalised frieze \(X'\) has the more general property that \(X'(\tau x)X'(x)-X'(y)\in \{0,1\}\) . The canonical example of a generalised frieze is the modified Caldero–Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps \(\alpha \) and \(\beta \) in the modified Caldero–Chapoton map, and in the case when \(\mathsf {C}\) is 2-Calabi–Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero–Chapoton map, which significantly simplifies its computation in practice. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9538-y Issue No:Vol. 26, No. 6 (2018)

Authors:Jorge Picado; Aleš Pultr Pages: 1305 - 1324 Abstract: We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one). PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9542-2 Issue No:Vol. 26, No. 6 (2018)

Abstract: We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and \(\varDelta \) -complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9552-0