Authors:Marcelo Aguiar; Mariana Haim; Ignacio López Franco Pages: 413 - 458 Abstract: We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which \(n\le 3\) . PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9497-8 Issue No:Vol. 26, No. 3 (2018)

Authors:Oghenetega Ighedo Pages: 459 - 476 Abstract: An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by \({{\mathrm{ZId}}}(\mathcal {R}L)\) the lattice of z-ideals of the ring \(\mathcal {R}L\) of continuous real-valued functions on L. It is a coherent frame, and it is known that \(L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)\) is the object part of a functor \(\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}\) , where \(\mathbf {CRFrm}\) is the category of completely regular frames and frame homomorphisms, and \(\mathbf {CohFrm}\) is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere. PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9498-7 Issue No:Vol. 26, No. 3 (2018)

Authors:B. Banaschewski Pages: 477 - 489 Abstract: This paper establishes various conditions characterizing the homomorphisms \(h: L \rightarrow M\) of 0-dimensional frames which induce an isomorphism between the rings of all integervalued continuous function, or their bounded parts, on L and M, based on the Lindelöf and the compact coreflection of 0-dimensional frames. This provides natural analogues of familiar results concerning the realvalued continuous functions on completely regular frames, albeit by rather different methods of proof from those originally used in that setting. In addition, it will be shown that the present approach also leads to alternative proofs for the latter. PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9499-6 Issue No:Vol. 26, No. 3 (2018)

Authors:Hiroyuki Nakaoka Pages: 491 - 544 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: 2018-06-01 DOI: 10.1007/s10485-017-9501-3 Issue No:Vol. 26, No. 3 (2018)

Authors:Scott Balchin Pages: 545 - 558 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: 2018-06-01 DOI: 10.1007/s10485-017-9502-2 Issue No:Vol. 26, No. 3 (2018)

Authors:Rory B. B. Lucyshyn-Wright Pages: 559 - 596 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: 2018-06-01 DOI: 10.1007/s10485-017-9503-1 Issue No:Vol. 26, No. 3 (2018)

Authors:Alexander Campbell Pages: 597 - 615 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: 2018-06-01 DOI: 10.1007/s10485-017-9504-0 Issue No:Vol. 26, No. 3 (2018)

Authors:Chris Heunen; Vaia Patta Pages: 205 - 237 Abstract: The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9490-2 Issue No:Vol. 26, No. 2 (2018)

Authors:Martin Brandenburg Pages: 287 - 308 Abstract: Let k be a commutative \(\mathbb {Q}\) -algebra. We study families of functors between categories of finitely generated modules which are defined for all commutative k-algebras simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to k-linear representations of symmetric groups. This result is closely related to Macdonald’s classification of polynomial functors. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9494-y Issue No:Vol. 26, No. 2 (2018)

Authors:Rory B. B. Lucyshyn-Wright Pages: 369 - 400 Abstract: Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of \(\mathscr {T}\) -algebra for an algebraic theory \(\mathscr {T}\) in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of left R -affine spaces for a ring or rig R, proving that this theory can be described as a commutant of the theory of pointed right R-modules. Further, we show that for a wide class of rigs R that includes all rings, these theories are commutants of one another in the full finitary theory of R in the category of sets. We define left R -convex spaces for a preordered ring R as left affine spaces over the positive part \(R_+\) of R. We show that for any firmly archimedean preordered algebra R over the dyadic rationals, the theories of left R-convex spaces and pointed right \(R_+\) -modules are commutants of one another within the full finitary theory of \(R_+\) in the category of sets. Applied to the ring of real numbers \(\mathbb {R}\) , this result shows that the connection between convex spaces and pointed \(\mathbb {R}_+\) -modules that is implicit in the integral representation of probability measures is a perfect ‘duality’ of algebraic theories. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9496-9 Issue No:Vol. 26, No. 2 (2018)

Authors:Dragan Mašulović Pages: 401 - 412 Abstract: In this paper we give a new proof of the Nešetřil–Rödl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate combinatorial strategies, this proof is spelled out in the language of category theory and the main result follows by applying several simple categorical constructions. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general principles. PubDate: 2018-04-01 DOI: 10.1007/s10485-017-9500-4 Issue No:Vol. 26, No. 2 (2018)

Authors:Jiří Adámek; Lurdes Sousa Abstract: For a functor F whose codomain is a cocomplete, cowellpowered category \(\mathcal {K}\) with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from \(\mathcal {K}(X, F-)\) to \(\mathcal {K}(s, F-)\) form a set. Moreover, the codensity monad has an explicit description using the above natural transformations. Concrete examples are presented, e.g., the codensity monad of the power-set functor \(\mathcal {P}\) assigns to every set X the set of all nonexpanding endofunctions of \(\mathcal {P}X\) . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from \(X^F\) to \(2^F\) form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of \({\mathsf {Set}}\) we prove that F has a density comonad iff F is accessible. PubDate: 2018-05-29 DOI: 10.1007/s10485-018-9530-6

Authors:Olivier De Deken; Wendy Lowen Abstract: We develop the basic theory of curved \(A_{\infty }\) -categories ( \(cA_{\infty }\) -categories) in a filtered setting, encompassing the frameworks of Fukaya categories (Fukaya et al. in Part I, AMS/IP studies in advanced mathematics, vol 46, American Mathematical Society, Providence, RI, 2009) and weakly curved \(A_{\infty }\) -categories in the sense of Positselski (Weakly curved \(A_\infty \) algebras over a topological local ring, 2012. arxiv:1202.2697v3). Between two \(cA_{\infty }\) -categories \(\mathfrak {a}\) and \(\mathfrak {b}\) , we introduce a \(cA_{\infty }\) -category \(\mathsf {qFun}(\mathfrak {a}, \mathfrak {b})\) of so-called \(qA_{\infty }\) -functors in which the uncurved objects are precisely the \(cA_{\infty }\) -functors from \(\mathfrak {a}\) to \(\mathfrak {b}\) . The more general \(qA_{\infty }\) -functors allow us to consider representable modules, a feature which is lost if one restricts attention to \(cA_{\infty }\) -functors. We formulate a version of the Yoneda Lemma which shows every \(cA_{\infty }\) -category to be homotopy equivalent to a curved dg category, in analogy with the uncurved situation. We also present a curved version of the bar-cobar adjunction. PubDate: 2018-05-28 DOI: 10.1007/s10485-018-9526-2

Authors:P.-A. Jacqmin; S. Mantovani; G. Metere; E. M. Vitale Abstract: We characterize fibrations and \(*\) -fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for \(*\) -fibrations from the internal version of the Gabriel–Zisman exact sequence. We also analyse fibrations and \(*\) -fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma. PubDate: 2018-05-26 DOI: 10.1007/s10485-018-9529-z

Authors:George Janelidze; Ross Street Abstract: In the original publication of the article, Eq. 3.24 was published incorrectly. The corrected equation is given in this correction article. The original article has been corrected. PubDate: 2018-05-12 DOI: 10.1007/s10485-018-9528-0

Authors:Wei Li; Dexue Zhang Abstract: The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective \(T_0\) approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the d-Scott topology and the generalized Scott topology; and that every injective \(T_0\) approach space is a cocomplete and continuous metric space equipped with its Scott distance. PubDate: 2018-05-09 DOI: 10.1007/s10485-018-9527-1

Authors:George Janelidze Abstract: We describe an alternative way of constructing some of the monads, recently introduced by E. Colebunders, R. Lowen, and W. Rosiers for the purposes of categorical topology. PubDate: 2018-05-07 DOI: 10.1007/s10485-018-9525-3

Authors:George Janelidze; Ross Street Abstract: We make various observations on infinitary addition in the context of the series monoids introduced in our previous paper on real sets. In particular, we explore additional conditions on such monoids suggested by Tarski’s Arithmetic of Cardinal Algebras, and present a monad-theoretic construction that generalizes our construction of paradoxical real numbers. PubDate: 2018-04-18 DOI: 10.1007/s10485-018-9524-4

Authors:John Frith; Anneliese Schauerte Abstract: The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance, \(\sigma \) -frames, \(\kappa \) -frames and some partial frames also turn out to be frames. The situation for congruences of a meet-semilattice is notably different. In this paper we analyze the meet-semilattice congruence lattices of arbitrary frames and compare them with the corresponding lattices of frame congruences. In the course of this, we provide a structure theorem as well as many examples and counter-examples. PubDate: 2018-04-13 DOI: 10.1007/s10485-018-9521-7

Authors:Dikran Dikranjan; Hans-Peter A. Künzi 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: 2018-04-07 DOI: 10.1007/s10485-018-9523-5