Abstract: This paper concerns the problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudo-distributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique [as known to be the case in the (co)KZ over KZ setting]. Moreover, we give a simple description of these distributive laws using Bunge and Funk’s notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2-category of KZ doctrines on a 2-category is biequivalent to a poset. We will also discuss here the problem of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that a bijection between oplax and lax structures is exhibited on the lifted “Yoneda structure” similar to Kelly’s doctrinal adjunction. We also briefly discuss how this bijection may be viewed as a coherence result for oplax functors out of the bicategories of spans and polynomials, but leave the details for a future paper. PubDate: 2019-12-01

Abstract: We give conditions on an inclusion \({\mathbb {C}}\hookrightarrow {\mathbb {D}}\) where \({\mathbb {C}}\) is a Mal’tsev (resp. protomodular) subcategory in order to produce on \({\mathbb {D}}\) a partial \(\Sigma \) -Mal’tsev (resp. \(\Sigma \) -protomodular) structure. PubDate: 2019-12-01

Abstract: We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a \({{\,\mathrm{Hom}\,}}\) -finite hereditary abelian category with enough projectives, we prove that the Auslander–Reiten–Smalø–Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers. PubDate: 2019-10-30

Abstract: We introduce compactly finite MV-algebras and continuous MV-algebras. We also investigate pro-compactly finite MV-algebras, which are the MV-algebras that are inverse limits of systems of compactly finite MV-algebras. We obtain that continuous MV-algebras as well as pro-compactly finite MV-algebras coincide with compact Hausdorff MV-algebras. In addition, further categorical properties of compact Hausdorff MV-algebras such as co-completeness, injective objects, (co)-Hopfian objects are considered and studied. PubDate: 2019-10-17

Abstract: The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes’ characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Böhm, of the power-set monad over the ultrafilter monad. PubDate: 2019-10-16

Abstract: We define and discuss the notions of additivity and idempotency for neighbourhood and interior operators. We then propose an order-theoretic description of the notion of convergence that was introduced by D. Holgate and J. Šlapal with the help of these two properties. This will provide a rather convenient setting in which compactness and completeness can be studied via neighbourhood operators. We prove, among other things, a Frolík-type theorem with the introduction of reflecting morphisms. PubDate: 2019-10-11

Abstract: When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of \(\hbox {C}^*\) -categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in \(\hbox {C}^*\) -categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of \(\hbox {W}^*\) -categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any \(\hbox {W}^*\) -category of normal representations of a \(\hbox {W}^*\) -algebra. Finding a universal property for the more general case of direct integrals remains an open problem. PubDate: 2019-10-08

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: 2019-10-01

Abstract: We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category. PubDate: 2019-10-01

Abstract: We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and \(L_\infty \) -algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. whose anchor map is zero. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids. PubDate: 2019-10-01

Abstract: The present note has three aims. First, to complement the theory of cofibrant generation of algebraic weak factorisation systems (awfss) to cover some important examples that are not locally presentable categories. Secondly, to prove that cofibrantly kz-generated awfss (a notion we define) are always lax orthogonal. Thirdly, to show that the two known methods of building lax orthogonal awfss, namely cofibrantly kz-generation and the method of “simple adjunctions”, construct different awfss. We study in some detail the example of cofibrant kz-generation that yields representable multicategories, and a counterexample to cofibrant generation provided by continuous lattices. PubDate: 2019-10-01

Abstract: We study the category of Reedy diagrams in a \(\mathscr {V}\) -model category. Explicitly, we show that if K is a small category, \(\mathscr {V}\) is a closed symmetric monoidal category and \(\mathscr {C}\) is a closed \(\mathscr {V}\) -module, then the diagram category \(\mathscr {V}^K\) is a closed symmetric monoidal category and the diagram category \(\mathscr {C}^K\) is a closed \(\mathscr {V}^K\) -module. We then prove that if further K is a Reedy category, \(\mathscr {V}\) is a monoidal model category and \(\mathscr {C}\) is a \(\mathscr {V}\) -model category, then with the Reedy model category structures, \(\mathscr {V}^K\) is a monoidal model category and \(\mathscr {C}^K\) is a \(\mathscr {V}^K\) -model category provided that either the unit 1 of \(\mathscr {V}\) is cofibrant or \(\mathscr {V}\) is cofibrantly generated. PubDate: 2019-10-01

Abstract: We introduce integral structure types as a categorical analogue of virtual combinatorial species. Integral structure types then categorify power series with possibly negative coefficients in the same way that combinatorial species categorify power series with non-negative rational coefficients. The notion of an operator on combinatorial species naturally extends to integral structure types, and in light of their ‘negativity’ we define the notion of the commutator of two operators on integral structure types. We then extend integral structure types to the setting of stuff types as introduced by Baez and Dolan, and then conclude by using integral structure types to give a combinatorial description for Chern classes of projective hypersurfaces. PubDate: 2019-09-12

Abstract: For any site of definition \(\mathcal {C}\) of a Grothendieck topos \(\mathcal {E}\) , we define a notion of a \(\mathcal {C}\) -ary Lawvere theory \(\tau : \mathscr {C} \rightarrow \mathscr {T}\) whose category of models is a stack over \(\mathcal {E}\) . Our definitions coincide with Lawvere’s finitary theories when \(\mathcal {C}=\aleph _0\) and \(\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}\) . We construct a fibered category \({{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}}\) of models as a stack over \(\mathcal {E}\) and prove that it is \(\mathcal {E}\) -complete and \(\mathcal {E}\) -cocomplete. We show that there is a free-forget adjunction \(F \dashv U: {{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}} \leftrightarrows \mathscr {E}\) . If \(\tau \) is a commutative theory in a certain sense, then we obtain a “locally monoidal closed” structure on the category of models, which enhances the free-forget adjunction to an adjunction of symmetric monoidal \(\mathcal {E}\) -categories. Our results give a general recipe for constructing a monoidal \(\mathcal {E}\) -cosmos in which one can do enriched \(\mathcal {E}\) -category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration. PubDate: 2019-09-09

Abstract: Let \({\mathcal {C}}\) be a finite tensor category, and let \({\mathcal {M}}\) be an exact left \({\mathcal {C}}\) -module category. The action of \({\mathcal {C}}\) on \({\mathcal {M}}\) induces a functor \(\rho : {\mathcal {C}} \rightarrow \mathrm {Rex}({\mathcal {M}})\) , where \(\mathrm {Rex}({\mathcal {M}})\) is the category of k-linear right exact endofunctors on \({\mathcal {M}}\) . Our key observation is that \(\rho \) has a right adjoint \(\rho ^{\mathrm {ra}}\) given by the end $$\begin{aligned} \rho ^{\mathrm {ra}}(F) = \int _{M \in {\mathcal {M}}} \underline{\mathrm {Hom}}(M, F(M)) \quad (F \in \mathrm {Rex}({\mathcal {M}})). \end{aligned}$$ As an application, we establish the following results: (1) We give a description of the composition of the induction functor \({\mathcal {C}}_{{\mathcal {M}}}^* \rightarrow {\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*)\) and Schauenburg’s equivalence \({\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*) \approx {\mathcal {Z}}({\mathcal {C}})\) . (2) We introduce the space \(\mathrm {CF}({\mathcal {M}})\) of ‘class functions’ of \({\mathcal {M}}\) and initiate the character theory for pivotal module categories. (3) We introduce a filtration for \(\mathrm {CF}({\mathcal {M}})\) and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that \(\mathrm {Ext}_{{\mathcal {C}}}^{\bullet }(1, \rho ^{\mathrm {ra}}(\mathrm {id}_{{\mathcal {M}}}))\) is isomorphic to the Hochschild cohomology of \({\mathcal {M}}\) . As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category. PubDate: 2019-09-06

Abstract: We compute the Ext group of the (filtered) Ogus category over a number field K. In particular we prove that the filtered Ogus realisation of mixed motives is not fully faithful. PubDate: 2019-08-28

Abstract: For a subfit frame L, let \({\mathcal {S}}_{\mathfrak {c}}(L)\) denote the complete Boolean algebra whose elements are the sublocales of L that are joins of closed sublocales. Identifying every element of L with the open sublocale it determines allows us to view L as a subframe of \({\mathcal {S}}_{\mathfrak {c}}(L)\) . With this backdrop, we say L is maximal Lindelöf if it is Lindelöf and whenever \(L\subseteq M\) , for some Lindelöf subframe M of \({\mathcal {S}}_{\mathfrak {c}}(L)\) , then \(L=M\) . Recall that a topological space \((X,\tau )\) is maximal Lindelöf if it is Lindelöf, and there is no strictly finer topology \(\rho \) on X such that \((X,\rho )\) is a Lindelöf space. We show that a space is maximal Lindelöf if and only if the frame of its open subsets is maximal Lindelöf. We then characterize maximal Lindelöf frames internally. Among regular frames, we show that the maximal Lindelöf frames are precisely the Lindelöf ones in which every \(F_\sigma \) -sublocale (meaning a join of countably many closed sublocales) is closed. PubDate: 2019-08-20

Abstract: We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spaces with closed relations, and the corresponding subcategories with continuous and interior relations. Various equivalences of these categories are given extending known equivalences of the category of compact Hausdorff spaces and continuous functions with compact regular frames, de Vries algebras, and also with a category of Gleason spaces that we introduce. Study of categories of compact Hausdorff spaces with relations is of interest as a general setting to consider Gleason spaces, for connections to modal logic, as well as for the intrinsic interest in these categories. PubDate: 2019-08-13

Abstract: This is the first part of a series of three strongly related papers in which three equivalent structures are studied: internal categories in categories of monoids, defined in terms of pullbacks relative to a chosen class of spans crossed modules of monoids relative to this class of spans simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category. PubDate: 2019-08-10