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: It follows from standard results that if \(\mathcal {A}\) and \(\mathcal {C}\) are locally \(\lambda \) -presentable categories and \(F : \mathcal {A}\rightarrow \mathcal {C}\) is a \(\lambda \) -accessible functor, then the comma category \(\mathsf {Id}_\mathcal {C}{\downarrow }{}F\) is locally \(\lambda \) -presentable. We show that, under the same hypotheses, \(F{\downarrow }{}\mathsf {Id}_\mathcal {C}\) is also locally \(\lambda \) -presentable. PubDate: 2019-08-10

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

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: 2019-08-01

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: 2019-08-01

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: 2019-08-01

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: 2019-08-01

Abstract: Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic settings, there is only one notion of differentiation. This paper also presents a number of separating examples for coalgebra modalities including examples which are and are not monoidal, as well as examples which do and do not support differential structure. Of particular interest is the observation that—somewhat counter-intuitively—differential algebras never induce a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota–Baxter algebras—which are usually associated with integration—provide an example of a differential category which has a non-monoidal coalgebra modality. PubDate: 2019-07-04