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We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads.

For a reductive group G, we study the Drinfeld–Gaitsgory functor of the category of conjugation-equivariant D-modules on G. We show that this functor is an equivalence of categories, and that it has a filtration with layers expressed via parabolic induction of parabolic restriction. We use this to provide a conceptual definition of the Deligne–Lusztig involution on the set of isomorphism classes of irreducible character D-modules, which was defined previously in Lusztig (Adv Math 57:266–315, 1985, §15).

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations., Comment: 28 pages, 6 figures. v5 processed for publication in Epiga

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a 'limiting case' of the category Hilb packaging quantum mechanics. Following from this assumption, quantum-classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics., Major revision of the previous draft; 8 pages

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO+(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.

This paper studies questions of coherence and strictification related to self-similarity - the identity $S\cong S\otimes S$ in a (semi-)monoidal category. Based on Saavedra's theory of units, we first demonstrate that strict self-similarity cannot simultaneously occur with strict associativity -- i.e. no monoid may have a strictly associative (semi-)monoidal tensor, although many monoids have a semi-monoidal tensor associative up to isomorphism. We then give a simple coherence result for the arrows exhibiting self-similarity and use this to describe a `strictification procedure' that gives a semi-monoidal equivalence of categories relating strict and non-strict self-similarity, and hence monoid analogues of many categorical properties. Using this, we characterise a large class of diagrams (built from the canonical isomorphisms for the relevant tensors, together with the isomorphisms exhibiting the self-similarity) that are guaranteed to commute., Significant revisions from previous version: proofs simplified and based on Saavedra units & idempotent splitting, monoidal equivalences made explicit, expository sections significantly revised and shortened, notation and terminology revised and clarified, a clearer criterion for coherence given

The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group.

For a smooth manifold X we show an equivalence of categories between the category of O X ♯ [ n ] -extensions of T X ♯ and the category of higher-dimensional exact Courant algebroids on X . In addition, for any object in the category of R [ n ] dg-principal bundles over X ♯ we construct its Atiyah algebroid which gives rise to an example of O X ♯ [ n ] -extensions of T X ♯ .

Let $G$ be a $\mathbb{Q}_p$-split reductive group with connected centre and Borel subgroup $B=TN$. We construct a right exact functor $D^\vee_\Delta$ from the category of smooth modulo $p^n$ representations of $B$ to the category of projective limits of finitely generated \'etale $(\varphi,\Gamma)$-modules over a multivariable (indexed by the set of simple roots) commutative Laurent-series ring. These correspond to representations of a direct power of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ via an equivalence of categories. Parabolic induction from a subgroup $P=L_PN_P$ corresponds to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component $L_P$. $D^\vee_\Delta$ is exact and yields finitely generated objects on the category $SP_A$ of finite length representations with subquotients of principal series as Jordan-H\"older factors. Lifting the functor $D^\vee_\Delta$ to all (noncommuting) variables indexed by the positive roots allows us to construct a $G$-equivariant sheaf $\mathfrak{Y}_{\pi,\Delta}$ on $G/B$ and a $G$-equivariant continuous map from the Pontryagin dual $\pi^\vee$ of a smooth representation $\pi$ of $G$ to the global sections $\mathfrak{Y}_{\pi,\Delta}(G/B)$. We deduce that $D^\vee_\Delta$ is fully faithful on the full subcategory of $SP_A$ with Jordan-H\"older factors isomorphic to irreducible principal series., Comment: 55 pages, revised, to appear in Selecta Mathematica

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided., Comment: To appear in Communications in Algebra

To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories., Comment: 16 pages, LaTex, this is a companion to arXiv:1506.05415

Given a compact K\"ahler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, \theta)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that any stable basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian metric that satisfies the Yang--Mills--Higgs equation., Comment: Final version; to appear in the Communications in Mathematical Physics

Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s 2 B(G) of full subcategories R(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (⌧,W ) of a compact open subgroup J < G. If the functor V 7! HomJ(W,V ) is an equivalence of categories from R(G) ! H(G, ⌧) mod we call (J, ⌧) a type. Given a Levi subgroup L < G and a type (JL, ⌧L) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, ⌧L) by constructing an object known as a cover. In particular, a cover implements induction of H(L, ⌧L)-modules in a manner compatible with parabolic induction of L-representations. In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In particular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.

Given a tagged triangulation of a once-punctured polygon $P^*$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-order $\Lambda$. Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathcal{C}$ of type $D_n$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulations of $P^*$. Moreover, it extends naturally the triangulated categorification by $\mathcal{C}$ of the cluster algebra of type $D_n$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a path algebra of type $D_n$., Comment: 50 pages. Several improvements after refereeing. arXiv admin note: text overlap with arXiv:1307.0676

Let A A be a dg category, F:A→A F : A → A be a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/F A / F be the associated dg orbit category. For every A 1 A 1 -homotopy invariant E ( e.g. homotopy K -theory, K -theory with coefficients, etale K -theory, and periodic cyclic homology), we construct a distinguished triangle expressing E(A/F) E ( A / F ) as the cone of the endomorphism E(F)−Id E ( F ) − Id of E(A) E ( A ) . In the particular case where F is the identity dg functor, this triangle splits and gives rise to the fundamental theorem. As a first application, we compute the A 1 A 1 -homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the A 1 A 1 -homotopy invariants of the dg orbit categories associated with Fourier–Mukai autoequivalences.

Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories.

We construct categorifications of tensor products of arbitrary finite-dimensional irreducible representations of sl k with subquotient categories of the BGG category O , generalizing previous work of Sussan and Mazorchuk–Stroppel. Using Lie theoretical methods, we prove in detail that they are tensor product categorifications in the sense of the recent definition of Losev and Webster. As an application we deduce an equivalence of categories between certain versions of category O and Webster's tensor product categories. Finally we indicate how the categorifications of tensor products of the natural representation of gl ( 1 | 1 ) fit into this framework.

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors., 21 pages; v2: 23 pages, introduced "rigid types"

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence., Comment: v1: 66 pages; v2: 67 pages, minor corrections; v3: 65 pages, final version, to appear in JEMS

We give an equivalence of categories between certain subcategories of modules of pro-$p$-Iwahori Hecke algebras and modulo $p$ representations., Comment: 21 pages

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.

The Dieudonn\'e crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p>2., Comment: 31 pages, sections 8-10 rewritten, title changed

Let [math] be an inner form of a general linear group over a nonarchimedean locally compact field of residue characteristic [math] , let [math] be an algebraically closed field of characteristic different from [math] and let [math] be the category of smooth representations of [math] over [math] . In this paper, we prove that a block (indecomposable summand) of [math] is equivalent to a level- [math] block (a block in which every simple object has nonzero invariant vectors for the pro- [math] -radical of a maximal compact open subgroup) of [math] , where [math] is a direct product of groups of the same type of [math] .

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known., Comment: 43 pages, many figures, best viewed in color. Accepted for publication in the Proceedings of the LMS

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them "saturated" or "univalent" categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack., Comment: 27 pages, ancillary files contain formalized proofs in the proof assistant Coq; v2: version for publication in Math. Struct. in Comp. Sci., incorporating suggestions by referees and Voevodsky

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups., Comment: 73 pages. We have deeply revised the paper to make it more accessible, it contains now explicit examples and computations. We have removed the part on localization, it was correct but we didn't want to make the paper longer and we thought this part was the less interesting one. Nevertheless it will remain here in the arXiv, in version 1. If you need it in your research, please let us know

Li-Bland's correspondence between linear Courant algebroids and Lie $2$-algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up to homotopy (Gracia-Saz and Mehta). Several classes of examples are discussed, yielding new examples of split Lie 2-algebroids. We prove that the bicrossproduct of a matched pair of $2$-representations is a split Lie $2$-algebroid and we explain this result geometrically, as a consequence of the equivalence of VB-Courant algebroids and Lie $2$-algebroids. This explains in particular how the two notions of double" of a matched pair of representations are geometrically related. In the same manner, we explain the geometric link between the two notions of double of a Lie bialgebroid., This is the improved and completed second part of the work arXiv:1504.00880 (N-manifolds of degree 2 and metric double vector bundles)

We show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although this result could be proved as a corollary of our previous article on realizations of algebra objects and discrete subfactors, we prove it here directly via categorical methods without passing through subfactor theory., 21 pages, many figures

We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ can be computed via the generalization of Herr's complex to multivariable $(\varphi,\Gamma)$-modules. Using Tate duality and a pairing for multivariable $(\varphi,\Gamma)$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ are overconvergent and, moreover, passing to overconvergent multivariable $(\varphi,\Gamma)$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups., Comment: 58 pages, final version, to appear in Journal of the Institute of Mathematics of Jussieu

We give a characterization of extremal irreducible discrete subfactors $(N\subseteq M, E)$ where $N$ is type ${\rm II}_1$ in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal $N-N$ bilinear ucp maps which preserve the state $\tau \circ E$, and the morphisms for W*-algebra objects are categorical ucp morphisms. As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions $(N\subseteq M, E)$, in particular, examples where $M$ is type ${\rm III}$ coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects., Comment: Fixed minor errors and added a section on standard invariants for extremal irreducible discrete subfactors. Comments welcome!

It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective \'etale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category., Comment: 41 pages; to appear in J. Math. Soc. Japan

In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions of the American Mathematical Society, 88) C.C. Chang proved that any totally ordered $MV$-algebra $A$ was isomorphic to the segment $A \cong \Gamma(A^*, u)$ of a totally ordered $l$-group with strong unit $A^*$. This was done by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also show that any such group $G$ can be recovered from its segment since $G \cong \Gamma(G, u)^*$, establishing an equivalence of categories. In "Interpretation of AF $C^*$-algebras in Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He takes the representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A \overset {} {\hookrightarrow} \prod_i G_i$ where $G_i = A_i^*$. Then he let $A^*$ be the $l$-subgroup generated by $A$ inside $\prod_i G_i$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang's result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product $l$-group $\prod_i G_i$, avoiding entirely the notion of good sequence., Comment: 6 pages

Let $${\mathcal{V}}$$ be a complete discrete valued ring of mixed characteristic (0, p), K its field of fractions, k its residue field which is supposed to be perfect. Let X be a separated k-scheme of finite type and Y be a smooth open of X. We check that the equivalence of categories sp (Y, X),+ (from the category of overconvergent isocrystals on (Y, X)/K to that of overcoherent isocrystals on (Y, X)/K) commutes with tensor products. Next, in Berthelot’s theory of arithmetic $${\mathcal{D}}$$ -modules, we prove the stability under tensor products of the devissability in overconvergent isocrystals. With Frobenius structures, we get the stability under tensor products of the overholonomicity.

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

This article proposes that meaningful social classification emerges from an ecological dynamic that operates in two planes: feature space and label space. It takes a dynamic view of classification, allowing objects' movements in both spaces to change the meaning of social categories. The first part of the theory argues that agents assign labels to objects based on perceptions of their similarities to existing members of a category. The second part of the theory shows that an object's perceived similarity to members of other categories reduces its typicality in a focal category. This means that for categories with a high degree of overlap with other categories in label space (lenient categories), the link between feature-based similarities and labeling weakens. The findings suggest that social classification will likely evolve to contain both constraining and lenient categories. The theory implies that this process is self-reinforcing, so that constraining categories become more constraining, whereas lenient categories become more lenient.

We introduce a new type of weakly enriched categories over a given symmetric monoidal model category M ; we call them co-Segal categories. Their definition derives from the philosophy of classical (enriched) Segal categories. The purpose of this paper is to expose the theory and give the first results on the homotopy theory of these structures. One of the motivations of developing such theory, was to have an alternative definition of higher linear categories following Segal-like methods.

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The canonical bases of the halves of quantum groups and supergroups are shown to match under the twistor up to powers of a square root of -1. We further show that the modified quantum group and supergroup are isomorphic over the rational function field adjoint with a square root of -1, by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows., Comment: v1 19 pages; v2 added acknowledgement, mild changes, to appear in CMP

We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish an equivalence of categories between module structures over a crossed module of groups and its respective crossed module of associative algebras.

We give a short proof of the existence of a cofibrantly generated Quillen model structure on the category of small topologically enriched categories, obtained by a transfer from Bergner’s model structure on simplicially enriched categories.

The notion of categories within the domain of syntax is fairly clearly circumscribed. We have an idea of what nouns, verbs, adjectives, and prepositions are. In this chapter, I present issues that arise, however, when specific questions are raised concerning these categories: What are the distinguishing characteristics between categories? Are categories primitives? Are category labels attached to lexical items or is their content derivable from syntactic environments? In the process of discussing these issues, noncanonical constructions such as crossover projections (e.g., gerunds) and multifunctional categories are presented. The discussion is placed within the context of natural language data that have been used to defend or refute particular claims. While the main goal of the chapter is to provide some background on the issues, I will defend a need for categorial information in lexical entries using data from St'at'imcets and Malagasy.

The correspondence between definable connected groupoids in a theory T and internal generalised imaginary sorts of T, established by Hrushovski in [“Groupoids, imaginaries and internal covers,” Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaborations on Hrushovki's original constructions are also included.

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller’s axioms defining Quillen’s exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen’s “obscure axiom”. We show that some homological results, such as the Short Five Lemma and the 3×3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.