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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).

As a valued domain, quantales have been applied to the study of enriched category, fuzzy algebra, fuzzy topology and fuzzy domain. One purpose of this paper is to investigate semitopological groups and semiclosure semigroups from the viewpoint of quantales. We first show that a semitopological group is completely determined by the topological ideal conuclei of the power-set quantale over a group. Next, we prove that the concepts of semitopological groups and semiclosure groups are dually equivalent. Another purpose of this paper is to investigate ordered semigroups, conditionally complete quantales and quantales from the viewpoint of semiclosure semigroups. As an important result of semiclosure semigroups, we prove that the category of complete semiclosure semigroups is isomorphic to the category of quantales.

We show that the Giry monad is not strong with respect to the canonical symmetric monoidal closed structure on the category Meas of all measurable spaces and measurable functions.

The article is concerned with homotopy in the category P whose objects are the pairs (X,∗) consisting of a Polish space X and a closed binary operation ∗. Homomorphisms in P are continuous maps compatible with the operations. The result showed that the category P admits the structure of a fibration category in the sense of H. Baues. The notions of fibration and weak equivalence are defined in the category P and showed to satisfy fundamental properties that the corresponding notions satisfy in the category Top of topological spaces. Keywords: Polish spaces, Homotopy theory, Fibration, Model category

We provide conditions on a monoidal model category $\mathcal{M}$ so that the category of commutative monoids in $\mathcal{M}$ inherits a model structure from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and only if it is so in $\mathcal{M}$. We then investigate properties of cofibrations of commutative monoids, rectification between $E_\infty$-algebras and commutative monoids, the relationship between commutative monoids and monoidal Bousfield localization functors, when the category of commutative monoids can be made left proper, and functoriality of the passage from a commutative monoid $R$ to the category of commutative $R$-algebras. In the final section we provide numerous examples of model categories satisfying our hypotheses., Version 2 adds material about rectification between strict commutative monoids and $E_\infty$-algebras, adds material about lifting Quillen equivalences to categories of commutative monoids, and adds several new examples: simplicial presheaves, diagram categories, and commutative Smith ideals of ring spectra

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group., Comment: 19 pages, minor modifications following refereeing

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids -- or in a straightforward generalisation, the category of modules over a commutative rig $k$. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad $Q$. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlach\'anyi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad $Q$ involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal $k$-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category -- thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area., Comment: 53 pages

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.

Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules HHH Y D is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.

A quasi-pointed category in the sense of D. Bourn is a finitely complete category \(\mathcal {C}\) having an initial object such that the unique morphism from the initial object to the terminal object is a monomorphism. When instead this morphism is an isomorphism, we obtain a (finitely complete) pointed category, and as it is well known, the structure of zero morphisms in a pointed category determines an enrichment of the category in the category of pointed sets. In this note we examine quasi-pointed categories through the structure formed by the zero morphisms (i.e. the morphisms which factor through the initial object), with the aim to compare this structure with an enrichment in the category of pointed sets.

This article shows that the axioms of a skew monoidal category are independent and that its unit is unique up to a unique isomorphism together with an analogue of this result for monoidal functors between skew monoidal categories. It is also noted that these results carry over to skew monoidales before some benefits of certain extra structure on the unit maps of a skew monoidal category are discussed.

In this paper, we construct an O -display theory and prove that, under certain conditions on the base ring, the category of nilpotent O -displays and the category of π-divisible formal O -modules are equivalent. Starting with this result, we then construct a Dieudonne O -display theory and prove a similar equivalence between the category of Dieudonne O -displays and the category of π-divisible O -modules. We also show that this equivalence is compatible with duality. These results generalize the corresponding results of Zink and Lau on displays and p-divisible groups.

In this expository paper, we give an explicit construction of an isomorphism between the category of homotopical quantum field theories with an underlying Eilenberg–MacLane space K(G, 1) and the category of G-topological quantum field theories, where G is a finite group. Additionally, in dimension two, we show that there is a monoidal equivalence between the category of G-topological quantum field theories and the category of G-Frobenius algebras.

Based on a completely distributive lattice L, the concept of L-fuzzy N-convergence structures is introduced. It is shown that the category of L-fuzzy topological spaces can be embedded in the category of L-fuzzy N-convergence spaces. It is also proved the category of (topological) pretopological L-fuzzy N-convergence spaces is isomorphic to the category of (topological) L-fuzzy neighborhood spaces and the former is a bireflective subcategory of the category of L-fuzzy N-convergence spaces.

In my Montreal lecture notes of 1988, it was suggested that the theory of linear quantum groups can be presented in the framework of the category of {\it quadratic algebras} (imagined as algebras of functions on "quantum linear spaces"), and quadratic algebras of their inner (co)homomorphisms. Soon it was understood (E. Getzler and J. Jones, V. Ginzburg, M. Kapranov, M. Kontsevich, M. Markl, B. Vallette et al.) that the class of {\it quadratic operads} can be introduced and the main theorems about quadratic algebras can be generalised to the level of such operads, if their components are {\it linear spaces} (or objects of more general monoidal categories.) When quantum cohomology entered the scene, it turned out that the basic tree level (genus zero) (co)operad of quantum cohomology not only is {\it quadratic} one, but its {\it components are themselves quadratic algebras.} In this short note, I am studying the interaction of quadratic algebras structure with operadic structure in the context of enriched category formalism due to G. M. Kelly et al., 13 pages

It is shown that for every monoidal bi-closed category ℂ left and right dualization by means of the unit object not only defines a pair of adjoint functors, but that these functors are monoidal as functors from , the dual monoidal category of ℂ into the transposed monoidal category ℂt. We thus generalize the case of a symmetric monoidal category, where this kind of dualization is a special instance of convolution. We apply this construction to the monoidal category of bimodules over a not necessarily commutative ring R and so obtain various contravariant dual ring functors defined on the category of R-corings. It becomes evident that previous, hitherto apparently unrelated, constructions of this kind are all special instances of our construction and, hence, coincide. Finally, we show that Sweedler's Dual Coring Theorem is a simple consequence of our approach and that these dual ring constructions are compatible with the processes of (co)freely adjoining (co)units.

We define the concept of a convergence class on an object of a given category by using certain generalized nets for expressing the convergence. The resulting topological category, whose objects are the pairs consisting of objects of the original category and convergence classes on them, is then investigated. We study the full subcategories of this category which are obtained by imposing on it some natural convergence axioms. In particular, we find sufficient conditions for the subcategories to be cartesian closed. We also investigate the behavior of the closure operator associated with the convergence in a natural way.

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts. The Mathematical Appendix is of general interest to category theorists as it is a defense of the use of heteromorphisms as a natural and necessary part of category theory.

In this paper we will introduce the notion of a weak Hom-Hopf algebra, generalizing both weak Hopf algebras and Hom-Hopf algebras. Then we study the category R e p ( H ) of Hom-modules with bijective Hom-structure maps over a weak Hom-Hopf algebra H and show that the tensor functor of a (weak) Hom-bialgebra is actually a (weak) bimonad on a Hom-type category. At last, we prove that if H is a quasitriangular weak Hom-bialgebra (resp. ribbon weak Hom-Hopf algebra), then R e p ( H ) is a braided monoidal category (resp. ribbon category).

The category Hopf ℂ of Hopf monoids in a symmetric monoidal category ℂ, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ℂ preserve directed colimits one has the following results: (1) If, in ℂ, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ℂ and comonadic over the category of monoids in C. (2) If, in ℂ, extremal monomorphisms are stable under tensor squaring, then Hopf ℂ is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ℂ.

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

We define a monoidal category of graphs, and define the notion of graph algebras using monoidal functors from the category of graphs. We construct a category of graph algebras and show its equivalence to the category of formally rational deformation operads, which is known to be isomorphic to the category of meromorphic field algebras.

In this paper we consider an internal representation of the lattice of membership values used by the relations in an arbitrary arrow category. This is achieved by defining a more general construction that pairs regular with membership values in one object. We will show basic properties of this construction including its relationship to the lattice of membership values. This characterization of the lattice of membership values allows to specify and reason about the lattice elements as elements of an object of the category. Furthermore, this representation also leads to an abstract approach to higher-order fuzziness based on arrow categories.

In this paper complexes with N-nilpotent differentials are considered. We describe the loop functor Ω and the suspension functor Σ in the category C N ( A ) of N-complexes on an abelian category A to provide an effective construction of left and right triangles in the homotopy category K N ( A ) of N-complexes, and prove that K N ( A ) together with the functors Ω, Σ and the classes of left and right triangles mentioned above is a pretriangulated category. We also investigate different homologies and get another example of pretriangulated categories, i.e., the derived category D N ( A ) of N-complexes. Some basic properties are given.

We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a 2-category enriched category of equipment-like structures. These abstract concepts are necessary to handle the presence of oplax constructs – the construction is only oplax functorial on a certain class of tripos morphisms. A by-product of our analysis is the decomposition of the tripos-to-topos construction into two steps, the intermediate step being a generalization of quasitoposes.

Given a locally presentable enriched category $\mathcal{E}$ together with a small dense full subcategory $\mathcal A$ of arities, we study the relationship between monads on $\mathcal E$ and identity-on-objects functors out of $\mathcal A$, which we call $\mathcal A$-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the $\mathcal A$-nervous monads---those for which the conclusions of Weber's nerve theorem hold---and the $\mathcal A$-theories, which we introduce here. The resulting equivalence between $\mathcal A$-nervous monads and $\mathcal A$-theories is best possible in a precise sense, and extends almost all previously known monad--theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak $\omega$-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of $\mathcal A$-nervous monads and $\mathcal A$-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melli\`es and Weber., Comment: 43 pages; v2: final journal version

Submitted by Dirk Hofmann (dirk@ua.pt) on 2020-07-22T15:05:19Z No. of bitstreams: 1 compact_V_cats.pdf: 619860 bytes, checksum: 2676a94d5af9176f4705f232b74789fd (MD5) Approved for entry into archive by Rita Gonçalves (ritaisabel@ua.pt) on 2020-07-30T18:00:35Z (GMT) No. of bitstreams: 1 compact_V_cats.pdf: 619860 bytes, checksum: 2676a94d5af9176f4705f232b74789fd (MD5) Made available in DSpace on 2020-07-30T18:00:35Z (GMT). No. of bitstreams: 1 compact_V_cats.pdf: 619860 bytes, checksum: 2676a94d5af9176f4705f232b74789fd (MD5) Previous issue date: 2018 published

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

For a pomonoid S, let us denote Pos-S the category ofS-posets and S-poset maps. In this paper, we consider the slice cate-gory Pos-S/B for an S-poset B,and study some categorical ingredients.We first show that there is no non-trivial injective object in Pos-S/B.Then we investigate injective objects with respect to the class of regu-lar monomorphisms in this category and show that Pos-S/B has enoughregular injective objects. We also prove that regular injective objects areretracts of exponentiable objects in this category. One of the main aimsof the paper is to draw attention to characterizing injectivity in the cate-gory Pos-S/Bunder a particular case where Bhas trivial action. Amongother things, we also prove that the necessary condition for a map (an ob-ject) here to be regular injective is being convex and present an exampleto show that the converse is not true, in general. 1. Introduction and preliminariesA slice category is a construction in category theory which provides anotherway of looking at morphisms: instead of simply relating objects of a categoryto one another, morphisms become objects in their own right. This notion wasintroduced in 1963 by F. W. Lawvere, although the technique did not becomegenerally known until many years later.One of the very useful notions in many categories as well as in homologicalalgebra is the injectivity of objects with respect to a class M of morphisms. Forexample, in the category Pos of partially ordered sets and monotone mappings,injective objects, with respectto the classofall order-embeddings,coincidewiththe complete lattices (see [4]). Injective objects in slice categories have beeninvestigated in detail (see [1, 16]), especially in relation with weak factorizationsystems, a concept used in homotopy theory, in particular for model categories.

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., 16 pages; v2: minor edits, final version; published in Applied Categorical Structures

Submitted by Dirk Hofmann (dirk@ua.pt) on 2017-06-27T19:23:14Z No. of bitstreams: 1 The_Rise_and_Fall_of_V-functors.pdf: 363554 bytes, checksum: 1cff8fce6de9f71b634c700418fa75bf (MD5) Approved for entry into archive by Bella Nolasco(bellanolasco@ua.pt) on 2017-07-14T11:18:35Z (GMT) No. of bitstreams: 1 The_Rise_and_Fall_of_V-functors.pdf: 363554 bytes, checksum: 1cff8fce6de9f71b634c700418fa75bf (MD5) Made available in DSpace on 2017-07-14T11:18:35Z (GMT). No. of bitstreams: 1 The_Rise_and_Fall_of_V-functors.pdf: 363554 bytes, checksum: 1cff8fce6de9f71b634c700418fa75bf (MD5) Previous issue date: 2017-08

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory. Two functors, the forward globalization functor and the backward globalization functor, that map a fuzzy set with a fuzzy preorder to a crisp set with a fuzzy preorder are considered. It is shown that both of them preserve cocompleteness with respect to several classes of weights.

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloid-enriched category theory. We discuss at length the definitions of quantaloid, quantaloid-enriched category, distributor and functor, always giving several examples that - hopefully - appeal to the intended readership. To indicate the strength of this general theory, we explain in considerable detail how (co)limits are dealt with, and particularly how the Yoneda embedding of a quantaloid-enriched category in its free (co)completion comes to be. Our insistence on quantaloid-enrichment (rather than quantale-enrichment) is duly explained by examples requiring a notion of "partial elements" (sheaves, partial metric spaces). A final section glosses over some further topics, providing ample references for the interested reader.

We introduce the notion of normalizer as motivated by the classical notion in the category of groups. We show for a semi-abelian category ℂ that the following conditions are equivalent: (a) ℂ is action representable and normalizers exist in ℂ; (b) the category Mono(ℂ) of monomorphisms in ℂ is action representable; (c) the category ℂ2 of morphisms in ℂ is action representable; (d) for each category \(\mathbb {D}\) with a finite number of morphisms the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable.

The following classes of categories are shown to be weakly Mal’tsev in the sense of the author: (i) a suitable class of algebras with cancellation; (ii) the dual of any quasi-adhesive category; (iii) the dual of any extensive category with pullback-stable epimorphisms; (iv) the dual of any solid quasi-topos. The examples in (i) include all the Mal’tsev varieties of algebras such as groups, rings, Lie algebras, etc., but also distributive lattices and commutative monoids with cancellation. The examples in (ii)-(iv) capture many of the familiar aspects of topological spaces.

Conformal field theories were first axiomatized by Segal (2004) as symmetric monoidal functors from a topological category of conformal cobordisms between compact oriented 1-dimensional manifolds to vector spaces. Costello (2007) later expanded the definition of the category to allow for cobordisms between manifolds with boundaries, and was able to use representations of this category to give a mirror partner for Gromov-Witten invariants.

This paper investigates the structure of the “missing part” from the category of coherent sheaves over a weighted projective line of weight type (2, 2, n) to the category of finitely generated right modules on the associated canonical algebra. By constructing a t-structure in the stable category of the vector bundle category, we show that the “missing part” is equivalent to the heart of the t-structure, hence it is abelian. Moreover, it is equivalent to the category of finitely generated modules on the path algebra of type \(\mathbb{A}_{n - 1}\).

We use directed algebraic topology for establishing categorical relationships between two geometrical models of concurrency, precubical sets and transitional Chu spaces. We study the universal dicovering functor from the category of precubical sets to the category of simply diconnected counterparts of precubical sets. We prove that the category of transitional Chu spaces is equivalent to the category of simply diconnected precubical sets and show that the openness of morphisms is preserved. Bibliography :1 9titles. Illustrations :7 figures.

In this paper, we discuss a relationship between representation theory of graded self-injective algebras and that of algebras of finite global dimension. For a positively graded self-injective algebra $A$ such that $A_0$ has finite global dimension, we construct two types of triangle-equivalences. First we show that there exists a triangle-equivalence between the stable category of $\mathbb{Z}$-graded $A$-modules and the derived category of a certain algebra $\Gamma$ of finite global dimension. Secondly we show that if $A$ has Gorenstein parameter $\ell$, then there exists a triangle-equivalence between the stable category of $\mathbb{Z}/\ell\mathbb{Z}$-graded $A$-modules and a derived-orbit category of $\Gamma$, which is a triangulated hull of the orbit category of the derived category., Comment: 32 pages

S.A. Solovyov (2008) has recently introduced the notion of a Q-topological space (and Q-continuous maps between them), where Q is a fixed member of a variety of Omega-algebras, which in turn gives rise to the category Q-TOP of such spaces. The purpose of this note is to give a characterization of this category (in a large class of categories), in terms of a 'Sierpinski-like' object, which is similar to the one given by E.G. Manes in 1976 for the category TOP of topological spaces.

PosM is a category whose objects are ample spaces and morphisms are possibility mappings. We study some properties of the Category PosM . So we show that Category PosM is a cartesian closed category, and it forms a topos with some condition.

We introduce the dendroidal analogues of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for ∞-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal’s quasi-categories, Rezk’s complete Segal spaces, and Segal categories.

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism.

We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.

In this paper we give detecting elements for the proper Lusternik–Schnirelmann category of any missing boundary 3-manifold. As a consequence, we determine the proper LS category of all orientable one-ended semistable open 3-manifolds. These results extend those in Cardenas et al. (2012) [10] .

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech's cohomology groups of monoids., 38 pages; corrected typos and made modifications suggested by the referee

Category theory was introduced by S. Eilenberg and S. MacLane in an article published in 1945, which also axiomatized the notions of functor and natural transformation . MacLane wrote that the notion of category was defined in order to define the notion of functor, which was in turn defined in order to define the notion of natural transformation. The archetypal example of a natural (or canonical ) isomorphism is the one that identifies finite-dimensional vector spaces with their biduals. The notion of structure gradually began to emerge at the end of the 19th Century, and was fully formalized in Elements de mathematique by N. Bourbaki (vector space structures, topological spaces, etc.). A vector space, for example, is a set equipped with a vector space structure. This “structuralist” perspective adopted by Bourbaki is based on set theory. In the category theory, however, the objects of a category are not always sets, and consequently the morphisms are not always mappings. A functor F F is a “large function” from a category C C to a category D D that sends each object A in C to an object F(A) F A in D D and each morphism f : A → B in C C to a morphism ℱ(f):ℱ(A)→ℱ(B) ℱ f : ℱ A → ℱ B in D D .