Ordinary Categories Research Articles

The behaviour of the $$\overline{W}$$ -construction on the homotopy theory of bisimplicial sets

The codiagonal functor $$\overline{W}$$ transfers a Quillen closed model structure on the bisimplicial set category from the ordinary model category of simplicial sets. This bisimplicial model structure is different from the so called Moerdijk model structure, which is similarly transferred from simplicial sets but through the diagonal functor. We show the mutual relationship of these two closed model structures on the category of bisimplicial sets.

Generalized Enrichments of Categories for Operads

Most enriched categories also have an ordinary category structure which is compatible with the enrichment on them. In this paper, enrichments in a monoidal category are generalized to arbitrary categories. These specialize to the classical enrichments when sets are regraded as discrete categories. We also generalize the definitions of PROs and PROPs as some generalized enrichments of categories. Then an operad in some monoidal category corresponds to a generalized PROP. Algebras of operads induce some special kind of monoidal functors. In the category of small categories, we construct several operads to define lax monoids and lax commutative monoids which are formal descriptions of natural associativity and commutativity. Using this identification, operads and their algebras can be studied by lax commutative monoids and morphisms between them.

A fuzzification of the category of M-valued L-topological spaces

A fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FTOP(L,M) extending the category TOP(L,M) of M-valued L- topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak’s sense. Basic properties of the fuzzy category FTOP(L,M) and its objects are studied.

Trois perspectives linguistiques sur la notion de genre discursif

This contribution examines how a complex entity as “genres” can be described in linguistic terms. It is assumed that genres are ordinary categories through which communication is analysed in speech communities and which organises everyday communication : we speak through genres, as M. Bakhtine used to write. In that sense, the study of genres (specially of the genres names) is part of folk linguistics. Genres can also be described as text types, which shows common features in structure, morpho-syntaxic realisations, tone, theme... from a comparative point of view and using the categories of text linguistics. Genres are also part of institutions and expression of social and ideological conflicts : this socio-historical dimension could explain some of their formal and semantic features in the theoretical framework of discourse analysis. It is argued that these three forms of linguistic approaches of genres are together necessary to their description.

Trois perspectives linguistiques sur la notion de genre discursif

Jean-Claude Beacco : Trois perspectives linguistiques sur la notion de genre discursif. This contribution examines how a complex entity as "genres" can be described in linguistic terms. It is assumed that genres are ordinary categories through which communication is analysed in speech communities and which organises everyday communication: we speak through genres, as M. Bakhtine used to write. In that sense, the study of genres (specially of the genres names) is part of folk linguistics. Genres can also be described as text types, which shows common features in structure, morpho-syntaxic realisations, tone, theme... from a comparative point of view and using the categories of text linguistics. Genres are also part of institutions and expression of social and ideological conflicts: this socio-historical dimension could explain some of their formal and semantic features in the theoretical framework of discourse analysis. It is argued that these three forms of linguistic approaches of genres are together necessary to their description.

Tangential Lusternik–Schnirelmann Category of Foliations

A new invariant of (integrable) homotopy type for foliations is introduced: the tangential category of a foliated manifold.The classical Lusternik–Schnirelmann theory is generalized to foliations and the relations of the tangential category with other known invariants such as the fibrewise and the equivariant category are studied. Cohomological lower bounds are provided in terms of foliated cohomology.If the foliation is a product, the tangential category coincides with the ordinary category of the leaves. In general it is just bounded below. Estimates are given of the tangential category for compact‐Hausdorff foliations and suspensions. Examples show that the lower and upper bounds are realized.

Some reflections on metaphoric processing: a move toward a meta-sign formulation

A metaphor always connects two domains in a way that goes beyond our ordinary category structure. It may be defined a tension - resolvable contextual anomaly with emphasis on mental imagery as an essential constituent of most everyday metaphors. It has been suggested that the main function of metaphor is not dependent solely on the structure or on the alignment of logical relations, but on the implicational and psycho-semiotic transmission of information which brings the direction of processing of the two sides of communication close to each other.

The Category of Markov Kernels

Markov kernels are fundamental objects in probability theory. One can define a category based on Markov kernels which has many of the formal properties of the ordinary category of relations. In the present paper we will examine the categorical properties of Markov kernels and stress the analogies and differences with the category of relations. We will show that this category has partially-additive structure and, as such, supports basic constructs like iteration. This allows one to give a probabilistic semantics for a language with while loops in the manner of Kozen. The category in question was originally defined by Giry following suggestions of Lawvere.

Unity and identity of opposites in calculus and physics

A significant fraction of dialectical philosophy can be modeled mathematically through the use of “cylinders” (diagrams of shape Δ) in a category, wherein the two identical subobjects (united by the third map in the diagram) are “opposite”. In a bicategory, oppositeness can be very effectively characterized in terms of adjointness, but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.

Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi

Lambek used categories with indeterminates to capture explicit variables in simply typed λ-calculus. He observed that such categories with indeterminates can be described as Kleisli categories for suitable comonads. They account for ‘functional completeness’ for Cartesian (closed) categories.Here we refine this analysis, by distinguishing ‘contextual’ and ‘functional’ completeness, and extend it to polymorphic λ-calculi. Since the latter are described as certain fibrations, we are lead to consider indeterminates, not only for ordinary categories, but also for fibred categories. Following a 2-categorical generalisation of Lambek's approach, such fibrations with indeterminates are presented as 'simple slices' in suitable 2-categories of fibrations; more precisely, as Kleisli objects. It allows us to establish contextual and functional completeness results for some polymorphic calculi.

Braided groups of Hopf algebras obtained by twisting

It is known that every quasitriangular Hopf algebra H can be converted by a process of transmutation into a braided group B(H, H). The latter is a certain braided-cocommutative Hopf algebra in the braided monoidal category of //-modules. We use this transmutation construction to relate two approaches to the quantization of enveloping algebras. Specifically, we compute B(H, H) in the case when H is the quasitriangular Hopf algebra (quantum group) obtained by Drinfeld's twisting construction on a cocommutative Hopf algebra H. In the case when H is triangular we recover the S-Hopf algebra HF previously obtained as a deformation-quantization of H. Here Hf is a Hopf algebra in a symmetric monoidal category. We thereby extend the definition of HF to the braided case where H is strictly quasitriangular. We also compute its structure to lowest order in a quantization parameter h. In this way we show that B(Uq(g), Uq(g)) is the quantization of a certain generalized Poisson bracket associated to the Drinfeld-Jimbo solution of the classical Yang-Baxter equations. 1. Introduction. Hopf algebras in braided monoidal categories have been introduced in [10] and [11] in the context of Tannaka-Krein reconstruction theorems. It is shown there that every quantum group gives rise to a Hopf algebra in a braided category by a process of transmutation. The category is that of representatio ns of the quantum group, and in this braided category the resulting Hopf algebra is in a certain sense braided-cocommutative, i.e. like a group algebra. Hence such Hopf algebras in braided categories are called braided groups. The process is called transmutation because it turns a quantum group in the ordinary category of Hopf algebras into a group-like object in a non-commutative category. Hopf algebras in symmetric monoidal categories, on the other hand, arise naturally in the deformation-quantization of triangular solutions of the classical Yang-Baxter equations (CYBE) [6]. They have been called S-Hopf algebras and are the enveloping algebras of S-Lie algebras and S-groups. Hence we are led to consider if braided groups, too, can arise as such deformation-quantization of some kind of Poisson structure. This is one motivation for the present paper. The

The Politics of Art: The Domination of Style and the Crisis in Contemporary Art

societal sense, but also questions of domination and subordination in matters aesthetic: in style, influence, creative direction, and aesthetic ideology as well as in the more mundane contexts of artistic production, artistaudience relations, and in the roles of artworld entrepreneurs (gallery owners, museum directors, collectors, and agents). The ordinary categories are transformed and specified by the peculiar functions and interests of the artworld, so that aesthetic desiderata come to appear as the defining terms for power, governance, and interests. The blunter political-economic categories take on the protective coloration of artworld concepts: production and consumption are mediated as creation and appreciation; of becomes cultural diffusion; buying becomes collecting. The duality is not simply a semantic overlay of aesthetic euphemisms on a ground of stark social realities. Rather, it exhibits a real duality within the artworld itself, both as a world of art-a world of creative and imaginative praxis defined by aesthetic norms and categories-and as a social and historical artworld within a larger society or culture-a world of production, exchange, and distribution in which goods are produced, livelihoods earned, and profits made. My paper attempts to analyze the interconnection between these two aspects of the artworld. It does this in order, first, to bring to bear an analysis of the crisis in the aesthetics of contemporary art upon the crisis in the political economy of the contemporary artworld; and second, to sketch the political economy of that artworld so that I can explain what I take to be the crucial symptom of the crisis, namely, the domination

A characterization of pie limits

It is well-known that limits in 2-categories are more complex than limits in ordinary categories. Most readers will at least be familiar with terms such as ‘lax limit’ and ‘pseudo-limit’. In the more modern treatments, these become special cases of a more general class of ‘weighted’ or ‘indexed’ limits (see Kelly [7] and Section 1 of this paper).

Languages for monoidal categories

Proofs of propositions about ordinary categories, e.g. the Yoneda Lemma, may often be reinterpreted to yield proofs of the equivalent propositions about enriched categories, without recourse to diagram-chasing. Elements of the homset are replaced by variables of the hom-type. The language may also be used to lift definitions to the enriched context in a natural way, e.g., natural transformation, strong monic.

Cocompleteness over coverings

AbstractFor enriched categories the correct notion of limit involves indexing by a module. This paper studies the question of cocompletion for a given set of indexing modules. As well as providing a simplified treatment of cocompleteness for ordinary categories, associated sheaves and associated stacks are also included as cocompletion processes for appropriate bases. In fact the saturation of a general set of indexing modules has properties which justify our use of the term “covering” for members of the saturation.

A notion of limit for enriched categories

For a V-category B, where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B0 ) preserved by the V-valued representable functors; cotensor products; ends; pointwise Kan extensions. It has further been recognized that, to be called complete, B should admit all of these; for which it suffices to demand the first two. Hitherto, however, there has been no single limit-notion of which all these are special cases, and particular instances of which may exist even when B is not complete or even cotensored. In consequence it has not been possible even to state, say, the representability criterion for a V-functor T: B → V, or even to define, say, pointwise Kan extensions into B, except for cotensored B. (It is somewhat as if, for ordinary categories, we had the notions of product and equalizer, but lacked that of general limit, and could not discuss pullbacks in the absence of products.) In this paper we provide such a general limit-notion for V-categories.

Note on monoidal localisation

If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a “change-of-universe” procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed.