Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces

The classical theorem of Cartier-Milnor-Moore-Quillen gives an equivalence between the category of connected cocommutative bialgebras and the category of Lie algebras. We establish an analogous equivalence between the category of connected dendriform bialegebras and the category of brace algebras. It is given by the primitive elements functor and the “enveloping dendriform algebra” of a brace algebra.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

A Cartan-Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.     

Monads in double categories

We extend the basic concepts of Street’s formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.     

INITIAL COMPLETIONS OF MONOTOPOLOGICAL CATEGORIES,AND CARTESIAN CLOSEDNESS

Abstract

Herrlich and Strecker [9] give examples of monotopological categories for which the MacNeille completion coincides with the universal initial completion. It is shown here that this situation always holds for monotopological categories. If the category is a proper monotopological c-category, then the MacNeille completion also coincides with the largest epi-reflective initial completion. During the course of the proof a lemma is given which characterizes monotopological categories (not necessarily c-categories) which are already topological. (Schwarz [14] gave such a characterization for the case of c-categories.)

It is also shown that a monotopological c-category is Cartesian closed if and only if its largest epi-reflective initial completion is Cartesian closed. A similar result holds for the case of a topological category which is not necessarily a c-category.     

The category of 3-computads is not cartesian closed

We show, using [A. Carboni, P.T. Johnstone, Connected limits, familial representability and Artin glueing, Math. Structures Comput. Sci. 5 (1995) 441-459] and Eckmann-Hilton argument, that the category of 3-computads is not cartesian closed. As a corollary we get that neither the category of all computads nor the category of n-computads, for n>2, do form locally cartesian closed categories, and hence elementary toposes.     

On the (non)vanishing of some “derived” categories of curved dg algebras

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

Categories with sums and right distributive tensor product

Models for parallel and concurrent processes lead quite naturally to the study of monoidal categories (Inform. Comput. 88 (2) (1990) 105). In particular a category Tree of trees, equipped with a non-symmetric tensor product, interpreted as a concatenation, seems to be very useful to represent (local) behavior of non-deterministic agents able to communicate (Enriched Categories for Local and Interaction Calculi, Lecture Notes in Computer Science, Vol. 283, Springer, Berlin, 1987, pp. 57-70). The category Tree is also provided with a coproduct (corresponding to choice between behaviors) and the tensor product is only partially distributive w.r.t. it, in order to preserve non-determinism. Such a category can be properly defined as the category of the (finite) symmetric categories on a free monoid, when this free monoid is considered as a 2-category. The monoidal structure is inherited from the concatenation in the monoid. In this paper we prove that for every alphabet A, Tree(A), the category of finite A-labeled trees is equivalent to the free category which is generated by A and enjoys the afore-mentioned properties. The related category Beh(A), corresponding to global behaviors is also proven to be equivalent to the free category which is generated by A and enjoys a smaller set of properties.     

INSTANCES AND RAMIFICATIONS OF THE SEMI-ADJOINT SITUATION II. THE COMPARISON FUNCTOR

Abstract

If a functor U has a left co-unadjoint then U can be factored through a category of semad algebras. An analogue of the Beck monadicity theory is obtained. If R is a ring without a left unit but satisfying R² = R then the category of unitary left R-modules need not be monadic over Set. The forgetful functor has, however, a left co-unadjoint for which a comparison functor is an equivalence of categories. Another example of a semadic functor is obtained by composing the forgetful functor from Abelian groups to Set with the doubling functor. The semi-adjoint situations in the senses of Medvedev and Davis are examined.     

Adjoining adjoints

We construct the 2-category obtained from a category by freely adjoining a right adjoint for each morphism and isolate its universal property. Some others basic properties are also studied. Some examples in which the category is freely generated by a graph are discussed in detail. For these categories, the 2-cells are given a geometric interpretation and shown to be similar to certain diagrams which have appeared in the literature on C∗-algebras.     

CONCRETE CATEGORIES FOR WHICH FIBRE COMPLETIONS INDUCE TOPOLOGICAL COMPLETIONS

Abstract

Every topological category over an arbitrary base category X may be considered as a category of T-models with respect to some theory (i.e., functor) T from X into a category of complete lattices. Using this model-theoretic correspondence as our basic tool, we study initial and final completions of (co)fibration complete categories. For an arbitrary concrete category (A, U) over X, the process of order-theoretically completing each fibre does not usually yield an initial/final completion of (A, U). It is shown in this paper that for concrete categories which are assumed to be fibration and/or cofibration complete, initial and final completions can be constructed by completing the fibres. These completions are further shown to exhibit some interesting external properties.