Manuela Sobral

Finite preorders and Topological descent I

Abstract It is shown that the descent constructions of finite preorders provide a simple motivation for those of topological spaces, and new counter-examples to open problems in Topological descent theory are constructed.

Finite preorders and topological descent II: étale descent

Abstract It is known that every effective (global-) descent morphism of topological spaces is an effective etale-descent morphism. On the other hand, in the predecessor of this paper we gave examples of: • a descent morphism that is not an effective etale-descent morphism; • an effective etale-descent morphism that is not a descent morphism. Both of the examples in fact involved only finite topological spaces, i.e. just finite preorders, and now we characterize the effective etale-descent morphisms of preorders/finite topological spaces completely.

RESTRICTING THE COMPARISON FUNCTOR OF AN ADJUNCTION TO PROJECTIVE OBJECTS

Abstract This paper deals with projectives (in the sense of K.A.Hardie [5] relative to a right adjoint functor U: A → K. We answer the question, raised by R.-E. Hoffmann [6] p. 135, of knowing under what conditions there exists an equivalence between Proj u and Proj Ur, induced by the comparison functor Φ: A → KT, where T denotes the monad induced by U. In the case, that U is an algebraic functor we also give necessary and sufficient conditions for the re gular projective objects to coincide with the U-projectives. Finally, we delineate how these results nay be applied in certain familiar situations.

Semidirect products and Split Short Five Lemma in normal categories

In this paper we study a generalization of the notion of categorical semidirect product, as defined in [6], to a non-protomodular context of categories where internal actions are induced by points, like in any pointed variety. There we define semidirect products only for regular points, in the sense we explain below, provided the Split Short Five Lemma between such points holds, and we show that this is the case if the category is normal, as defined in [12]. Finally, we give an example of a category that is neither protomodular nor Mal’tsev where such generalized semidirect products exist.

Descent for Regular Epimorphisms in Barr Exact Goursat Categories

We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an effective descent morphism.

Some aspects of topological descent

The paper deals with (effective) descent morphisms for subfibrations \(\mathbb{E}\)(X) of the basic fibration T op/X, for topological spaces X and classes \(\mathbb{E}\) of continuous functions stable under pullback. For a category with pullbacks, we prove the stability under pullback of effective e-descent morphisms for a class \(\mathbb{E}\) satisfying some suitable conditions. This plays a role in relating effective \(\mathbb{E}\)-descent to effective global descent and enables us to obtain a criterion for effective etale-descent. We also show that the inclusion of the class of effective global-descent maps in the class surjective effective etale-descent is strict.

Descent for Priestley Spaces

A characterization of descent morphism in the category of Priestley spaces, as well as necessary and sufficient conditions for such morphisms to be effective are given. For that this category is embedded in suitable categories of preordered topological spaces where descent and effective morphisms are described using the monadic description of descent.

Another approach to topological descent theory

In the category Top of topological spaces and continuous functions, we prove that surjective maps which are descent morphisms with respect to the class E of continuous bijections are exactly the descent morphisms, providing a new characterization of the latter in terms of subfibrations E(X) of the basic fibration given by Top/X which are, essentially, complete lattices. Also effective descent morphisms are characterized in terms of effective morphisms with respect to continuous bijections. For classes E satisfying suitable conditions, we show that the class of effective descent morphisms coincides with the one of effective E-descent morphisms.

CABool is monadic over almost all categories

We give a description of the Eilenberg-Moore category of algebras induced by an adjunction F ⊣ U : Setop → C and prove that, under mild conditions, U is weakly monadic. This enables us to sketch another proof of the well-known fact that CABool is equivalent to Setop, which justifies our title. Next, we characterize the category of algebras TopT, T being the monad induced in Top by the Sierpinski space, in terms of the topology of the spaces underlying T-algebras, concluding that TopT is, up to isomorphism, and so CABool is, up to equivalence, a reflective subcategory of Top. Finally, we show that the conclusion still holds for the skeleton of each category of algebras CT, whenever T is induced by a C-object A satisfying some conditions, if C has no unnatural isomorphisms between any two powers of A, in the sense of V. Trnkova.

Descent for Discrete (Co)fibrations

We characterize the (effective) E-descent morphisms in the category Cat of small categories, when E is the class of discrete fibrations or the one of discrete cofibrations, and prove that every effective global-descent morphism is an effective E-descent morphism while its converse fails.