| Jirí Adámek | Well-pointed coalgebras and minimal nondeterministic automata: In the coalgebraic description of systems terminal coalgebras serve as a formalization of the concept of behaviour of individual states. We present a new description of terminal coalgebras by introducing well-pointed coalgebras as their elements. In the example of deterministic automata in a category C these are precisely the minimal automata. By taking as C boolean algebras (or semilattices) we obtain a construction of a "small" nondeterministic automaton for every reguar language called a'tomaton (or jiromaton, resp.). |
| Maria Manuel Clementino | On fibrewise injectivity in T0-spaces: It is well known that injective topological T0-spaces can be characterized as retracts of powers of the Sierpinski space, or as continuous lattices (cf. [6]). In this talk we analyse their fibrewise counterpart, that is injective continuous maps between T0-spaces. We present fibrewise versions of the characterizations mentioned, obtained in [2, 4, 5], and recent results that were inspired by the characterizations of injective monotone maps in the category of partially ordered sets of [7, 1]. This is ongoing joint work with F. Cagliari and S. Mantovani [3].
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| Nelson Martins-Ferreira | Top as a quotient category from fibrous preorders: It is well known that the category of finite topological spaces is equivalent to the category of finite preorders. It is also well known that this equivalence can be extended to the category of preorders (not necessarily finite), establishing a categorical equivalence between preorders and Alexandrov spaces (the ones in which the intersection of any collection of open sets is open). In this talk we will see how to extend the previous equivalence to the context of arbitrary topological spaces. To do that we introduce a new notion, called fibrous preorder, together with the appropriate notion of morphism, called fibrous morphism (a pair (f,f*) in which f is said to be the underlying map while f* is denoted by the fibrousness of the morphism). We will show that under the identification (f,f*)~(g,g*) if and only if f=g the category Top, of topological spaces, is obtained as the quotient category from the category of fibrous preorders. This generalizes the categorical equivalen ce from preorders and Alexandrov spaces to fibrous preorders and topological spaces. Examples and counter-examples are also provided. |
| Andrea Montoli | Semidirect products of topological semi-abelian algebras: F.Borceux and M.M. Clementino [1] proved that all topological models of a semi-abelian variety admit semidirect products in the categorical sense introduced by D. Bourn and G. Janelidze [2]. Using some techniques developed in [3], we can show that, in any semi-abelian variety, the semidirect product of two objects X and B always appears as a subset of a certain cartesian product built using X and B. This allows us to give an explicit description of the topology of the semidirect product in topological semi-abelian algebras. Joint work with Maria Manuel Clementino and Lurdes Sousa.
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| Jorge Picado | Normality and extremal disconnectedness in parallel: Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. We will discuss the source of this 'duality' and will show that each pair of parallel results can be framed by a single proof. This enables us to unify under a single pointfree result a significative list of classical insertion and extension results. The key ingredients will be our localic approach to general real-valued functions [1] and new relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales) that bring and extend to pointfree topology a variety of well known classical variants of normality and upper and lower semicontinuities in a nice unified manner. This is ongoing joint work with Javier Gutiérrez García [2,3].
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| Aleš Pultr | Subfitness, fitness and nearness: Subfitness as a handy separation axiom. Its merits and its faults. Fitness as hereditary sub tness. Comparing some properties of fitness and subfitness. Nearness and regular nearness in classical topology. Unlike uniformity it requires different types of admissibility. Pointfree nearness as standardly used is in fact the regular one. How to represent the general nearness: subfitness enters the story. Non-symmetric nearness and what happens with the subfitness. More about heredity. Another description of fitness: prefitness. Joint work with Jorge Picado. |
| Diana Rodelo | 2-star-permutability, a generalisation of subtractivity and the Mal'tsev property: 2-star-permutable categories were introduced in a joint work with Z. Janelidze and A. Ursini [1] as a common generalisation of regular Mal'tsev categories and normal subtractive categories. In the present talk we first characterise these categories in terms of what we call star-regular pushouts. We then show that the 3x3 Lemma characterising normal subtractive categories [3] and the Cuboid Lemma characterising regular Mal'tsev categories [2] are special instances of a more general homological lemma for star-exact sequences. We show that 2- star-permutability is equivalent to the validity of this lemma for a star-regular category. (Joint work with Marino Gran.)
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| Lurdes Sousa | On the Kan-injective Subcategory Problem: In a poset enriched category, an object X is said to be left Kan-injective w.r.t. a morphism h: A-->B, if for every f:A-->X the left Kan-extension f/h of f along h exists and fulfils f=(f/h)h. In [1] we defined left Kan-injectivity also for morphisms: suppose X and Y are left Kan-injective w.r.t. h:A-->B. Then the morphism g:X-->Y is called left Kan-injective w.r.t. h provided that the equality (gf)/h=g(f/h) holds for all f:A-->X. We thus have, for every class H of morphisms, a (non-full, in general) subcategory LInj(H) of all objects and morphisms that are left Kan-injective w.r.t. every member of H. Every Eilenberg-Moore category of a Kock-Zoberlein monad over a poset enriched category is a left Kan-injective subcategory, that is, of the form LInj(H) for some H. So we may ask whether the converse is also true. Equivalently, we have the Kan-injective Subcategory Problem: When is LInj(H) a Kock-Zoberlein reflective subcategory? In this talk I present an answer to this problem obtained in recent joint work with Jirí Adámek and Jirí Velebil.
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| João Xarez | An attempt to extend the classical monotone-light factorization to ordered spaces: The semi-left-exact subreflections into Priestley spaces of Nachbin's ordered compact spaces will be identified. In order to do so it will be needed the simplification given in [5] to the pullback preservation conditions in the definition of a semi-left-exact reflection (see [3]). Then the proofs in [1, 5.6, 5.7] will be generalized; in particular, an appropriate notion of connected component and a non-symmetrical generalization of entourage will be introduced. It will also be shown that these semi-left-exact subreflections necessarily have the stronger property of stable units, and monotone maps in such cases will be characterized. These results provide a good ground for the project of extending the classical monotone-light factorization of compact Hausdorff spaces via Stone spaces (itself an extension of Eilenberg's factorization for metric spaces; see [2]) to non-trivial ordered spaces.
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