Estudo GeralThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.https://estudogeral.sib.uc.pt2019-05-26T03:00:34Z2019-05-26T03:00:34Z50101Morita equivalence of many-sorted algebraic theoriesAdámek, JiríSobral, ManuelaSousa, Lurdeshttp://hdl.handle.net/10316/46162018-05-23T21:59:51Z2006-01-01T00:00:00ZTitle: Morita equivalence of many-sorted algebraic theories
Authors: Adámek, Jirí; Sobral, Manuela; Sousa, Lurdes
Abstract: Algebraic theories are called Morita equivalent provided that the corresponding varieties of algebras are equivalent. Generalizing Dukarm's result from one-sorted theories to many-sorted ones, we prove that all theories Morita equivalent to an S-sorted theory are obtained as idempotent modifications of . This is analogous to the classical result of Morita that all rings Morita equivalent to a ring R are obtained as idempotent modifications of matrix rings of R.
2006-01-01T00:00:00ZKZ-monadic categories and their logicAdámek, JiříSousa, Lurdeshttp://hdl.handle.net/10316/438152018-05-22T00:06:08Z2017-01-01T00:00:00ZTitle: KZ-monadic categories and their logic
Authors: Adámek, Jiří; Sousa, Lurdes
Abstract: Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic for Kan-injectivity consequences and prove that in "reasonable" categories (such as Pos or Top_0) it is also complete for every set H of squares.
2017-01-01T00:00:00ZOn Final Coalgebras of Power-Set Functors and Saturated TreesAdámek, JiříLevy, Paul B.Milius, StefanMoss, Lawrence S.Sousa, Lurdeshttp://hdl.handle.net/10316/438102018-05-22T00:05:03Z2014-01-01T00:00:00ZTitle: On Final Coalgebras of Power-Set Functors and Saturated Trees
Authors: Adámek, Jiří; Levy, Paul B.; Milius, Stefan; Moss, Lawrence S.; Sousa, Lurdes
Abstract: The final coalgebra for the finite power-set functor was described by Worrell who also proved that the final chain converges in ω+ω steps. We describe the step ω as the set of saturated trees, a concept equivalent to the modally saturated trees introduced by K. Fine in the 1970s in his study of modal logic. And for the bounded power-set functors P_λ, where λ is an infinite regular cardinal, we prove that the construction needs precisely λ+ω steps. We also generalize Worrell’s result to M-labeled trees for a commutative monoid M, yielding a final coalgebra for the corresponding functor ℳ_f studied by H.-P. Gumm and T. Schröder. We describe the final chain of the power-set functor by introducing the concept of i-saturated tree for all ordinals i, and then prove that for i of cofinality ω, the i-th step in the final chain consists of all i-saturated, strongly extensional trees.
2014-01-01T00:00:00ZKan injectivity in order-enriched categoriesAdamek, JiriSousa, LurdesVelebil, Jirihttp://hdl.handle.net/10316/438872018-05-22T00:17:38Z2015-01-01T00:00:00ZTitle: Kan injectivity in order-enriched categories
Authors: Adamek, Jiri; Sousa, Lurdes; Velebil, Jiri
Abstract: Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1.
For every class H of morphisms, we study the subcategory of all objects that are Kan-injective with respect to H and all morphisms preserving Kan extensions. For categories such as Top_0 and Pos, we prove that whenever H is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings in Top_0 for which Kan-injectivity does not yield a monadic category.
2015-01-01T00:00:00ZWell-Pointed CoalgebrasAdámek, JiříMilius, StefanMoss, Lawrence SSousa, Lurdeshttp://hdl.handle.net/10316/438982018-05-22T00:52:42Z2013-08-09T00:00:00ZTitle: Well-Pointed Coalgebras
Authors: Adámek, Jiří; Milius, Stefan; Moss, Lawrence S; Sousa, Lurdes
Abstract: For endofunctors of varieties preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. The initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Osius and Taylor. And initial algebras are precisely the final well-founded coalgebras. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems.
2013-08-09T00:00:00ZOn functors which are lax epimorphismsAdámek, JiríBashir, Robert ElSobral, ManuelaVelebil, Jiríhttp://hdl.handle.net/10316/114602018-05-24T00:48:30Z2001-01-01T00:00:00ZTitle: On functors which are lax epimorphisms
Authors: Adámek, Jirí; Bashir, Robert El; Sobral, Manuela; Velebil, Jirí
Abstract: We show that lax epimorphisms in the category Cat are precisely the functors P : Ε → B for which the functor P* : [B, Set] → [E, Set] of composition with P is fully faithful. We present two other characterizations. Firstly, lax epimorphisms are precisely the ``absolutely dense'' functors, i.e., functors P such that every object B of B is an absolute colimit of all arrows P(E) → B for E in E. Secondly, lax epimorphisms are precisely the functors P such that for every morphism f of B the category of all factorizations through objects of P[E] is connected.
A relationship between pseudoepimorphisms and lax epimorphisms is discussed.
2001-01-01T00:00:00ZThe orthogonal subcategory problem and the small object argumentAdámek, JiríHébert, MichelSousa, Lurdeshttp://hdl.handle.net/10316/112772018-05-24T04:42:19Z2007-01-01T00:00:00ZTitle: The orthogonal subcategory problem and the small object argument
Authors: Adámek, Jirí; Hébert, Michel; Sousa, Lurdes
Abstract: A classical result of P. Freyd and M. Kelly states that in “good” categories,
the Orthogonal Subcategory Problem has a positive solution for all classes H
of morphisms whose members are, except possibly for a subset, epimorphisms. We
prove that under the same assumptions on the base category and on H, the generalization
of the Small Object Argument of D. Quillen holds - that is, every object of
the category has a cellular H-injective weak reflection. In locally presentable categories,
we prove a sharper result: a class of morphisms is called quasi-presentable if
for some cardinal ë every member of the class is either ë-presentable or an epimorphism.
Both the Orthogonal Subcategory Problem and the Small Object Argument
are valid for quasi-presentable classes. Surprisingly, in locally ranked categories
(used previously to generalize Quillen’s result), this is no longer true: we present a
class H of morphisms, all but one being epimorphisms, such that the orthogonality
subcategory H? is not reflective and the injectivity subcategory InjH is not weakly
reflective. We also prove that in locally presentable categories, the Injectivity Logic
and the Orthogonality Logic are complete for all quasi-presentable classes.
2007-01-01T00:00:00ZA logic of implications in algebra and coalgebraAdámek, JiríSobral, ManuelaSousa, Lurdeshttp://hdl.handle.net/10316/112992018-05-24T05:03:34Z2007-01-01T00:00:00ZTitle: A logic of implications in algebra and coalgebra
Authors: Adámek, Jirí; Sobral, Manuela; Sousa, Lurdes
Abstract: Implications in a category can be presented as epimorphisms: an ob-
ject satis¯es the implication i® it is injective w.r.t. that epimorphism. G. Ro»cu
formulated a logic for deriving an implication from other implications. We present
two versions of implicational logics: a general one and a ¯nitary one (for epimor-
phisms with ¯nitely presentable domains and codomains). In categories Alg § of
algebras on a given signature our logic specializes to the implicational logic of R.
Quackenbush. In categories Coalg H of coalgebras for a given accessible endofunctor
H of sets we derive a logic for implications in the sense of P. Gumm.
2007-01-01T00:00:00ZOn quasi-equations in locally presentable categories II: a logicAdámek, JiríSousa, Lurdeshttp://hdl.handle.net/10316/111772018-05-24T03:44:06Z2009-01-01T00:00:00ZTitle: On quasi-equations in locally presentable categories II: a logic
Authors: Adámek, Jirí; Sousa, Lurdes
Abstract: Quasi-equations given by parallel pairs of finitary morphisms represent
properties of objects: an object satisfies the property if its contravariant homfunctor
merges the parallel pair. Recently Ad´amek and H´ebert characterized subcategories
of locally finitely presentable categories specified by quasi-equations. We
now present a logic of quasi-equations close to Birkhoff’s classical equational logic.
We prove that it is complete in all locally finitely presentable categories with effective
equivalence relations.
2009-01-01T00:00:00ZLogic of implicationsAdámek, JiríSobral, ManuelaSousa, Lurdeshttp://hdl.handle.net/10316/113802018-05-23T23:54:08Z2005-01-01T00:00:00ZTitle: Logic of implications
Authors: Adámek, Jirí; Sobral, Manuela; Sousa, Lurdes
Abstract: A sound and complete logic for implications (or quasi-equations) is
presented, extending naturally Birkhoff’s equational logic. This is based on a general
logic for injectivity, following an idea of G. Ro¸su.
2005-01-01T00:00:00Z