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On quasi-equations in locally presentable categories II: a logic
Adámek, Jirí; Sousa, LurdesQuasi-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...
The orthogonal subcategory problem and the small object argument
Adámek, Jirí; Hébert, Michel; Sousa, LurdesA 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 logic of implications in algebra and coalgebra
Adámek, Jirí; Sobral, Manuela; Sousa, LurdesImplications 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 give...
Morita equivalence of many-sorted algebraic theories
Adámek, Jirí; Sobral, Manuela; Sousa, LurdesAlgebraic 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 obtaine...
Logic of implications
Adámek, Jirí; Sobral, Manuela; Sousa, LurdesA 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. ; Centre for Mathematics of University of Coimbra/ FCT; International Center for Mathematics; Ministry of Education of the Czech Republic, Project MSM 6840770014; School of Technology of Viseu
On functors which are lax epimorphisms
Adámek, Jirí; Bashir, Robert El; Sobral, Manuela; Velebil, JiríWe 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 epim...
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