## Algebraic theories of quasivarieties (1998)

Venue: | Journal of Algebra |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Porst98algebraictheories,

author = {Hans E. Porst},

title = {Algebraic theories of quasivarieties},

journal = {Journal of Algebra},

year = {1998},

volume = {208},

pages = {379--398}

}

### OpenURL

### Abstract

Analogously to the fact that Lawvere’s algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of many–sorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebraic theories of many–sorted Horn classes are precisely the small left exact categories with enough M–injectives, where M is a class of monomorphisms closed under finite products and containing all regular monomorphisms. We also present a Gabriel–Ulmer–type duality theory for quasivarieties and Horn classes. 1 Quasivarieties and Horn Classes The aim of the present paper is to describe, via algebraic theories, classes of finitary algebras, or finitary structures, which are presentable by implications. We work with finitary many–sorted algebras and structures, but we also mention the restricted version to the one–sorted case on the one hand, and the generalization to infinitary structures on the other hand. Recall that Lawvere’s thesis [11] states that Lawvere–theories of varieties, i.e., classes of algebras presented by equations, are precisely the small categories with finite products, (in the one sorted case moreover product–generated by a single object; for many–sorted varieties the analogous statement can be found in [4, 3.16, 3.17]). More in detail: If we denote, for small categories A, by P rodωA the full subcategory of Set A formed by all functors preserving finite products, we obtain the following: (i) If K is a variety, then its Lawvere–theory L(K), which is the full subcategory of K op of all finitely generated free K–algebras, is essentially small, and has finite products. The variety K is equivalent to P rodωL(K).

### Citations

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Locally presentable and accessible categories
- Adámek, Rosický
- 1994
(Show Context)
Citation Context ...y, basically the same characterization theorem holds in the many–sorted case; here only “generating object” has to be replaced by “generator”. The crucial equivalence of (i) and (iv) is formulated in =-=[4]-=- as Theorem 3.24. A similar result appears in [5] (Theorem 2.3) where, however, it is incorrectly claimed that every (finitary) quasivariety has a finite regular generator; the category Set A , where ... |

143 |
Handbook of Categorical Algebra I
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(Show Context)
Citation Context ... to natural isomorphism”) nor isomorphism dense (though it is “isomorphism dense up to equivalence of categories”). The proper setting for the duality is that of 2–categories and bifunctors (see e.g. =-=[6]-=- for basic notions). Recall from [16] that if K and L are 2–categories, then a 2–functor (or, more generally, a lax functor) R : K −→ L is called a biequivalence if (a) for each object X of L there ex... |

69 |
Lokal präsentierbare Kategorien
- Gabriel, Ulmer
- 1971
(Show Context)
Citation Context ...ch is the dual of the full subcategory Kfp of K formed by all finitely presentable algebras. Remark 1 The notion of theory can be introduced more generally, when recalling first some basic facts from =-=[8]-=- needed throughout this paper: (a) An object K of a category K is called finitely presentable provided that K(K, −) preserves directed colimits; more explicitely: given a directed colimit di (Di −→ D)... |

9 |
Subobjects, adequacy, completeness and categories of algebras
- Isbell
- 1964
(Show Context)
Citation Context ...ith or without minor modifications) tend to be reinvented now and again, we include a brief account. Apparently Isbell was the first one to characterize quasivarieties in categorical terms as follows =-=[9]-=-: A category K is equivalent to a quasivariety iff K satisfies the following conditions: • K is cocomplete and has equalizers, • K has an object P which is (i) extremally projective, (ii) extremally (... |

4 |
Models of Horn theories
- Barr
- 1989
(Show Context)
Citation Context ...lds in the many–sorted case; here only “generating object” has to be replaced by “generator”. The crucial equivalence of (i) and (iv) is formulated in [4] as Theorem 3.24. A similar result appears in =-=[5]-=- (Theorem 2.3) where, however, it is incorrectly claimed that every (finitary) quasivariety has a finite regular generator; the category Set A , where A is an infinite discrete category, is a countere... |

4 |
Abstract horn theories
- Keane
- 1975
(Show Context)
Citation Context ...uivalent to the theory of some one–sorted Horn class iff A is left exact and has an object I satisfying a. and b. for some left exact class M of monomorphisms. This is essentially the result of Keane =-=[10]-=- and follows by modifications of the arguments in the proof of Theorem 5 analogously as in Remark 6. Remark 10 The generalization to infinitary structures is straightforward: let λ be a regular cardin... |

2 |
Kennzeichnung von primitiven und quasi-primitiven Kategorien von Algebren
- Felscher
- 1968
(Show Context)
Citation Context ...) the somewhat weaker notion “abstractly finite” since he allowed for implications slightly more general than in (*) above. Basically the same result was obtained by Linton [12] and later by Felscher =-=[7]-=-. Their characterizations are essentially translations of the properties of P in Isbell’s theorem into properties of the associated hom–functor K(P, −), by weakening at the same time Isbell’s (co–)com... |

1 |
How to sketch quasivarieties
- Adámek
- 1996
(Show Context)
Citation Context ...mits. We will prove that algebraic theories of quasivarieties are precisely the small left exact categories with enough regular injectives, or, somewhat more explicitely (improving the main result of =-=[1]-=-) that the following hold: (i’) If K is a quasivariety, then T h(K) is essentially small, left exact, and has enough regular injectives. The quasivariety K is equivalent to LexT h(K). (ii’) If A is a ... |

1 |
Cocompleteness almost implies completeness
- Adámek, Herrlich, et al.
- 1989
(Show Context)
Citation Context ...K is a regular epimorphism. An equivalent formulation for (a) and (b) is: every object is an (extremal) quotient of a coproduct of G–objects. This simplification does not work for (c) in general, see =-=[2]-=-, but this does not matter in the realm of quasivarieties, due to the following Lemma 3 In any cocomplete category K with a generator P consisting of regularly projective objects, an epimorphism is re... |