![]() Rosický (to appear) as an equational hull of the 2-category LFP (of locally finitely presentable categories), are characterized by the above properties (2) and (3). Analogously, precontinuous categories, introduced in J. This can be viewed as a nonadditive generalization of the classical Roos Theorem characterizing essential localizations of categories of modules. (2) commutativity of filtered colimits with finite limits, (3) distributivity of filtered colimits over arbitrary products, and (4) product-stability of regular epimorphisms. ![]() We will show that algebraically exact categories with a regular generator are precisely the essential localizations of varieties and that, in this case, algebraic exactness is equivalent to (1) exactness. Rosický (to appear), as an equational hull of the 2-category VAR of all varieties of finitary algebras. This result is new, even in the case where is empty and -filtered colimits are just arbitrary (small) colimits.Īlgebraically exact categories have been introduced in J. When consists of the finite discrete categories, these are the finitary varieties.As a by-product of this theory, we prove that the free completion under -filtered colimits distributes over the free completion under limits. When consists of the finite categories, these are precisely the locally finitely presentable categories of Gabriel and Ulmer. A surprising number of the main results from the theory of accessible categories remain valid in the -accessible context.The locally -presentable categories are defined as the cocomplete -accessible categories. Every -accessible category is accessible thus the choice of different sound provides a classification of accessible categories, as referred to in the title. The -accessible categories are then the categories with -filtered colimits and a small set of -presentable objects which is “dense with respect to -filtered colimits”.We suppose always that satisfies a technical condition called “soundness”: this is the “suitable” case mentioned above. An object of a category is called -presentable when the corresponding representable functor preserves -filtered colimits. A small category is called -filtered when -colimits commute with -limits in the category of sets. KGĬhristian Dzierzon Number of pages: 152 Published on: Stock: Available Category: Mathematics Price: 59.For a suitable collection of small categories, we define the -accessible categories, generalizing the λ-accessible categories of Lair, Makkai, and Paré here the λ-accessible categories are seen as the -accessible categories where consists of the λ-small categories. Moreover, polynomial set-endofunctors are characterized by the property that the corresponding category of coalgebras is concretely equivalent to some presheaf category. A non-trivial example is given by the category of coalgebras for a polynomial set-endofunctor, which turns out to be equivalent to some variety of unary algebras without equations. In addition to that it presents a new approach to the known characterization of quasivarieties. This thesis provides an intuitive proof of the mentioned fact, which covers existing examples, and can be generalized to the non-finitary case under mild assumptions. Unfortunately, the existing approaches in literature are either unsatisfactory - with respect to existing examples and to the number of sorts needed - or even wrong. The fact, that a category is locally finitely presentable iff it is equivalent to the category of models of some essentially algebraic, finitary theory, is widely known. ![]() Eligible for voucher ISBN-13: 978-3-8364-6416-1 ISBN-10: 3836464160 EAN: 9783836464161 Book language:īlurb/Shorttext: Local presentability has turned out to be one of the most fruitful concepts in category theory.
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