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Barr's Embedding Theorem for Enriched Categories
Abstract
We generalize Barr's embedding theorem for regular categories to the context of enriched categories. Comment: 11 pages
... Our aim is to extend the other three theorems, finding a common setting that includes both the ordinary and the additive context. Note that an enriched version of Barr's Embedding Theorem has already been considered in [10], but the notion of regularity appearing there is more restrictive than ours: see Remark 5.2. ...
... Remark 5.2. A different notion of regularity appeared before in [10]; there, in a regular V-category, regular epimorphisms need to be stable under all finite powers, instead of just finite projective ones like in our case. At the same time the base for enrichment can be assumed to be only locally finitely presentable as a closed category, and one can still prove the analogue of 7.3. ...
... At the same time the base for enrichment can be assumed to be only locally finitely presentable as a closed category, and one can still prove the analogue of 7.3. We chose to consider a different approach to recover the usual notions of regularity and exactness for V = Ab; in fact Ab itself is not regular as an additive category in the sense of [10], but it is regular in our sense. ...
Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosick\'y, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosick\'y.
... This resulted in the notion of a regular category [58,145,17], which has sparked considerable amounts of attention (e.g. [99,98] and [42,34] in the enriched setting). In this setting the homsets are partially ordered [104] like in the standard category of sets and binary relations. ...
A dagger category is a category equipped with a functorial way of reversing morphisms, i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories with additional structure have been studied under different names e.g. in categorical quantum mechanics and algebraic field theory. In this thesis we study the dagger in its own right and show how basic category theory adapts to dagger categories. We develop a notion of a dagger limit that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isomorphism; a wide class of dagger limits can be built from a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor; dagger limits can be built from ordinary limits in the presence of polar decomposition; dagger limits commute with dagger colimits in many cases. Using cofree dagger categories, the theory of dagger limits can be leveraged to provide an enrichment-free understanding of limit-colimit coincidences in ordinary category theory. We formalize the concept of an ambilimit, and show that it captures known cases. As a special case, we show how to define biproducts up to isomorphism in an arbitrary category without assuming any enrichment. Moreover, the limit-colimit coincidence from domain theory can be generalized to the unenriched setting and we show that, under suitable assumptions, a wide class of endofunctors has canonical fixed points. The theory of monads on dagger categories works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-Eilenberg-Moore algebras, which again have a dagger.
Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosický, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Working over a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosický.
The localisations of locally finitely presentable categories are characterised as those categories which admit small colimits, finite limits and a small strong generator, and have filtered colimits commuting with finite limits. Moreover, this is done in the context of enriched categories.
Let C be a small Barr-exact category, Reg(C, Set) the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e:C→[Reg(C, Set), Set] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and filtered colimits. The concept of κ-Barr-exact category is introduced, for κ any infinite regular cardinal, and the natural generalization to κ-Barr-exact categories of the above result is proved. The treatment combines methods of model theory and category theory. Some applications to module categories are given.
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