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Abstract
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.
... In the literature, many so-called "exactness properties" have been shown to be stable under this construction: if C satisfies the given property, so does Lex(C, Set) op . Among examples of such properties are the following (in each case, the cited reference is where the corresponding "stability" result was first established): being regular [6], coregular [34], additive [35], abelian [35], exact Mal'tsev with pushouts [11], coregular co-Mal'tsev [45], coextensive with pushouts [29], and extensive [29]. We prove in this paper a general stability theorem, which includes all of the above examples and establishes stability of other fundamental exactness properties, such as being semi-abelian, regular Mal'tsev, coherent with finite coproducts, and many more. ...
... The generality brings in heavy technicalities; these we have tackled using 2-categorical calculus of natural transformations. As it can be expected, we use a generalization of the set-based case of a lemma from [34] called the "uniformity lemma" (see also Lemma 5.1 in [73]); its detailed proof forms, in fact, a substantial part of the proof of our general stability theorem. And of course, we rely on classical results about pro-completion found in [5,39]. ...
... Subsection 1.9 provides many examples of exactness properties to which Theorems 2.2 and 2.3 can be applied. As mentioned in the Introduction, several particular instances of these theorems can already be found in the literature for particular exactness properties of a category, see [6,11,29,34,35,45]. The property of being cartesian closed has also been proved [35] to transfer from a small finitely cocomplete category C to Lex(C op , Set), but we have not been able to deduce this fact from Theorem 2.3, which perhaps suggests that our theorem could be further generalized. ...
In this paper we formulate and prove a general theorem of stability of exactness properties under the pro-completion, which unifies several such theorems in the literature and gives many more. The theorem depends on a formal approach to exactness properties proposed in this paper, which is based on the theory of sketches. Our stability theorem has applications in proving theorems that establish links between exactness properties, as well as in establishing embedding (representation) theorems for categories defined by exactness properties.
... We adopt a notion of regularity for enriched categories suitable for our setting. Regular enriched categories have been considered before by B. Day, R. Street in [6]. ...
... Let us prove that the pushout of a regular mono is a regular mono. Lemma in [6] proves that each diagram in L the indexing category of which has finite homsets can be written as a filtered colimit of diagrams landing in the subcategory of representables. An instance of this lemma is that each pushout diagram in L can be written as a colimit of representable pushout diagrams. ...
... By the Lemma in [6], D is filtered and ...
We generalize Barr's embedding theorem for regular categories to the context of enriched categories. Comment: 11 pages
... In recent years there has been a considerable interest in expressing properties of an essentially algebraic category (i.e. a locally finitely presentable category) in terms of properties of the corresponding essentially algebraic theory. From this point of view, complete answers have been given with respect to basic properties such as regularity, exactness, extensivity , cartesian closedness and so on (see [8], [9], [4] and [7]). In this note we analyse the condition of n-permutability of the composition of equivalence relations for a regular locally finitely presentable category K = Lex[C op , Set]. ...
... The dual category of C is called the essentially algebraic theory, while Lex[C op , Set] is the category of models of the theory. Many properties of a locally finitely presentable category can be expressed just in terms of its essentially algebraic theory C and various results in this direction can be found in the literature, for instance in [7], [8],[9] and [4]. In this paper we are interested in the property of n-permutability of the composition of the equivalence relations; the regularity of Lex[C op , Set] will be always required in order to express this kind of property. ...
. We characterize n-permutable locally finitely presentable categories Lex[C op ; Set] by a condition on the dual of the essentially algebraic theory C op . We apply these results to exact Maltsev categories as well as to n-permutable quasivarieties and varieties. Introduction In recent years there has been a considerable interest in expressing properties of an essentially algebraic category (i.e. a locally finitely presentable category) in terms of properties of the corresponding essentially algebraic theory. From this point of view, complete answers have been given with respect to basic properties such as regularity, exactness, extensivity, cartesian closedness and so on (see [8], [9], [4] and [7]). In this note we analyse the condition of n-permutability of the composition of equivalence relations for a regular locally finitely presentable category K = Lex[C op ; Set]. It is known [5] that n-permutability can be equivalently stated by saying that, for any reflexive relation...
... 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.
... In order for a functor F : A → B between locally finitely presentable categories that preserves filtered colimits to be left exact it is enough to preserve kernels of arrows between finitely presentable objects. This is a consequence of the distributivity of filtered colimits over finite limits in any locally finitely presentable category, and a short proof can be produced by using the uniformity lemma of [7]. OPEZ FRANCO 2.2. ...
The purpose of this article is to study the existence of Deligne's tensor
product of abelian categories by comparing it with the well-known ten- sor
product of finitely cocomplete categories. The main result states that the
former exists precisely when the latter is an abelian category, and moreover in
this case both tensor products coincide. An example of two abelian categories
whose Deligne tensor product does not exist is given.
In this paper we formulate and prove a general theorem of stability of exactness properties under the pro-completion, which unifies several such theorems in the literature and gives many more. The theorem depends on a formal approach to exactness properties proposed in this paper, which is based on the theory of sketches. Our stability theorem has applications in proving theorems that establish links between exactness properties, as well as in establishing embedding (representation) theorems for classes of categories defined by exactness properties.
Given a small exact category E with finite colimits, we prove that the category Lex(E) of left exact presheaves on E is exact precisely when in E, the equivalence relation generated by a reflexive symmetric relation R is a finite iterate of R. This is in particular the case when E is Noetherian, that is, every ascending chain of subobjects is stationary. When this condition is satisfied and moreover E is a pretopos, Lex(E) becomes a topos. Various examples are given, distinguishing the possible situations.
Localisations of a locally finitely presentable category A are shown to all arise from Grothendieck topologies on the full subcategory C of finitely presentable objects. Twelve further good structural aspects of C are shown to transfer to A.
Algebraically exact categories have been introduced in J. Adámek, F. W. Lawvere, and J. Rosický (to appear), as an equational hull of the 2-category VAR of all varieties of finitary algebras. 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, (2) commutativity of filtered colimits with finite limits, (3) distributivity of filtered colimits over arbitrary products, and (4) product-stability of regular epimorphisms. This can be viewed as a nonadditive generalization of the classical Roos Theorem characterizing essential localizations of categories of modules. Analogously, precontinuous categories, introduced in J. Adámek, F. W. Lawrence, and J. 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). Essential localizations of locally finitely presentable categories and presheaf categories are fully described.
An abstract approach is made to recent theory on “Tannaka” recovery of coalgebras, bialgebras, and Hopf algebras. It is hoped that the methods used shed some light upon the more quantitative aspects of the mathematics involved.
This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category.
Localisations of a geometric category
- Borceux
F. Borceux, Localisations
of a geometric category,
Preprint No. 71, June 1985, Inst. Math. Pure
- P Gabriel
- F Ulmer
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P. Gabriel and F. Ulmer, Lokal prasentierbare
Kategorien,
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(Springer,
Berlin, 1971).