×

A classification of accessible categories. (English) Zbl 1010.18005

Let \(\mathbb{D}\) be a collection of small categories then the authors say that a (small) category \(\mathcal C\) is \(\mathbb{D}\)-filtered if \(\mathcal C\)-colimits commute in \(\mathcal{S}et\) with \(\mathbb{D}\)-limits. If a category \(\mathcal K\) has \(\mathbb{D}\)-filtered colimits then an object \( A\) of \(\mathcal K\) is called \(\mathbb{D}\)-presentable if the hom-functor \(\mathcal K(A,-):\mathcal K \to \mathcal{S}et\) preserves \(\mathbb{D}\)-filtered colimits. The authors say that a collection \(\mathbb{D}\) is sound if any category \(\mathcal C\) is \(\mathbb{D}\)-filtered whenever the category of cocones for any functor \(S:\mathcal D^{op}\to \mathcal C\) with \(\mathcal D\in \mathbb{D}\) is connected. For a sound collection \(\mathbb{D}\) a characterization of \(\mathbb{D}\)-continuous functors into \(\mathcal{S}et\) is given. A category \(\mathcal K\) is said to be \(\mathbb{D}\)-accessible if \(\mathbb{D}\) is sound, \(\mathcal K\) has \(\mathbb{D}\)-filtered colimits and there is a small set \(\mathcal A\) of \(\mathbb{D}\)-presentable objects of \(\mathcal K\) such that every object of \( \mathcal K\) is a \(\mathbb{D}\)-filtered colimit of objects from \(\mathcal A\).
The paper develops a theory of \(\mathbb{D}\)-accessible categories that refines a theory of accessible (and locally presentable) categories. Since any \(\mathbb{D}\)-accessible category is also accessible, \(\mathbb{D}\)-accessible categories give a classification of accessible categories. The relation between \(\mathbb{D}\)-accessible categories and sketches is discussed. It is proved that the free completion under colimits in the quasi-category \(\mathcal{C}at\) of all categories distributes over free completion under limits in \(\mathcal{C}at\). Many examples illustrating the new concept are given.

MSC:

18C35 Accessible and locally presentable categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18C30 Sketches and generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Adàmek, F.W. Lawvere, J. Rosický, A duality between varieties and algebraic theories, Alg. Univ., to appear.; J. Adàmek, F.W. Lawvere, J. Rosický, A duality between varieties and algebraic theories, Alg. Univ., to appear.
[2] J. Adàmek, F.W. Lawvere, J. Rosický, Continuous categories revisited, submitted for publication.; J. Adàmek, F.W. Lawvere, J. Rosický, Continuous categories revisited, submitted for publication.
[3] Adàmek, J.; Lawvere, F. W.; Rosický, J., How algebraic is algebra?, Theory Appl. Categ., 8, 253-283 (2001) · Zbl 0978.18006
[4] Adàmek, J.; Koubek, V.; Velebil, J., A duality between infinitary varieties and algebraic theories, Comment. Math. Univ. Carol., 41, 529-541 (2000) · Zbl 1035.08004
[5] Adàmek, J.; Rosický, J., Locally Presentable and Accessible Categories (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0795.18007
[6] Adàmek, J.; Rosický, J., On sifted colimits and generalized varieties, Theory Appl. Categ., 8, 33-53 (2001) · Zbl 0971.18004
[7] Adàmek, J.; Rosický, J., Finitary sketches and finitely accessible categories, Math. Str. Comput. Sci., 5, 315-322 (1995) · Zbl 0838.18001
[8] Ageron, P., Limites inductives point par point dans les catégories accessibles, Theory Appl. Categ., 8, 313-323 (2001) · Zbl 0983.18005
[9] Artin, M.; Grothendieck, A.; Verdier, J. L., Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, Vol. 269 (1972), Springer: Springer Berlin · Zbl 0234.00007
[10] Carboni, A.; Johnstone, P., Connected limits, familial representability and Artin glueing, Math. Str. Comput. Sci., 5, 441-459 (1995) · Zbl 0849.18002
[11] Diers, Y., Catégories localement multiprésentables, Arch. Math., 34, 344-356 (1980) · Zbl 0432.18006
[12] Foltz, F., Sur la commutation de limites, Diagrammes, 5, F1-F33 (1981)
[13] P. Freyd, Several new concepts: lucid and concordant functors, pre-limits, pre-cocompleteness, the continuous and concordant completion of categories, Lecture Notes in Mathematics, Vol. 99, Springer, Berlin, 1969, pp. 196-241.; P. Freyd, Several new concepts: lucid and concordant functors, pre-limits, pre-cocompleteness, the continuous and concordant completion of categories, Lecture Notes in Mathematics, Vol. 99, Springer, Berlin, 1969, pp. 196-241. · Zbl 0225.18006
[14] Gabriel, P.; Ulmer, F., Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, Vol. 221 (1971), Springer: Springer Berlin · Zbl 0225.18004
[15] Hu, H., Dualities for accessible categories, Canad. Math. Soc. Conf. Proc., 13, 211-242 (1992) · Zbl 0792.18004
[16] Kelly, G. M., Basic Concepts of Enriched Categories, LMS Lecture Note Series, Vol. 64 (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0709.18501
[17] Lair, C., Catégories modelables et catégories esquissables, Diagrammes, 6, 1-20 (1981) · Zbl 0522.18008
[18] Lair, C., Sur le genre d’esquissabilité des catégories modelables (accessibles) possédant les produits de deux, Diagrammes, 35, 25-52 (1996) · Zbl 0899.18003
[19] Linder, H., Morita equivalences of enriched categories, Cahiers Topol. Géom. Diff., 15, 377-397 (1974) · Zbl 0319.18006
[20] M. Makkai, R. Paré, Accessible Categories, Contemporary Mathematics, Vol. 104, 1989, American Mathematical Society, Providence, RI, 1989.; M. Makkai, R. Paré, Accessible Categories, Contemporary Mathematics, Vol. 104, 1989, American Mathematical Society, Providence, RI, 1989.
[21] Marmolejo, F., Doctrines whose structures form a fully faithful adjoint string, Theory Appl. Categories, 3, 24-44 (1997) · Zbl 0878.18004
[22] Marmolejo, F., Distributive laws for pseudomonads, Theory Appl. Categories, 5, 91-147 (1999) · Zbl 0919.18004
[23] F. Marmolejo, Distributive laws for pseudomonads II, preprint.; F. Marmolejo, Distributive laws for pseudomonads II, preprint. · Zbl 1055.18002
[24] M.C. Pedicchio, R. Wood, A simple characterization of theories of varieties, to appear.; M.C. Pedicchio, R. Wood, A simple characterization of theories of varieties, to appear. · Zbl 0989.18006
[25] Street, R., Fibrations in bicategories, Cahiers Topol. Géom. Diff., 21, 111-160 (1980) · Zbl 0436.18005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.