Structures defined by finite limits in the enriched context. I. (English) Zbl 0538.18006

Kelly, G. M.

Cah. Topol. Géom. Différ. 23, 3-42 (1982).

In this article the author gives an excellent concise report how to treat locally finitely presentable (l.f.p.) categories in the sense of P. Gabriel and F. Ulmer [Lokal präsentierbare Kategorien, Lect. Notes Math. 221 (1971; Zbl 0225.18004)] in the context of enriched category theory. He points out that the major part of the set-based results can be carried over to \({\mathfrak V}\)-categories under the hypothesis that \({\mathfrak V}\) is l.f.p. as a (symmetric monoidal) closed category, that is, the underlying ordinary category \({\mathfrak V}_ o\) is l.f.p. and its subcategory \({\mathfrak V}_{of}\) of finitely presentable (f.p.) objects is closed under the monoidal structure in the sense that \(I\in {\mathfrak V}_{of}\) and \(x\otimes y\in {\mathfrak V}_{of}\) when \(x,y\in {\mathfrak V}_{of}.\) An object G of a \({\mathfrak V}\)-category \({\mathfrak A}\) is f.p. if the \({\mathfrak V}\)-functor \({\mathfrak A}(G,-):{\mathfrak A}\to {\mathfrak V}\) is finitary, that is, preserves (small) filtered colimits; let \({\mathfrak A}_ f\) be the full subcategory of f.p. objects. \({\mathfrak A}\) is said to be l.f.p. if it is cocomplete and if \({\mathfrak A}_ f\) contains a strong generator. It turns out that for \({\mathfrak A}\) cocomplete the following are equivalent: (i) \({\mathfrak A}\) is l.f.p., (ii) \({\mathfrak A}\) is a full reflective subcategory of some [\({\mathfrak T,V}]\) with \({\mathfrak T}\) small and with the inclusion \({\mathfrak A}\to [{\mathfrak T,V}]\) finitary, (iii) \({\mathfrak A}_ o\) is l.f.p. and \({\mathfrak A}_{of}={\mathfrak A}_{fo}\). In the representation (ii), \({\mathfrak T}\) can be taken as \({\mathfrak A}_ f^{op}\) and then, \({\mathfrak A}\) is equivalent to the subcategory \(Lex[{\mathfrak A}_ f^{op},{\mathfrak V}]\) of left exact functors \({\mathfrak A}_ f^{op}\to {\mathfrak V}\). This gives the interpretation of \({\mathfrak A}\) as the category \({\mathfrak T}-Alg\) of the models in \({\mathfrak V}\) of a finitary essentially algebraic \({\mathfrak V}\)-theory \({\mathfrak T}\), that is, a small finitely complete \({\mathfrak V}\)-category \({\mathfrak T}\). For such a theory \({\mathfrak T}\), any functor S:\({\mathfrak T}\)-Al\(g\to {\mathfrak B}\) into a cocomplete category \({\mathfrak B}\) is left adjoint if and only if it is cocontinuous (attention: there is an obvious misprint in the author’s Theorem 9).
The paper is rather self-contained and gives a lot of improvements even for the classical set-based situation. In 8.6 the author gives an elegant proof of the Freyd-Isbell-Gabriel-Ulmer result that a (set-based) locally presentable category is co-wellpowered.

MSC:

18D20	Enriched categories (over closed or monoidal categories)
18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
18A35	Categories admitting limits (complete categories), functors preserving limits, completions
18C10	Theories (e.g., algebraic theories), structure, and semantics
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

Keywords:

locally finitely presentable categories; symmetric monoidal closed category; finitary algebraic theory; enriched category theory; cocomplete; strong generator; locally presentable category; co- wellpowered

Citations:

Zbl 0225.18004

References:

[1]	1 A. ( Bastiani ) and C. Ehresmann , Categories of sketched structures , Cahiers Topo. et Géom. Diff. 13 ( 1972 ), 103 - 214 . Numdam | MR 323856 | Zbl 0263.18009 · Zbl 0263.18009
[2]	2 J.B. Enabou , Structures algébriques dans les catégories , Cahiers Topo. et Géom. Diff. 10 ( 1968 ), 1 - 126 . Numdam | MR 244335 | Zbl 0162.32602 · Zbl 0162.32602
[3]	3 M. Coste , Localisation dans les catégories de modèles , Thèse, Université Pari s-Nord , 1977 , 4 Y. Diers , Type de densité d’une sous-catégorie pleine , Ann. Soc. Scient. Bruxelles 90 ( 1976 ), 25 - 47 . MR 396716 | Zbl 0318.18005 · Zbl 0318.18005
[4]	5 C. Ehresmann , Esquisses et types de structures algébriques , Bul. Inst. Polit. Iaşi 14 ( 1968 ), 1 - 14 . MR 238918 | Zbl 0196.03102 · Zbl 0196.03102
[5]	6 P.J. Freyd , Aspects of topoi , Bull. Austral. Math. Soc. 7 ( 1976 ), 1 - 76 and 467 - 480 . Zbl 0252.18001 · Zbl 0252.18001
[6]	7 P. Gabriel and F. Ulmer , Lokal präsentierbare Kategorien , Lecture Notes in Math. 221 , Springer ( 1971 ). MR 327863 | Zbl 0225.18004 · Zbl 0225.18004
[7]	8 G.M. Kelly , Monomorphisms, epimorphisms, and pullbacks , J. Austral. Math. Soc. 9 ( 1969 ), 124 - 142 . MR 240161 | Zbl 0169.32604 · Zbl 0169.32604
[8]	9 G.M. Kelly , A unified treatment of transfinite constructions , Bull. Austral. Math. Soc. 22 ( 1980 ), 1 - 83 . MR 589937 | Zbl 0437.18004 · Zbl 0437.18004
[9]	G.M. Kelly , The basic concepts of enriched category theory , Cambridge Univ. Press , 1982 . MR 651714 | Zbl 0478.18005 · Zbl 0478.18005
[10]	10. G.M. Kelly , Categories with structure - biadjoints for algebraic functors (to appear).
[11]	11 F.W. Lawvere , Functorial semantics of algebraic theories , Proc. Nat. Acad. Sci. USA 50 ( 1963 ), 869 - 872 . MR 158921 | Zbl 0119.25901 · Zbl 0119.25901
[12]	12 F.E.J. Linton , Autonomous equational categories , J. Math. Mech. 15 ( 1966 ), 637 - 642 . MR 190205 | Zbl 0146.25104 · Zbl 0146.25104
[13]	13 M. Makkai and G.E. Reyes , First Order Categorical Logic , Lecture Notes in Math. 611 , Springer ( 1977 ). MR 505486 | Zbl 0357.18002 · Zbl 0357.18002