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

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.)

Zbl 0225.18004
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### References:

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