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Grothendieck quantaloids for allegories of enriched categories. (English) Zbl 1285.18014
A quantaloid $$\mathcal{Q}$$ is a category which is enriched in the symmetric monoidal closed category Sup of $$\bigvee$$-semilattices and $$\bigvee$$-preserving maps [I. Stubbe, Theory Appl. Categ. 14, 1–45 (2005; Zbl 1079.18005)]. In [R. F. C. Walters, J. Pure Appl. Algebra 24, 95–102 (1982; Zbl 0497.18016)], it is shown that the topos of sheaves on a small site $$(\mathbf{C},\mathcal{J})$$ is equivalent to the category of Cauchy-complete symmetric categories enriched in the (small) quantaloid of closed cribles $$\mathcal{R}(\mathbf{C},\mathcal{J})$$. The present paper refines the just mentioned result (Theorem 3.14 on page 879), employing the axiomatic description of $$\mathcal{R}(\mathbf{C},\mathcal{J})$$ given in [H. Heymans and I. Stubbe, J. Pure Appl. Algebra 216, No. 8–9, 1952–1960 (2012; Zbl 1279.18007)]. The authors, moreover, provide its further generalization, showing that given an involutive quantaloid $$\mathcal{Q}$$, the quantaloid $$\mathbf{Rel}(\mathcal{Q})$$ of $$\mathcal{Q}$$-sheaves (and relations) is equivalent to the quantaloid $$\mathbf{Rel}(\mathcal{T})$$ of internal relations in a Grothendieck topos $$\mathcal{T}$$ if and only if $$\mathcal{Q}$$ is a modular, locally localic, and weakly semi-simple quantaloid, i.e., $$\mathcal{Q}$$ is the so-called (by the authors) Grothendieck quantaloid (Theorem 4.7 on page 884). One can think then of Grothendieck quantaloids as those for which $$\mathcal{Q}$$-sheaves and relations form an “interesting allegory”. Recall (from, e.g., [P. J. Freyd and A. Scedrov, Categories, allegories. Amsterdam etc.: North-Holland (1990; Zbl 0698.18002)]) that an allegory A is a modular locally ordered 2-category the hom-posets of which have binary intersections. Taking left adjoints (also known as “maps”) in an allegory A, provides a category $$\mathrm{Map}(\mathbf{A})$$. One is interested in the case when the category $$\mathrm{Map}(\mathbf{A})$$ is a topos. In one word, “allegories $$\ldots$$ are to binary relations between sets as categories are to functions between sets”. Additionally, the authors of the current paper present an analogue of their achievement, replacing Grothendieck quantaloids with Grothendieck quantales (Corollary 4.10 on page 887), illustrating both of them with a number of examples (Examples 4.11–4.13 on pages 887–889).
The paper is well written (just a couple of typos) and reasonably self-contained (most of the preliminaries are contained in Section 2), but requires from its reader a certain background on quantaloid-enriched categories.

##### MSC:
 18D20 Enriched categories (over closed or monoidal categories) 06F07 Quantales 18B10 Categories of spans/cospans, relations, or partial maps 18B25 Topoi 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 18F10 Grothendieck topologies and Grothendieck topoi
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