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Gabriel–Ulmer Duality for Topoi and its Relation with Site Presentations

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Abstract

Let \(\kappa \) be a regular cardinal. We study Gabriel–Ulmer duality when one restricts the 2-category of locally \(\kappa \)-presentable categories with \(\kappa \)-accessible right adjoints to its locally full sub-2-category of \(\kappa \)-presentable Grothendieck topoi with geometric \(\kappa \)-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of \(\kappa \)-small cocomplete categories with \(\kappa \)-small colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally \(\kappa \)-presentable Grothendieck topoi with geometric \(\kappa \)-accessible morphisms is a reflective sub-bicategory of the 2-category of weakly \(\kappa \)-ary sites [in the sense of Shulman (Theory Appl Categ 27:97–173, 2012)] with morphisms of sites.

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Notes

  1. Observe that the passage to the conjugate category \(\mathsf {Rex}^{\text {co}}\) is needed just because \( [\mathsf {C}^{\circ },\mathsf {D}^{\circ }] \cong [\mathsf {C},\mathsf {D}]^{\circ }\).

  2. Observe that actually more is true. If \(\kappa > \aleph _0\) the equivalence holds for every category \(\mathsf {I}\) with cardinality smaller than \(\kappa \). Nonetheless, for our purposes is enough to consider \(\mathsf {I}\) with a finite set of objects.

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Acknowledgements

The authors are very grateful to the organisers of the summer school and conference Toposes in Como (Universitá degli Studi dell’Insubria - Como, June 2018) during which the initial discussions that led to this paper took place. We would also like to thank an anonymous referee for the useful comments. The first author is also indebted with Jiří Rosický for pointing out some very useful references. The second author would like to thank Jens Hemelaer for very interesting discussions on topos theory and in particular for his help in fully understanding the embedding \({\text {Ind}}_{\kappa }(\mathsf {C}) \rightarrow \mathsf {C}^{\circ }\)-\({\text {Alg}}\), which appears in the proof of Lemma 3.7. She is also grateful to Wendy Lowen for the reading of this manuscript and for her useful suggestions, as well as for pointing out reference [10]. Finally, she would like to thank the organisers of the Séminaire de théorie des catégories (Université catholique de Louvain, Université libre de Bruxelles and Vrije Universiteit Brussel) for giving her the opportunity to present this work during the seminar, which resulted in an improvement of the presentation of the paper.

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, Prague, 110 00, Czech Republic

    Ivan Di Liberti

  2. Department of Mathematics, Faculty of Sciences, University of Antwerp, Middelheimcampus, Middelheimlaan 1, 2020, Antwerp, Belgium

    Julia Ramos González

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  1. Ivan Di Liberti
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Correspondence to Julia Ramos González.

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I. Di Liberti: The first named author is supported by the grant 19-00902S from the Grant Agency of the Czech Republic. J. Ramos González: The second named author is a Postdoctoral Fellow of the Research Foundation - Flanders (FWO) under Grant No. 12T2619N and acknowledges the support of the Research Foundation - Flanders (FWO) under Grant No. G.0D86.16N during the first months of working on this paper.

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Di Liberti, I., Ramos González, J. Gabriel–Ulmer Duality for Topoi and its Relation with Site Presentations. Appl Categor Struct 28, 935–962 (2020). https://doi.org/10.1007/s10485-020-09605-x

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