×

zbMATH — the first resource for mathematics

Augmented virtual double categories. (English) Zbl 1440.18044
A. Burroni [Cah. Topologie Géom. Différ. Catégoriques 12, 215–321 (1971; Zbl 0246.18007)] generalized the notion of double category to that of virtual double category [G. S. H. Cruttwell and M. A. Shulman, Theory Appl. Categ. 24, 580–655 (2010; Zbl 1220.18003)], though Burroni himself called it multicatégory, in which cells have a horizontal multi-source and single horizontal target.
This paper introduces the notion of augmented virtual double category as a generalization of virtual double category by admitting cells to be of empty horizontal targets. The principal objective in introducing augmented virtual double categories is to internalize the notion of Yoneda embedding, just as Cruttwell and Shulman [loc. cit.] have internalized the notion of fully faithful morphism in the unital virtual double category \(\mathbb{M}\mathrm{od}(\mathbb{X})\) of modules in a virtual double category \(\mathbb{X}\) as well as E. Riehl and D. Verity [Algebr. Geom. Topol. 17, No. 1, 189–271 (2017; Zbl 1362.18020)] have internalized the notions of fully faithful morphism, exact square and pointwise Kan extension in the unital virtual double category \(\underline{\mathrm{Mod}}_{\mathcal{K}}\) of modules between \(\infty\)-categories in the homotopy \(2\)-category of a \(\infty\)-cosmos \(\mathcal{K}\).
MSC:
18N10 2-categories, bicategories, double categories
PDF BibTeX Cite
Full Text: Link
References:
[1] A. Burroni.T-cat´egories (Cat´egories dans un triple).Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, 12(3):215-321, 1971.
[2] A. Carboni, S. Kasangian, and R. Street. Bicategories of spans and relations. Journal of Pure and Applied Algebra, 33(3):259-267, 1984. · Zbl 0577.18005
[3] G. S. H. Cruttwell and M. A. Shulman. A unified framework for generalized multicategories.Theory and Applications of Categories, 24(21):580-655, 2010. · Zbl 1220.18003
[4] B. Day. On closed categories of functors. InReports of the Midwest Category Seminar IV, volume 137 ofLecture Notes in Mathematics, pages 1-38. SpringerVerlag, 1970.
[5] B. J. Day and S. Lack. Limits of small functors.Journal of Pure and Applied Algebra, 210(3):651-663, 2007. · Zbl 1120.18001
[6] R. J. MacG. Dawson, R. Par´e, and D. A. Pronk. Paths in double categories. Theory and Applications of Categories, 16(18):460-521, 2006. · Zbl 1120.18003
[7] C. Ehresmann. Cat´egories structur´ees: III. Quintettes et applications covariantes. InTopologie et G´eom´etrie Diff´erentielle: S´eminaire Ehresmann, volume 5, pages 1-21. Institut H. Poincar´e, 1963.
[8] P. Freyd and R. Street. On the size of categories.Theory and Applications of Categories, 1(9):174-178, 1995. · Zbl 0854.18003
[9] M. Grandis and R. Par´e. Limits in double categories.Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, 40(3):162-220, 1999.
[10] M. Grandis and R. Par´e. Adjoint for double categories.Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques, 45(3):193-240, 2004.
[11] M. Grandis and R. Par´e. Kan extensions in double categories (On weak double categories, Part III).Theory and Applications of Categories, 20(8):152-185, 2008. · Zbl 1141.18006
[12] C. Hermida.Representable multicategories.Advances in Mathematics, 151(2):164-225, 2000. · Zbl 0960.18004
[13] D. Hofmann, G. J. Seal, and W. Tholen, editors.Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, volume 153 ofEncyclopedia of Mathematics and Its Applications. Cambridge University Press, 2014. · Zbl 1297.18001
[14] G. B. Im and G. M. Kelly. A universal property of the convolution monoidal structure.Journal of Pure and Applied Algebra, 43(1):75-88, 1986. · Zbl 0604.18004
[15] G. M. Kelly.Basic Concepts of Enriched Category Theory, volume 64 ofLondon Mathematical Society Lecture Note Series. Cambridge University Press, 1982. · Zbl 0478.18005
[16] S. R. Koudenburg. On pointwise Kan extensions in double categories.Theory and Applications of Categories, 29(27):781-818, 2014. · Zbl 1442.18046
[17] S. R. Koudenburg. A double-dimensional approach to formal category theory. Draft, available asarXiv:1511.04070, November 2015.
[18] S. R. Koudenburg. A categorical approach to the maximum theorem.Journal of Pure and Applied Algebra, 222(8):2099-2142, 2018. · Zbl 1406.49005
[19] S. R. Koudenburg. Augmented virtual double categories as monoids in skew monoidal categories. In preparation, 2019.
[20] F. W. Lawvere. Metric spaces, generalized logic, and closed categories.Rendiconti del Seminario Mat´ematico e Fisico di Milano, 43:135-166, 1973.
[21] T. Leinster.Higher Operads, Higher Categories, volume 298 ofLondon Mathematical Society Lecture Note Series. Cambridge University Press, 2004. · Zbl 1160.18001
[22] S. Mac Lane.Categories for the Working Mathematician, volume 5 ofGraduate Texts in Mathematics. Springer, second edition, 1998. · Zbl 0906.18001
[23] C. Pisani. Sequential multicategories.Theory and Applications of Categories, 29(19):496-541, 2014. · Zbl 1305.18027
[24] E. Riehl and D. Verity. Kan extensions and the calculus of modules for∞categories.Algebraic & Geometric Topology, 17(1):189-271, 2017. · Zbl 1362.18020
[25] M. Shulman. Framed bicategories and monoidal fibrations.Theory and Applications of Categories, 20(18):650-738, 2008. · Zbl 1192.18005
[26] R. Street. Fibrations and Yoneda’s lemma in a 2-category. InProceedings of the Sydney Category Theory Seminar 1972/1973, volume 420 ofLecture Notes in Mathematics, pages 104-133. Springer-Verlag, 1974.
[27] R. Street and R. Walters. Yoneda structures on 2-categories.Journal of Algebra, 50(2):350-379, 1978. · Zbl 0401.18004
[28] K. Szlach´anyi. Skew-monoidal categories and bialgebroids.Advances in Mathematics, 231(3-4):1694-1730, 2012. · Zbl 1283.18006
[29] C. Walker. Yoneda structures and KZ doctrines.Journal of Pure and Applied Algebra, 222(6):1375-1387, 2018. · Zbl 1382.18001
[30] M. Weber. Yoneda structures from 2-toposes.Applied Categorical Structures, 15(3):259-323, 2007. · Zbl 1125.18001
[31] R. J. Wood.Abstract proarrows I.Cahiers de Topologie et G´eom´etrie Diff´erentielle, 23(3):279-290, 1982.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.