On lifting of biadjoints and lax algebras

Document Type: Research Paper

Author

CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.

Abstract

Given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give  general results on lifting of biadjoints. As a consequence, we get a biadjoint triangle theorem which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. In the context of lax algebras, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} _\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A}\to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study descent objects and lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.

Keywords


[1] Barr, M., The point of the empty set, Cah. Topol. Geom. Differ. Categ. 13 (1972), 357-368.
[2] Barr, M. and Wells, C., “Toposes, Triples and Theories”, Repr. Theory Appl. Categ. 21 (2005), 1-288.
[3] Benabou, J., Introduction to bicategories, Lecture Notes in Math. 47 (1967), 1-77.
[4] Bird, G.J., Kelly, G.M., Power, A.J., and Street, R.H., Flexible limits for 2-categories, J. Pure Appl. Algebra 61(1) (1989), 1-27.
[5] Blackwell, R., Kelly, G.M., and Power, A.J., Two-dimensional monad theory, J. Pure Appl. Algebra 59(1) (1989), 1-41.
[6] Börger, R., Tholen, W., Wischnewsky, M.B., and Wolff, H., Compact and hypercomplete categories, J. Pure Appl. Algebra 21 (1981), 219-144.
[7] Bourke, J., Two-dimensional monadicity, Adv. Math. 252 (2004), 708-747.
[8] Deligne, P., Action du groupe des tresses sur une categorie, Invent. Math. 218(1) (1997), 159-175.
[9] Dubuc, E., Adjoint triangles, Reports of the Midwest Category Seminar (1968), 69-91.
[10] Dubuc, E., Kan extensions in enriched category theory, Lecture Notes in Math. 145, Springer-Verlag, 1970.
[11] Fujii, S., Katsumata, S., Melliès, P., Towards a formal theory of graded monads, Foundations of software science and computation structures, 513-530, Lecture Notes in Comput. Sci. 9634, Springer, 2016.
[12] Gurski, N., “An algebraic theory of tricategories”, PhD Thesis, The University of Chicago, 2006.
[13] Hermida, C., Descent on 2-fibrations and strongly regular 2-categories, Appl. Categ. Structures 21 (2004), 427-459.
[14] Hermida, C., From coherent structures to universal properties, J. Pure Appl. Algebra 165(1) (2000), 7-61.
[15] Janelidze, G., and Tholen, W., Facets of descent II, Appl. Categ. Structures 5(3) (1997), 229-248.
[16] Johnstone, P.T., Adjoint lifting theorems for categories of algebras, Bull. Lond. Math. Soc. 7 (1975), 294-297.
[17] Kelly, G.M., “Basic Concepts of Enriched Category Theory”, London Math. Soc. Lecture Note Ser. 64., Cambridge University Press, 1982.
[18] Kelly, G.M., Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 39(2) (1989), 301-317.
[19] Kelly, G.M. and Lack, S., On property-like structures, Theory Appl. Categ. 3(9) (1997), 213-250.
[20] Kock, A., Monads for which structures are adjoint to units, J. Pure Appl. Algebra 104(1) (1995), 41-59.
[21] Lack, S., A coherent approach to pseudomonads, Adv. Math. 152(2) (2000), 179-220.
[22] Lack, S., Codescent objects and coherence, Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra 175(1-3) (2002), 223-241.
[23] Lack, S., Icons, Appl. Categ. Structures 18(3) (2000), 289-307.
[24] Lack, S., A 2-categories companion, in “Towards Higher Categories”, Springer, The IMA Volumes in Mathematics and its Applications 152 (2000), 105-191.
[25] Le Creurer, I.J., Marmolejo, F., and Vitale, E.M., Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173(3) (2020), 293-313.
[26] Leinster, T., “Higher Operads, Higher Categories”, London Math. Soc. Lecture Note Ser. 298, Cambridge University Press, 2004.
[27] Leinster, T., “Basic Category Theory”, Cambridge Studies in Advanced Mathematics 143, Cambridge University Press, 2004.
[28] Lucatelli Nunes, F., On biadjoint triangles, Theory Appl. Categ. 31(9) (2016), 217-256.
[29] Lucatelli Nunes, F., Pseudo-Kan extensions and descent theory, arXiv:1606.04999 or Preprints-CMUC (16-30).
[30] Marmolejo, F., Doctrines whose structure forms a fully faithful adjoint string, Theory Appl. Categ. 3(2) (1997), 24-44.
[31] Marmolejo, F., Distributive laws for pseudomonads, Theory Appl. Categ. 5(5) (1999), 91-147.
[32] Marmolejo, F. andWood, R.J., Coherence for pseudodistributive laws revisited, Theory Appl. Categ. 20(5) (2008), 74-84.
[33] Power, A.J., A unified approach to the lifting of adjoints, Cah. Topol. Geom. Diver. Categ. 29(1) (1988), 67-77.
[34] Power, A.J., A general coherence result, Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra 57 (1989), 165-173.
[35] Power, A.J., Cattani, G.L., and Winskel, G., A representation result for free cocompletions, J. Pure Appl. Algebra 151(3) (2000), 273-286.
[36] Street, R.H., The formal theory of monads, J. Pure Appl. Algebra 2(2) (1972), 149-168.
[37] Street, R.H., Fibrations and Yoneda’s lemma in a 2-category, Category Sem., Proc., Sydney 1972/1973, Lecture Notes Math. 420 (1974), 104-133.
[38] Street, R.H., Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8(2) (1976), 149-181.
[39] Street, R.H., Wischnewsky, M., Wolf, H., and Tholen, W., Semi-topological functors III: Lifting of monads and adjoint functors, J. Pure Appl. Algebra 16 (1980), 299-314.
[40] Street, R.H., Fibrations in bicategories, Cah. Topol. Geom. Diver. Categ. 21(2) (1980), 111-160.
[41] Street, R.H., Correction to: “Fibraer. Categ. 28(1) (1987), 53-56.
[42] Street, R.H., Categorical structures, in the “Handbook of Algebra, Volume 1”, Elsevier (1996), 529-577.
[43] Day, B. and Street, R., Monoidal bicategories and Hopf algebroids, Adv. Math. 219(1) (1997), 99-157.
[44] Street, R.H., Categorical and combinatorial aspects of descent theory, Appl. Categ. Structures 21(5-6) (2004), 537-576.
[45] Tholen, W., Adjungierte dreiecke, colimites und Kan-erweiterungen, Math. Ann. 217(2) (1975), 211-219.