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On the formal theory of pseudomonads and pseudodistributive laws. (English) Zbl 1457.18023
Monads are one of the fundamental notions of category theory. Beck’s theorem on distributive laws between monads [J. Beck, Lect. Notes Math. 80, 119–140 (1969; Zbl 0186.02902)] describes concisely the structure that is necessary and sufficient so as to combine two algebraic structures so that the operations of one distribute over those of the other. The formal theory of monads, having been introduced in [R. Street, J. Pure Appl. Algebra 2, 149–168 (1972; Zbl 0241.18003)] and having later been developed in [S. Lack and R. Street, J. Pure Appl. Algebra 175, No. 1–3, 243–265 (2002; Zbl 1019.18002)], offers an elegant account of the theory of monads, starting with the observation that the notion of a monad is to be defined within any \(2\)-category and providing a characterization of the existence of categories of Eilenberg-Moore algebras as a completeness property and a simple account of Beck’s theorem on distributive laws.
Pseudomonads are the counterparts of monads in \(2\)-dimensional category theory obtained by requiring the axioms for a monad to hold only up to coherent isomorphisms rather than strictly [M. C. Bunge, Trans. Am. Math. Soc. 197, 355–390 (1974; Zbl 0358.18004)]. Knowing how the formal theory of monads offers a simple proof of Beck’s theorem on distributive laws, it is natural to try to develop a formal theory of pseudomonads, confronting a notoriously hard problem that, just as the formal theory of monads is formulated within \(2\)-dimensional category theory [G. M. Kelly and R. Street, Lect. Notes Math. 420, 75–103 (1974; Zbl 0334.18016)], the formal theory of pseudomonads is to be developed within \(3\)-dimensional category theory [R. Gordon et al., Coherence for tricategories. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0836.18001); N. Gurski, Coherence in three-dimensional category theory. Cambridge: Cambridge University Press (2013; Zbl 1314.18002); T. Leinster, Bull. Lond. Math. Soc. 47, No. 3, 550–553 (2015; Zbl 1321.00040)], where it is convenient to work with Gray-categories, i.e., semistrict tricategories. So far there has not been a direct counterpart of the \(2\)-category \(\boldsymbol{Mnd}\left( \mathcal{K}\right) \) of monads, monad morphisms and monad transformations.
This paper aims to take some further steps in the development of the formal theory of pseudomonads. The main results are as follows.
Theorem 2.5 answers the question in [S. Lack, Adv. Math. 152, No. 2, 179–202 (2000; Zbl 0971.18008)], claiming that, for every Gray-category \(\mathcal{K}\), there is a Gray-category \(\boldsymbol{Psm} \left( \mathcal{K}\right) \) of pseudomonads, pseudomonad morphisms, pseudomonad transformations and pseudomonad modifications in \(\mathcal{K}\).
Theorem 3.4 is the analogue of a fundamental result of the formal theory of monads, assering that \(\boldsymbol{Psm}\left( \mathcal{K}\right) \) is equivalent, in a suitable \(3\)-categorical sense, to the Gray-category \(\boldsymbol{Lift}\left( \mathcal{K}\right) \).
Proposition 4.4 gives an identification of pseudodistributive laws with an object in \(\boldsymbol{Psm}\left( \boldsymbol{Psm}\left( \mathcal{K} \right) \right) \).
Following naturally from Theorem 2.5 and Theorem 3.4,Theorem 4.5 claims the equivalence between pseudodistributive laws and liftings of pseudomonads to \(2\)-categories of pseudoalgebras, which has already been established in [F. Marmolejo, Theory Appl. Categ. 5, 91–147 (1999; Zbl 0919.18004), Theorems 6.2, 9.3 and 10.2], The authors take a modular, abstract approach to the verification of the coherence conditions, avoiding completely the notion of a composite of pseudomonads with compatible structure.
MSC:
18N10 2-categories, bicategories, double categories
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18C20 Eilenberg-Moore and Kleisli constructions for monads
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References:
[1] [Barr and Wells, 1985] M. Barr and C. Wells.Toposes, triples, and theories. Springer, 1985. · Zbl 0567.18001
[2] [Beck, 1969] J. Beck. Distributive laws. InSeminar on Triples and Categorical Homology Theory (ETH, Z¨urich, 1966/67), pages 119-140. Springer, 1969.
[3] [Bunge, 1974] M. Bunge. Coherent extensions and relational algebras.Transactions of the American Mathematical Society, 197:355-390, 1974. · Zbl 0358.18004
[4] [Cattani and Winskel, 2005] G. L. Cattani and G. Winskel. Profunctors, open maps, and bisimulation.Mathematical Structures in Computer Science, 15(3):553-614, 2005. · Zbl 1169.68537
[5] [Cheng et al., 2003] E. Cheng, M. Hyland, and J. Power. pseudodistributive laws.Electronic Notes in Theoretical Computer Science, 83, 2003.
[6] [Curien, 2012] P.-L. Curien. Operads, clones, and distributive laws. Proc. of the International Conference on Operads and Universal Algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, Vol. 9, World Scientific, Singapore, 25-50, 2012. · Zbl 1305.18031
[7] [Dost´al, 2018] M. Dost´al. Two-dimensional universal algebra. PhD Thesis, Czech Technical University in Prague, 2018.https://math.feld.cvut.cz/dostamat/research/ phd-thesis.pdf.
[8] [Eilenberg and Kelly, 1966] S. Eilenberg and G. M. Kelly. Closed categories. InProceedings of La Jolla Conference on Categorical Algebra, pages 421-562. Springer-Verlag, 1966.
[9] [Fiore et al., 2008] M. Fiore, N. Gambino, M. Hyland, and G. Winskel. The cartesian closed bicategory of generalised species of structures.Journal of the London Mathematical Society, 77(2):203-220, 2008. · Zbl 1137.18003
[10] [Fiore et al., 2018] M. Fiore, N. Gambino, M. Hyland, and G. Winskel. Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures.Selecta Mathematica New Series, 24: 2791-2830, 2018. · Zbl 1427.18012
[11] [Gambino and Joyal, 2017] N. Gambino and A. Joyal. On operads, bimodules and analytic functors.Memoirs of the American Mathematical Society, 289(1184), (v) + 110pp, 2017.
[12] [Garner, 2008] R. Garner. Polycategories via pseudodistributive laws.Advances in Mathematics, 218(3):781-827, 2008. · Zbl 1146.18002
[13] [Garner and Gurski, 2009] R. Garner and N. Gurski.The low-dimensional structures formed by tricategories.Mathematical Proceedings of the Cambridge Philosophical Society, 146(3):551-589, 2009. · Zbl 1169.18001
[14] [Gordon et al., 1995] R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. Memoirs of the American Mathematical Society, 117(558), 1995. · Zbl 0836.18001
[15] [Gurski, 2013] N. Gurski.Coherence in Three-Dimensional Category Theory. Cambridge University Press, 2013. · Zbl 1314.18002
[16] [Kelly, 1974] G. M. Kelly. Coherence theorems for lax algebras and for distributive laws. In G. M. Kelly and R. H. Street, editors,Category Seminar (Proc. Sem., Sydney, 1972/1973), volume 420 ofLecture Notes in Mathematics, pages 281-375. Springer, 1974.
[17] [Kelly, 1982] G. M. Kelly.Basic Concepts of Enriched Category Theory. Cambridge University Press, 1982. · Zbl 0478.18005
[18] [Kelly and Street, 1974] G. M. Kelly and R.H. Street.Review of the elements of 2categories. In G. M. Kelly and R. H. Street, editors,Category Seminar (Proc. Sem. Sydney 1972/1973), volume 420 ofLecture Notes in Mathematics, pages 75-103. Springer, 1974.
[19] [Lack, 2000] S. Lack. A coherent approach to pseudomonads.Advances in Mathematics, 152:179-202, 2000. · Zbl 0971.18008
[20] [Lack, 2007] S. Lack. Bicat is not triequivalent to Gray.Theory and Applications of Categories, 18(1):1-3, 2007. · Zbl 1119.18002
[21] [Lack and Street, 2002] S. Lack and R. Street. The formal theory of monads. II.Journal of Pure and Applied Algebra, 175(1-3):243-265, 2002. · Zbl 1019.18002
[22] [Mac Lane, 1998] S. Mac Lane.Categories for the working mathematician.2nd ed., Springer, 1998. · Zbl 0906.18001
[23] 56N. GAMBINO AND G. LOBBIA
[24] [Marmolejo, 1997] F. Marmolejo. Doctrines whose structure forms a fully faithful adjoint string.Theory and applications of categories, 3(2):24-44, 1997. · Zbl 0878.18004
[25] [Marmolejo, 1999] F. Marmolejo. Distributive laws for pseudomonads.Theory and Applications of Categories, 5(5):91-147, 1999. · Zbl 0919.18004
[26] [Marmolejo, 2004] F. Marmolejo. Distributive laws for pseudomonads II.Journal of Pure and Applied Algebra, 195(1-2):169-182, 2004. · Zbl 1055.18002
[27] [Marmolejo and Wood, 2008] F. Marmolejo and R. J. Wood. Coherence for pseudodistributive laws revisited.Theory and Applications of Categories, 20(6):74-84, 2008. · Zbl 1153.18005
[28] [Power, 1990] A. J. Power. A 2-Categorical Pasting Theorem.Journal of Algebra, volume 129, pages 439-445, 1990. · Zbl 0698.18005
[29] [Street, 1972] R. Street. The formal theory of monads.Journal of Pure and Applied Algebra, 2(2):149-168, 1972. · Zbl 0241.18003
[30] [Tanaka, 2004] M. Tanaka.Pseudodistributive laws and a unified framework for variable binding. PhD thesis, Laboratory for the Foundations of Computer Science, School of Informatics, University of Edinburgh, 2004. Available as LFCS Technical Report ECS-LFCS-04-438.
[31] [Tanaka and Power, 2006a] M. Tanaka and J. Power. Pseudodistributive laws and axiomatics for variable binding.Higher-order and Symbolic Computation, 19(2-3):305- 337, 2006. · Zbl 1105.68077
[32] [Tanaka and Power, 2006b] M. Tanaka and J. Power. A unified category-theoretic semantics for binding signatures in substractural logics.Journal of Logic and Computation, 16:5-25, 2006. · Zbl 1105.03072
[33] [Walker, 2019] C. Walker. Distributive laws via admissibility.Applied Categorical Structures, 27(6) 567-617, 2019. · Zbl 1444.18004
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