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Cofibrantly Generated Lax Orthogonal Factorisation Systems

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Abstract

The present note has three aims. First, to complement the theory of cofibrant generation of algebraic weak factorisation systems (awfss) to cover some important examples that are not locally presentable categories. Secondly, to prove that cofibrantly kz-generated awfss (a notion we define) are always lax orthogonal. Thirdly, to show that the two known methods of building lax orthogonal awfss, namely cofibrantly kz-generation and the method of “simple adjunctions”, construct different awfss. We study in some detail the example of cofibrant kz-generation that yields representable multicategories, and a counterexample to cofibrant generation provided by continuous lattices.

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References

  1. Adámek, J., Sousa, L., Velebil, J.: Kan injectivity in order-enriched categories. Math. Struct. Comput. Sci. 25(1), 6–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. J. Pure Appl. Algebra 59(1), 1–41 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourke, J., Garner, R.: Algebraic weak factorisation systems I: accessible AWFS. J. Pure Appl. Algebra 220(1), 108–147 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cagliari, F., Clementino, M.M., Mantovani, S.: Fibrewise injectivity and Kock–Zöberlein monads. J. Pure Appl. Algebra 216(11), 2411–2424 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clementino, M.M., Franco, I.López: Lax orthogonal factorisation systems. Adv. Math. 302, 458–528 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clementino, M.M., Franco, I.L.: Lax orthogonal factorisations in monad-quantale-enriched categories. Log. Methods Comput. Sci. 13 (2017)

  7. Clementino, M. M., Franco, I.L.: Lax orthogonal factorisations in ordered structures (2017). arXiv:1702.02602

  8. Day, A.: Filter monads, continuous lattices and closure systems. Can. J. Math. 27, 50–59 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Escardó, M.H.: Properly injective spaces and function spaces. Topol. Appl. 89(1–2), 75–120 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Freyd, P.J., Kelly, G.M.: Categories of continuous functors. I. J. Pure Appl. Algebra 2, 169–191 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garner, R.: Understanding the small object argument. Appl. Categ. Struct. 17(3), 247–285 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grandis, M., Tholen, W.: Natural weak factorization systems. Arch. Math. (Brno) 42(4), 397–408 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Gray, J. W.: Fibred and cofibred categories. In: Proceedings Conference on Categorical Algebra (La Jolla, CA, 1965), pp. 21–83 (1966)

  14. Hermida, C.: Representable multicategories. Adv. Math. 151(2), 164–225 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hermida, C.: Fibrations for abstract multicategories. Galois Theory, Hopf Algebras, and Semiabelian Categories. Series in Fields Institute Communications, vol. 43, pp. 281–293. American Mathematical Society, Providence, RI (2004)

    Google Scholar 

  16. Kelly, G. M.: Coherence theorems for lax algebras and for distributive laws. In: Category Seminar (Proceeding Seminar, Sydney, 1972/1973). Lecture Notes in Mathematics, vol. 420, pp. 281–375. (1974)

  17. Kelly, G.M.: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Aust. Math. Soc. 22(1), 1–83 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kelly, G.M.: Basic Concepts of Enriched Category Theory. Repr. Theory Appl. Categ. 10, 137 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Kelly, G.M., Lack, S.: On property-like structures. Theory Appl. Categ. 3(9), 213–250 (1997)

    MathSciNet  MATH  Google Scholar 

  20. Kelly, G. M., Street, R.: Review of the elements of 2-categories. In: Category Seminar, Proceeding, Sydney 1972/1973. Lecture Notes in Mathematics, vol. 420, pp. 75–103 (1974)

  21. Kelly, G.M.: Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Note Series, vol. 64. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  22. Kock, A.: Monads for which structures are adjoint to units. J. Pure Appl. Algebra 104(1), 41–59 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lambek, J.: Deductive systems and categories. II. Standard constructions and closed categories. In Category Theory, Homology Theory and Their Applications, Proceeding Conference on Seattle Research Center of Battelle Memorial Institute 1968, vol. 1, pp. 76–122 (1969)

  24. Leinster, T. (ed.): Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  25. Marmolejo, F.: Doctrines whose structure forms a fully faithful adjoint string. Theory Appl. Categ. 3(2), 24–44 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Quillen, D.G.: Homotopical Algebra. Lecture Notes in Mathematics, No. 43. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  27. Scott, D.: Continuous Lattices. In: Toposes, Algebraic Geometry and Logic (Conference, Dalhousie University, Halifax, NS, 1971). Lecture Notes in Mathematic, vol. 274, pp. 97–136 (1972)

  28. Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zöberlein, V.: Doctrines on 2-categories. Math. Z. 148(3), 267–279 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Departamento de Matemática y Aplicaciones, CURE, Universidad de la República, Tacuarembó s/n, Maldonado, Uruguay

    Ignacio López Franco

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  1. Ignacio López Franco
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Correspondence to Ignacio López Franco.

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Communicated by M. M. Clementino.

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The author gratefully acknowledges the support of the following institutions during the long gestation of this article: a Research Fellowship of Gonville and Caius College, Cambridge; the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge; sni-anii, pedeciba and Universidad de la República.

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López Franco, I. Cofibrantly Generated Lax Orthogonal Factorisation Systems. Appl Categor Struct 27, 463–492 (2019). https://doi.org/10.1007/s10485-019-09561-1

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