Regular pairings of functors and weak (co)monads

R. Wisbauer

Algebra and discrete mathematics, Tome 15 (2013) no. 1, pp. 127-154.

Résumé

For functors $L:\mathbb{A}\to \mathbb{B}$ and $R:\mathbb{B}\to \mathbb{A}$ between any categories $\mathbb{A}$ and $\mathbb{B}$, a pairing is defined by maps, natural in $A\in \mathbb{A}$ and $B\in \mathbb{B}$, $$ \xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@0.5ex>[r]^{\alpha} {\rm Mor}_\mathbb{A} (A,R(B))\ar@0.5ex>[l]^{\beta}}. $$ $(L,R)$ is an adjoint pair provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\mathbb{A}$, $LR$ defines a comonad on the category $\mathbb{B}$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors $(L,R,\alpha,\beta)$ with $\alpha = \alpha\cdot \beta\cdot \alpha$ and $\beta = \beta \cdot\alpha\cdot\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\mathbb{A}$ gives rise to a regular pairing between $\mathbb{A}$ and the category of compatible (co)modules.

Détail

Keywords: pairing of functors; adjoint functors; weak monads and comonads; $r$-unital monads; $r$-counital comonads; lifting of functors; distributive laws.

Bibliographie

[1] Alonso Álvarez J. N., Fernández Vilaboa J. M., González Rodríguez R., Rodríguez Raposo A. B., “Crossed products in weak contexts”, Appl. Categ. Struct., 18:3 (2010), 231–258 | DOI | MR | Zbl

[2] Beck J., “Distributive laws”, Seminar on Triples and Categorical Homology Theory, LNM, 80, ed. B. Eckmann, Springer, 1969, 119–140 | MR

[3] Böhm G., “The weak theory of monads”, Adv. Math., 225:1 (2010), 1–32 | DOI | MR | Zbl

[4] Böhm G., Lack S., Street R., “On the 2-category of weak distributive laws”, Commun. Algebra, 39:12 (2011), 4567–4583 | DOI | MR | Zbl

[5] Böhm G., Lack S., Street R., “Idempotent splittings, colimit completion, and weak aspects of the theory of monads”, J. Pure Appl. Algebra, 216 (2012), 385–403 | DOI | MR | Zbl

[6] Böhm G., Nill F., Szlachányi K., “Weak Hopf algebras. I: Integral theory and $C^*$-structure”, J. Algebra, 221:2 (1999), 385–438 | DOI | MR | Zbl

[7] Brzeziński T., “The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties”, Alg. Rep. Theory, 5 (2002), 389–410 | DOI | MR | Zbl

[8] Brzeziński T., Wisbauer R., Corings and Comodules, London Math. Soc. Lecture Note Series, 309, Cambridge University Press, 2003 | MR | Zbl

[9] Caenepeel S., De Groot E., “Modules over weak entwining structures”, New trends in Hopf algebra theory, Proc. Coll. quantum groups and Hopf algebras (La Falda, Argentina, 1999), Contemp. Math., 267, eds. Andruskiewitsch N. et al., Amer. Math. Soc., Providence, 2000, 31–54 | DOI | MR | Zbl

[10] Eilenberg S., Moore J. C., “Adjoint functors and triples”, Ill. J. Math., 9 (1965), 381–398 | MR | Zbl

[11] Fernández Vilaboa J. M., González Rodríguez R., Rodríguez Raposo A. B., “Preunits and weak crossed products”, J. Pure Appl. Algebra, 213:12 (2009), 2244–2261 | DOI | MR | Zbl

[12] Fernández Vilaboa J. M., González Rodríguez R., Rodríguez Raposo A. B., Weak Crossed Biproducts and Weak Projections, 2009, arXiv: 0906.1693

[13] Johnstone P. T., “Adjoint lifting theorems for categories of modules”, Bull. Lond. Math. Soc., 7 (1975), 294–297 | DOI | MR | Zbl

[14] Kasch F., Mader A., Regularity and substructures of Hom, Frontiers in Mathematics, Birkhäuser, Basel, 2009 | MR | Zbl

[15] Lack S., Street R., “The formal theory of monads, II”, J. Pure Appl. Algebra, 175:1–3 (2002), 243–265 | DOI | MR | Zbl

[16] Sib. Mat. Zh., 15 (1974), 952–956 | DOI | DOI | MR

[17] Mesablishvili B., Wisbauer R., “Bimonads and Hopf monads on categories”, J. K-Theory, 7:2 (2011), 349–388 | DOI | MR | Zbl

[18] Mesablishvili B., Wisbauer R., On Rational Pairings of Functors, 2010, arXiv: ; Appl. Cat. Struct. (to appear) 1003.3221 | MR | DOI

[19] Pareigis B., Kategorien und Funktoren, Mathematische Leitfäden, Teubner Verlag, Stuttgart, 1969 | MR | Zbl

[20] Street R., “The formal theory of monads”, J. Pure Appl. Algebra, 2 (1972), 149–168 | DOI | MR | Zbl

[21] Wisbauer R., “Weak corings”, J. Algebra, 245:1 (2001), 123–160 | DOI | MR | Zbl

[22] Wisbauer R., “Algebras versus coalgebras”, Appl. Categ. Struct., 16:1–2 (2008), 255–295 | DOI | MR | Zbl

[23] Wisbauer R., “Lifting theorems for tensor functors on module categories”, J. Algebra Appl., 10:1 (2011), 129–155 | DOI | MR | Zbl