On bornological semi-abelian algebras

Document Type : Research Paper


1 Universit\'e de Louvain, Belgium.

2 Department of Mathematics, University of Coimbra, Portugal.


If $\Bbb T$ is a semi-abelian algebraic theory, we prove that the category ${\rm Born}^{\Bbb T}$ of bornological $\Bbb T$-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in ${\rm Born}^{\Bbb T}$ and, for a bornological $\Bbb T$-algebra $X$, we investigate the relation between the representability of actions on $X$ as a $\Bbb T$-algebra and as a bornological $\Bbb T$-algebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of ${\rm Born}^{\Bbb T}$ and prove in particular that both properties hold in the case of bornological groups.


[1] Adamek, J. and Herrlich, H., Cartesian closed categories, quasitopoi and topological universes, Comment. Math. Univ. Carolin. 27(2) (1986), 235-257.
[2] Adamek, J. and Rosicky, J., "Locally presentable and accessible categories", London Math. Soc. Lecture Notes Ser. 189, Cambridge University Press, 1994.
[3] Bambozzi, F., "On a generalization of affnoid varieties", Universita degli Studi di Padova, Dottorato di Ricerca in Matematica, 2013.
[4] Barr, M., Exact categories, Exact categories and categories of sheaves, 1-120, Lecture Notes in Math. 236, Springer, 1971.
[5] Borceux, F. and Bourn, D., "Mal'cev, Protomodular, Homological and Semi-Abelian Categories", Mathematics and its Applications 566, Kluwer Academic Publishers, 2004.
[6] Borceux, F. and Clementino, M.M., Topological semi-abelian algebras, Adv. Math. 90 (2005), 425-453.
[7] Borceux, F., Clementino, M.M., and Montoli, A., On the representability of actions for topological algebras, Categorical methods in algebra and topology, 41-66, Textos Mat./Math. Texts 46, University of Coimbra, 2014.
[8] Borceux, F., Janelidze, G., and Kelly, G.M., On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14(11) (2005), 244-286.
[9] Borceux, F. and Pedicchio, M.C., A characterization of Quasi-Toposes, J. Algebra 139 (1991), 505-526.
[10] Cagliari, F. and Clementino, M.M., Topological groups have representable actions, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), 519-526.
[11] Cigoli, A.S., Gray, J.R.A., and Van der Linden, T., Algebraically coherent categories, Theory Appl. Categ. 30(54) (2015), 1864-1905.
[12] Clementino, M.M., Montoli, A., and Sousa, L., Semi-direct products of (topological) semi-abelian algebras, J. Pure Appl. Algebra 219 (2015), 183-197.
[13] Espa~nol, L. and Lamban, L., On bornologies, locales and toposes of M-sets, J. Pure Appl. Algebra 176 (2002), 113-125.
[14] Gray, J.R.A., Algebraic exponentiation in general categories, Appl. Categ. Structures 22 (2014), 305-310.
[15] Janelidze, G., Marki, L. and Tholen, W., Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), 367-386.
[16] Johnstone, P.T., "Sketches of an Elephant: A Topos Theory Compendium", Volume 1, Clarendon Press, 2002.
[17] Orzech, G., Obstruction theory in algebraic categories I, J. Pure Appl. Algebra 2 (1972), 287-314.
[18] Penon, J., Sur les quasi-topos, Cah. Topol. Geom. Differ. Categ. 18(2) (1977), 181- 218.
[19] Wyler, O., On the categories of general topology and topological algebra, Arch. Math. (Basel) 22 (1971), 7-17.