From torsion theories to closure operators and factorization systems

Document Type: Research Paper


1 Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, 16146-Genova, Italy

2 Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.


Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].


[1] Beck, J. , Distributive laws, in: Seminar on triples and categorical homology, Lecture Notes in Math. 80 (1969), 119-140.
[2] Borceux, F., "Handbook of Categorical Algebra" 1, Cambridge University Press, 1994.
[3] Borceux, F., "Handbook of Categorical Algebra" 2, Cambridge University Press, 1994.
[4] Bourke, J. and Garner, R., Algebraic weak factorisation systems I: Accessible AWFS, J. Pure Appl. Algebra 220 (2016), 108-147.
[5] Bourke, J. and Garner, R., Algebraic weak factorisation systems II: Categories of weak maps, J. Pure Appl. Algebra 220 (2016), 148-174.
[6] Dikranjan, D. and Tholen, W., "Categorical Structure of Closure Operators: With Applications to Topology, Algebra and Discrete Mathematics", Mathematics and its Applications 346, Springer, 1995.
[7] Ehresmann, C., Cohomologie à valeurs dans une catégorie dominée, Extraits du Colloque de Topologie, Bruxelles 1964, in: C. Ehresmann, Oeuvres completes et commentées, Partie III-2, 531-590, Amiens 1980.
[8] Eilenberg, S. and Steenrod, N., "Foundations of Algebraic Topology", Princeton University Press, 1952.
[9] Gardner, B.J., Morphic orthogonality and radicals in categories, in: Rings and radicals (Shijiazhuang, 1994), 178-206, Pitman Res. Notes Math. Ser. 346, Longman, 1996.
[10] Garner, R., Understanding the small object argument, Appl. Categ. Structures 17 (2009), 247-285.
[11] Grandis, M., A categorical approach to exactness in algebraic topology, in: Atti del V Convegno Nazionale di Topologia, Lecce-Otranto 1990, Rend. Circ. Mat. Palermo 29 (1992), 179-213.
[12] Grandis, M., On the categorical foundations of homological and homotopical algebra, Cah. Topol. Géom. Différ. Catég. 33 (1992), 135-175.
[13] Grandis, M., "Homological Algebra in Strongly Non-Abelian Settings", World Scientific Publishing Co., Singapore, 2013.
[14] Grandis, M. and Tholen, W., Natural weak factorization systems, Arch. Math. (Brno) 42 (2006), 397-408.
[15] Grandis, M. and Janelidze, G., Márki, L., Non-pointed exactness, radicals, closure operators, J. Aust. Math. Soc. 94 (2013), 348-361.
[16] Janelidze, G. and Tholen, W., Characterization of torsion theories in general categories, Contemp. Math. 431 (2007), 249-256.
[17] Lavendhomme, R., Un plongement pleinement fidele de la catégorie des groupes, Bull. Soc. Math. Belgique 17 (1965), 153-185.
[18] Mantovani, S., Torsion theories for crossed modules, Workshop in Category Theory and Algebraic Topology, Louvain-la-Neuve 2015, Unpublished talk.
[19] Mitchell, B., "Theory of Categories", Academic Press, 1965.
[20] Puppe, D., Korrespondenzen in abelschen Kategorien, Math. Ann. 148 (1962), 1-30.
[21] Rosický, J. and Tholen, W., Lax factorization algebras, J. Pure Appl. Algebra 175 (2002), 355-382.
[22] Tholen, W., Factorizations, fibres and connectedness, in: Categorical Topology (Toledo, Ohio, 1983), 549–566, Sigma Ser. Pure Math. 5, Heldermann, 1984.