nLab torsion theory

Definition

A torsion theory in an abelian category $A$ is a pair $(T,F)$ of additive subcategories, called the torsion class $T$ and the torsion free class $F$, such that the following conditions hold:

• $Hom(T,F) = 0$

(in other words, $A(X,Y) = 0$ if $X \in \mathop{Ob} T$ and $Y\in \mathop{Ob} F$).

• $Hom(T,Y) = 0 \Rightarrow Y\in \mathop{Ob} F$

• $Hom(X,F) = 0 \Rightarrow X\in \mathop{Ob} T$

• for all $X\in \mathop{Ob} A$, there exists $Y\subset X$, $Y\in \mathop{Ob} T$ and $X/Y\in \mathop{Ob} F$

Equivalently, a torsion theory in $A$ is a pair $(T,F)$ of strictly full subcategories of $A$ such that the first and last conditions in the above list hold. Alternatively, we can require the last condition and the following 3: $T\cap F=\{0\}$, $T$ is closed under quotients and $F$ under subobjects. It follows also that $T$ and $F$ are stable under extensions.

Torsion part of an object

If the abelian category $A$ satisfies Gabriel‘s property (sup) then for every object $X$ there exists the largest subobject $t(X)\subset X$ which is in $T$, which is called the torsion part of $X$ (sometimes written as $X_T$). Under the axiom of choice, $t: X\to t(X)$ can be extended to a functor.

Hereditary torsion theories

A torsion theory is called hereditary if $T$ is closed under subobjects, or equivalently, $t$ is left exact functor. For some authors (e.g. Golan) torsion theory is assumed to be hereditary.

Properties

If $(T,F)$ is a torsion theory then $T$ and $F$ both contain the zero object and are closed under biproducts (Borceux II 1.12.3). Presentation of an object $X$ in $\mathop{Ob} A$ as an extension $0\to Y\to X\to X/Y\to 0$, $Y$ in $\mathop{Ob} T$ by $X/Y$ in $\mathop{Ob} F$ is unique up to an isomorphism of short exact sequences (Borceux II 1.12.4).

Given an abelian category $A$ there is a bijection between universal closure operations on $A$, hereditary torsion theories in $A$ (Borceux II 1.12.8) and, if $A$ is a locally finitely presentable category, also with left exact localizations of $A$ admitting a right adjoint and with localizing subcategories of $A$ (Borceux II 1.13.15).

Examples

The basic example of a torsion class is the class of torsion abelian groups within the category $A =$ Ab of all abelian groups. Torsion theories are often used as a means to formulate localization theory in abelian categories.

Literature

The notion originates with:

• Spencer Ernst Dickson, Torsion theories for abelian categories, Thesis, New Mexico State University (1963) (ProQuest)

• Spencer Ernst Dickson, A Torsion Theory for Abelian Categories, Transactions of the American Mathematical Society 121 1 (1966) 223-235 (doi:10.2307/1994341, jstor:1994341)

Comprehensive accounts:

• Francis Borceux, Handbook of Categorical Algebra, vol. 2

• N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

• Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, with app. by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics 177, Springer-Verlag 1971, vi+94 pp. MR284459

For a unified treatment in abelian and in triangulated categories see

• Apostolos Beligiannis, Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. pdf

As explained there, in the triangulated context, torsion pairs are in 1-1 correspondence with t-structures. One could also study a relation between torsion theories on an abelian category with tilting theory and $t$-structures on the derived category:

• Dieter Happel, Idun Reiten, Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88

• Riccardo Colpi, Luisa Fiorot, Francesco Mattiello, On tilted Giraud subcategories, arxiv/1307.1987

Other references in the abelian context include

• Lia Vaš, Differentiability of torsion theories, (pdf)

For analogues in nonadditive contexts see

• Michael Barr, Non-abelian torsion theories, Canad. J. Math. 25 (1973) 1224–1237

• Basil A. Rattray, Torsion theories in non-additive categories, Manuscripta Math. 12 (1974), 285–305 MR340360 doi

• Jiří Rosický, Walter Tholen, Factorization, fibration and torsion, arxiv/0801.0063, Journal of homotopy and Related Structures

• M. M. Clementino, D. Dikranjan, Walter Tholen, Torsion theories and radicals in normal categories, J. of Algebra 305 (2006) 92-129

• Dominique Bourn, Marino Gran, Torsion theories in homological categories, J. of Algebra 305 (2006) 18–47 MR2007k:18018 doi

Presenting a pretorsion theory on Cat whose torsion(-free) objects are the groupoids (skeletal categories, respectively), hence whose “trivial objects” are the skeletal groupoids:

• Francis Borceux, Federico Campanini, Marino Gran, Walter Tholen, Groupoids and skeletal categories form a pretorsion theory in $Cat$, Advances in Math. 426 (2023) 109110, [arXiv:2207.08487]