Torsion theories as short exact sequences

Ülo Reimaa

Joint work with: Graham Manuell, Nelson-Martins Ferreira, Fosco Loregian

A torsion theory allows one to decompose objects of a category into pairs of objects falling into two fixed subcategories. We will make the point that torsion theories can be seen as short exact sequences of categories and explore how this idea works with different notions of torsion theory [1, 3].

To illustrate the point, consider the following notion of torsion theory on a category $\mathcal{A}$ . Equip $\mathcal{A}$ with full subcategories $\mathcal{T}$ and $\mathcal{F}$ , along with an object $Z$ that lies in both subcategories, such that:

•

there exists precisely one arrow $T\to F$ whenever $T\in\mathcal{T}$ and $F\in\mathcal{F}$ ;
•

for every object $A\in\mathcal{A}$ , there exists a square (1) that is bicartesian, meaning it is simultaneously a pullback and a pushout, with $T\in\mathcal{T}$ and $F\in\mathcal{F}$ .

The interpretation is that the square (1) exhibits the decomposition of an object $A$ into an (essentially unique) “short exact sequence” $T\to A\to F$ . Indeed, in the special case where $Z$ is the zero object, asking the square (1) to be a pullback is the same as requiring $(T\to A)\cong\mathsf{ker}(A\to F)$ and asking the square (1) to be a pushout is the same as requiring $(A\to F)\cong\mathsf{coker}(T\to A)$ . In general, $Z$ need not be the zero object, although it can be thought of as lying between the initial object and the terminal object.

It turns out that $\mathcal{T}$ is coreflective and $\mathcal{F}$ is reflective in $\mathcal{A}$ , with the arrows $T\to A$ and $A\to F$ in the square (1) giving the coreflection into $\mathcal{T}$ and reflection into $\mathcal{F}$ . Therefore, our data builds the diagram

(2)

which can be viewed as a short exact sequence, in the sense that the composite is the zero adjunction (the essentially unique adjunction that factors through the terminal category), in addition to

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0.0}{-2.56943pt}{-2.43748pt}\pgfsys@invoke{ }\hbox{{\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }% \pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\dashv}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }% \pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\!\right),

in a sense that is appropriate for adjunctions. The main observation here is the following.

Observation.

Exact sequences of the form (2) are essentially the same as torsion theories on $\mathcal{A}$ .

Given a short exact sequence of categories in the above sense, the central category $\mathcal{A}$ can be viewed as a category of extensions with kernel coming from $\mathcal{T}$ and cokernel coming from $\mathcal{F}$ . This idea can also be used to generate a cofree torsion theory on a pair of full subcategories of a category.

Example.

If the categories are pointed then $Z$ is necessarily the zero object and one recovers torsion theories in the classical sense.

Example.

A large class of non-pointed examples is provided by Artin glueings of toposes [2]. The open subtopos and its closed complement form a torsion theory, with $Z$ the corresponding subterminal.

We can interpret (2) as a split extension if the functor $\mathcal{A}\to\mathcal{F}$ additionally has a left adjoint. Such “split extensions” admit a nice theory of semi-direct products, which is especially well-behaved if the categories in question are protomodular. For instance, when dealing with opposites of toposes or semi-abelian categories.

[1] A. Facchini, C. Finocchiaro, M. Gran, Pretorsion theories in general categories, J. Pure Appl. Algebra 225 (2021), no. 2, Paper No. 106503, 21 pp.
[2] P. Faul, G. Manuell, J. Siqueira, Artin glueings of toposes as adjoint split extensions, J. Pure Appl. Algebra 227 (2023), no. 5, Paper No. 107273, 40 pp.
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