Mathematics

M. S. Tsalenko

CORRESPONDENCES OVER A QUASI-EXACT CATEGORY

(Presented by Academician A. I. Mal’tsev on 6 XII 1963)

1. In the paper \((^{1})\) an axiomatic characterization was given of the category of correspondences over an Abelian category. The principal role in this characterization is played by the notion of a category with involution, or an \(I\)-category (for all definitions see §§ 2, 3). It was shown that the subcategory of all proper mappings of an \(I\)-category satisfying axioms (K1)—(K3) (see § 2) is quasi-exact. In the present note, for each quasi-exact category \(K\), an \(I\)-category \(\mathfrak R(K)\) of correspondences over \(K\) is constructed which satisfies axioms (K1)—(K3). The subcategory of all proper mappings of the category \(\mathfrak R(K)\) turns out to be isomorphic to the category \(K\).

2. Let us recall some definitions (see \((^{1})\)). A category \(\mathfrak R\) is called a category with involution or an \(I\)-category if the following conditions are satisfied:

a) the set of mappings \(\mathfrak R(a,b)\) of an object \(a\) into an object \(b\), for any objects \(a,b \in \mathfrak R\), is partially ordered by the relation \(\subset\);

b) if \(\alpha_1,\alpha_2 \in \mathfrak R(a,b)\), \(\beta \in \mathfrak R(b,c)\), and \(\alpha_1 \subset \alpha_2\), then \(\alpha_1\beta \subset \alpha_2\beta\);

c) for any objects \(a,b\) there is given a mapping of the set \(\mathfrak R(a,b)\) into the set \(\mathfrak R(b,a)\), called involution (\(\alpha \in \mathfrak R(a,b)\) goes into \(\alpha^* \in \mathfrak R(b,a)\)), with the following properties: c1) \((\alpha\beta)^*=\beta^*\alpha^*\); c2) \(\alpha^{**}=\alpha\); c3) from \(\alpha_1 \subset \alpha_2\) it follows that \(\alpha_1^* \subset \alpha_2^*\) (for any mappings \(\alpha,\alpha_1,\alpha_2 \in \mathfrak R(a,b)\), \(\beta \in \mathfrak R(b,c)\)).

We now formulate axioms (K1)—(K3), concerning \(I\)-categories.

(K1). In the \(I\)-category \(\mathfrak R\) there exists an object \(0\) such that the set \(\mathfrak R(0,0)\) consists of the identity mapping \(\varepsilon_0\), and the sets \(\mathfrak R(0,a)\), for any object \(a \in \mathfrak R\), have a greatest and a least element \(\Omega_a\) and \(\omega_a\), respectively.

For a mapping \(\alpha:a \to b\), put \(I\alpha=\omega_a\alpha\), \(B\alpha=\Omega_a\alpha\), \(K\alpha=\omega_b\alpha^*\), \(D\alpha=\Omega_b\alpha^*\). The mapping \(\alpha\) is called \(I\)-regular (\(B\)-, \(K\)-, \(D\)-regular) if \(I\alpha=\omega_b\) (\(B\alpha=\Omega_b\), \(K\alpha=\omega_a\), \(D\alpha=\Omega_a\)); \(\alpha\) is called a proper mapping if it is \(I\)-regular and \(D\)-regular. A proper mapping is called a projection if it is \(B\)-regular, and is called an injection if it is \(K\)-regular.

(K2). If \(\alpha \in \mathfrak R(a,b)\), \(\beta \in \mathfrak R(c,b)\), then: a) from \(I\alpha \subset I\beta\) it follows that \(\beta\alpha^*\alpha \subset \beta\); b) from \(B\alpha \supset B\beta\) it follows that \(\beta\alpha^*\alpha \supset \beta\).

(K3). For any mapping \(\alpha \in \mathfrak R(0,a)\), where \(a\) is an arbitrary object of \(\mathfrak R\), there exist an injection \(\mu:u \to a\) and a projection \(\nu:a \to v\) such that \(B\mu=\alpha\), \(K\nu=\alpha\).

3. Let us recall the definition of a quasi-exact category \((^{1})\), and for the terminology see \((^{2})\). A category \(K\) is called quasi-exact if the following conditions are satisfied: 1) every mapping has a kernel and a cokernel; 2) every mapping is representable in the form of a product of a normal epimorphism and a normal monomorphism; 3) the subobjects of any object form a set.

Everywhere in what follows we shall be speaking of a fixed quasi-exact category \(K\). In contrast to \((^{2})\) (see \((^{1})\)), we shall assume that in each class of equivalent pairs of the form \((u,\mu)\), where \(\mu:u \to a\) is a monomorphism, there has been chosen

representative, called a subobject of the object \(a\). An analogous supposition is made about factor-objects. Let now \(\mu_1:u_1\to a\) and \(\mu_2:u_2\to a\) be two monomorphisms. A monomorphism \(\mu\) will be called the intersection of the monomorphisms \(\mu_1\) and \(\mu_2\), \(\mu=\mu_1\cap\mu_2\), if \(\mu=\mu'_1\mu_1=\mu'_2\mu_2\) and if every monomorphism \(\bar\mu\), representable in the form \(\bar\mu=\bar\mu_1\mu_1=\bar\mu_2\mu_2\), is representable in the form \(\bar\mu=\mu'\mu\). The monomorphism \(\mu\) is determined up to multiplication on the left by an invertible mapping. Let now \((u_1,\mu_1)\) and \((u_2,\mu_2)\) be subobjects of the object \(a\), and let \((u,\mu)\) be their intersection. The monomorphism \(\mu\) is the intersection of \(\mu_1\) and \(\mu_2\) in the sense of the definition just given, and therefore the subobject \((u,\mu)\) will be denoted by \((u_1\cap u_2,\mu_1\cap\mu_2)\). In what follows one must also bear in mind the dual definitions.

Fig. 1

1. Let \(K\) be a quasi-exact category. Consider the class of all triples of the form
   \[ u \xrightarrow{\nu} x \xleftarrow{\nu'} u', \]
   where \(\nu,\nu'\) are epimorphisms. In this class introduce an equivalence relation: the triple
   \[ u \xrightarrow{\nu} x \xleftarrow{\nu'} u' \]
   is regarded as equivalent to the triple
   \[ v \xrightarrow{\pi} y \xleftarrow{\pi'} v', \]
   if there exists an invertible mapping \(\xi:x\to y\) such that \(\nu\xi=\pi\), \(\nu'\xi=\pi'\). A class of equivalent triples \(\bar\alpha\) with representative
   \[ u \xrightarrow{\nu} x \xleftarrow{\nu'} u' \]
   will be called a correspondence of the object \(a\) with the object \(b\) over the category \(K\), if there exist such monomorphisms \(\mu:u\to a\), \(\mu':u'\to a\), that the pairs \((u,\mu)\), \((u',\mu')\) are subobjects of the objects \(a\) and \(b\), respectively (notation:
   \[ \bar\alpha=\langle u \xrightarrow{\nu} x \xleftarrow{\nu'} u',\mu,\mu'\rangle). \]
   The correspondences of the object \(a\) with the object \(b\), as is easy to see, form a set, which we shall denote by \(\mathfrak R(a,b)\). In this set introduce a partial order: the correspondence
   \[ \bar\alpha=\langle u \xrightarrow{\nu} x \xleftarrow{\nu'} u',\mu,\mu'\rangle \]
   precedes the correspondence
   \[ \bar\beta=\langle v \xrightarrow{\pi} y \xleftarrow{\pi'} v',\sigma,\sigma'\rangle, \]
   \(\bar\alpha,\bar\beta\in\mathfrak R(a,b)\), \(\bar\alpha\subset\bar\beta\), if \((u,\mu)\leqslant(v,\sigma)\), \((u',\mu')\leqslant(v',\sigma')\), i.e. \(\mu=\mu_1\sigma\), \(\mu'=\mu'_2\sigma'\), and there exists a mapping \(\varphi:x\to y\) such that \(\nu\varphi=\mu_1\pi\), \(\nu'\varphi=\mu'_1\pi'\).

If
\[ \bar\alpha=\langle u \xrightarrow{\nu} x \xleftarrow{\nu'} u',\mu,\mu'\rangle\in\mathfrak R(a,b), \]
then set
\[ \bar\alpha^{*}=\langle u' \xrightarrow{\nu'} x \xleftarrow{\nu} u,\mu',\mu\rangle\in\mathfrak R(b,a). \]
The indicated mapping of the set \(\mathfrak R(a,b)\) into the set \(\mathfrak R(b,a)\) will be called involution.

Let now
\[ \bar\alpha=\langle u \xrightarrow{\nu} x \xleftarrow{\nu'} u',\mu,\mu'\rangle\in\mathfrak R(a,b),\qquad \bar\beta=\langle v \xrightarrow{\pi} y \xleftarrow{\pi'} v',\sigma,\sigma'\rangle\in\mathfrak R(b,c). \]
Put
\[ (u',\mu')\cap(v,\sigma)=(u'\cap v,\mu'\cap\sigma). \]
Then \(\mu'\cap\sigma=\mu_1\mu'=\sigma_1\sigma\). Let
\[ \mu_1\nu'=\rho_1\tau_1,\qquad \sigma_1\pi=\rho_2\tau_2, \]
where \(\rho_1\rho_2\) are epimorphisms, \(\tau_1,\tau_2\) are monomorphisms. Choose such maximal subobjects
\[ (\bar u,\bar\mu)\leqslant(u,\mu),\qquad (\bar v,\bar\sigma)\leqslant(v',\sigma') \]
of the objects \(a,c\), respectively, that
\[ \mu_1\nu=\lambda_1\tau_1,\qquad \sigma_1\pi'=\lambda_2\tau_2, \]
where \(\bar\mu_1\mu=\bar\mu\), \(\bar\sigma_1\sigma'=\bar\sigma\). It is known that the subobjects \((\bar u,\bar\mu)\), \((\bar v,\bar\sigma)\) are uniquely determined and \(\lambda_1,\lambda_2\) are epimorphisms. If

\(\rho=\rho_1\cap\rho_2: u'\cap v\to z,\ \rho=\rho_1\rho'=\rho_2\rho''\), then the correspondence
\(\widetilde{\gamma}=\langle \bar u \xleftarrow{\lambda_1\rho'} z \xrightarrow{\lambda_2\rho''} \bar v,\ \mu,\sigma\rangle\) will be called the product of the correspondences \(\bar\alpha\) and \(\bar\beta\): \(\bar\gamma=\bar\alpha\circ\bar\beta\). It is not difficult to verify that \(\bar\gamma\) does not depend on the choice of representatives of the correspondences \(\bar\alpha,\bar\beta\), on the choice of representations of the mappings \(\mu_1\nu',\sigma_1\pi\) in the form of a product of an epimorphism and a monomorphism, or on the choice of the epimorphism \(\rho\).

In constructing the correspondence \(\bar\gamma\), a commutative diagram arises; Fig. 1 makes this construction clear.

Theorem. Correspondences over a quasiexact category \(K\) form an \(I\)-category \(\mathfrak K(K)\), with respect to the multiplication, partial order, and involution introduced above, satisfying axioms (K1)—(K3). The category \(K\) is isomorphically embedded in the category \(\mathfrak K(K)\) as the subcategory of all proper mappings.

From this theorem and the results of [1] it follows that between quasiexact categories and \(I\)-categories in which axioms (K1)—(K3) are satisfied there is a one-to-one correspondence, under which each \(I\)-category corresponds to the subcategory of all its proper mappings, and conversely.

Moscow State University
named after M. V. Lomonosov

Received
27 XI 1963

CITED LITERATURE

1. D. Puppe, Math. Ann., 148, 1 (1962).
2. A. G. Kurosh, A. Kh. Livshits, E. G. Shulgeifer, UMN, 15, No. 6 (96), 3 (1960).