UDC 519.48

MATHEMATICS

M. S. BURGIN

CATEGORIES OF CORRESPONDENCES OVER SEMI-ABELIAN CATEGORIES

(Presented by Academician P. S. Novikov on 22 V 1969)

An axiomatic description is given of the categories of correspondences over semi-abelian categories considered in (⁴). In doing so, first the case of categories without direct sums and products—β-categories—is studied, and then the semi-abelian categories themselves.

For the notions of category theory and the properties of various categories used in this paper without references, see (¹, ⁴, ⁵).

1. We shall call a category \(\mathcal K\) a β-category if it satisfies the following axioms:

SA1(A3). In the category \(\mathcal K\) there exists a zero object \(O\).

SA2. For any morphism \(\alpha\) of \(\mathcal K\) there exists a cokernel \(\operatorname{Coker}\alpha\).

A4.a) For a diagram
\[ \xrightarrow{\alpha}\ \uparrow_{\mu}, \]
where \(\mu\) is a monomorphism, there exists a couniversal square

\[ \begin{array}{ccc} & \alpha & \\ \mu' \uparrow & \dashrightarrow & \uparrow \mu\\ & \alpha' & \end{array} \tag{1} \]

b) moreover, if in the couniversal square (1) \(\alpha\) is a normal epimorphism, then \(\alpha'\) is also a normal epimorphism.

From axioms SA1 and A4 it follows that:

1) for any morphism \(\alpha\) of \(\mathcal K\) there exists a kernel \(\operatorname{Ker}\alpha\);

2) for any two monomorphisms \(\mu_i:B_i\to A,\ i=1,2\), there exists their intersection \(\mu=\mu_1\cap\mu_2:B\to A\).

SA3. For any morphism \(\alpha\) of \(\mathcal K\) there exists a canonical decomposition
\[ \alpha=(\operatorname{Coker}\operatorname{Ker}\alpha)\,\dot{\alpha}\,(\operatorname{Ker}\operatorname{Coker}\alpha), \]
in which \(\dot{\alpha}\) is a bimorphism.

A1. \(\mathcal K\) is a locally small category on the left, i.e. the subobjects \(\bigcup(A)\) of an object \(A\in\mathcal K\) form a set.

A7(SA4). If in the diagram (2) the rows are exact in the sense of (⁴), then \(\alpha\) is an equivalence

\[ \begin{array}{cccccc} 0 & \to & \cdot & \xrightarrow{\mu} & \cdot & \xrightarrow{\pi}\ \cdot \to 0\\ & & 1\uparrow & & \uparrow \alpha & \uparrow 1\\ 0 & \to & \cdot & \to & \cdot & \to \cdot \to 0\\ & & \rho & & \nu & \end{array} \tag{2} \]

It is easy to see that, for β-categories, properties 1–6 from (⁴) are valid. In addition, in β-categories axiom SA6 always holds, as do such consequences of it as 12–15 (see (⁴)), and also the properties:

I. Any pair of morphisms \(\varphi_1,\varphi_2\) from \(\mathcal K\) has a left equalizer.

II. The left equalizer is a normal monomorphism.

For the proof of these properties, axioms A4 and SA3 are used.

Let \(R\) be a category with involution (see (¹)), satisfying the axioms:

K1. In \(R\) there exists an \(I\)-zero object \(O\).

K2. a) If \(If\subset Ig\), then \(gf^{*}f\subset g\).

b) If \(Bf\supset Bg\), then \(gf^{*}f\supset g\).

K3. For every morphism \(u \in R(0,A)\) there exists an injection \(m \in R(U,A)\) for which \(Bm=u\).

Various properties of such categories were obtained in the papers \((^{1,3})\). In particular, the subcategory \(G(R)\) of proper morphisms of the category \(R\) satisfies axioms A1, A3, A4, A7 (see \((^3)\)), and for any morphisms \(u,v \in R(0,A)\) there exists an intersection \(u \cap v \in R(0,A)\).

A morphism \(u \in R(0,A)\) will be called strict if, for every \(D\)-regular morphism \(f \in R(A,B)\) and every \(v \in R(0,A)\),
\[ ((u \subset Kf \Rightarrow v \subset Kf) \Rightarrow v \subset u), \]
where \(R(A,B)\) is the set of morphisms from an object \(A \in R\) to an object \(B \in R\). We note that a morphism equal to \(Kf\) for some \(D\)-regular morphism \(f\) is always strict.

K7\(^0\). For every strict morphism \(u \in R(0,A)\) there exists a projection \(p \in R(A,U)\) for which \(Kp=u\).

K8. For every morphism \(v \in R(0,A)\) there exists a minimal strict morphism \(u \in R(0,A)\) for which \(u \supset v\).

K9. If an injection \(m\) is a left equalizer in the category \(G(R)\), then \(Bm\) is a strict morphism.

In a \(\beta\)-category \(\mathcal K\) one can define a correspondence between objects \(A,B \in \mathcal K\) by means of a construction analogous to the construction of correspondences in quasi-exact and \(\gamma\)-categories (see \((^{2,3})\)), and show, as in the paper \((^3)\), that the following is true.

Theorem 1. For an arbitrary \(\beta\)-category \(\mathcal K\) there exists an extension to a category with involution \(R(\mathcal K)\), satisfying axioms K1—K3, K7\(^0\)—K9, and the category \(\mathcal K\) is isomorphic to the subcategory \(G(R(\mathcal K))\) of proper morphisms of the category \(R(\mathcal K)\).

Let \(R\) be a category with involution satisfying axioms K1, K2.

Lemma. If \(m,n\) are injections, and \(Bm,Bn\) are strict morphisms, then \(Bmn\) is a strict morphism \((m \in R(A,B),\ n \in R(B,C))\).

With the help of this lemma and axioms K7\(^0\)—K9 it is verified that the category \(G(R)\) satisfies axioms SA2 and SA3, i.e.

Theorem 2. If a category with involution \(R\) satisfies axioms K1—K3, K7\(^0\)—K9, then its subcategory of proper morphisms \(G(R)\) is a \(\beta\)-category.

Theorem 3. Between \(\beta\)-categories and categories with involution satisfying axioms K1—K3, K7\(^0\)—K9, there exists a one-to-one correspondence up to equivalence of categories.

The proof is carried out in the same way as for \(\gamma\)-categories (see \((^3)\)), relying on Theorems 1, 2 and the following

Proposition. If categories with involution \(R\) and \(R'\) satisfy axioms K1—K3, K7\(^0\)—K9, then:

1) an \(I\)-functor (see \((^1)\)) \(Q:R \to R'\) induces an exact functor
\[ BQ:G(R)\to G(R'), \]
preserving intersections of subobjects;

2) every exact functor \(F:G(R)\to G(R')\), preserving intersections of subobjects, can be uniquely extended to an \(I\)-functor
\[ EF:R\to R'. \]

1. Let the \(\beta\)-category \(\mathcal K\) satisfy axiom A8. For any objects \(A,B \in \mathcal K\) there exists a direct product
   \[ A\times B \in \mathcal K. \]

Then, just as in the case of abelian categories (see \((^1)\)), one can show that the \(\beta\)-category \(\mathcal K\) will be additive (in the sense of Grothendieck \((^6)\)), i.e. it satisfies axiom SA5 and is semiabelian (see \((^4)\)). In this case, instead of axiom A8 one may require the existence in the category \(\mathcal K\) of a couniversal square for an arbitrary diagram

or the fulfillment of the axiom dual to A8:

A8′. For any objects \(A,B \in \mathcal K\) there exists a direct sum (free product)
\[ A\oplus B \in \mathcal K. \]

Let a category with involution \(R\) satisfy axiom K1. We introduce some additional conditions (see (1)):

K4. In the category \(R\), for any pair of morphisms \(f, g \in R(A,B)\), \(A, B \in R\), there exist:

a) their intersection \(f \cap g \in R(A,B)\);

b) their union \(f \cup g \in R(A,B)\).

Remark. The intersection of strict morphisms (see § 1) is a strict morphism.

K5. Let \(f \in R(A,B)\), \(g_1, g_2 \in R(B,C)\). Then:

a) from \(If \subset Kg_1\) it follows that
\[ f(g_1 \cap g_2) \supset fg_1 \cap fg_2; \]

b) from \(Bf \supset Dg_1\) it follows that
\[ f(g_1 \cup g_2) \subset fg_1 \cup fg_2. \]

We note that the relations
\[ f(g_1 \cap g_2) \subset fg_1 \cap fg_2 \]
and
\[ f(g_1 \cup g_2) \subset fg_1 \cup fg_2 \]
follow from the monotonicity of multiplication and from the properties of intersection and union of morphisms.

K6. For any objects \(A_1, A_2 \in R\) there exist:

a) an object \(P \in R\) and proper morphisms \(g_i : P \to A_i\), \(i=1,2\), for which
\[ g_1^{*}g_2=\Omega_{A_1A_2}; \]

b) an object \(S \in R\) and proper morphisms \(h_i : A_i \to S\), \(i=1,2\), for which
\[ h_1h_2^{*}=\omega_{A_1A_2}. \]

Theorem 4. 1) In a category with involution \(R\) satisfying axioms K1—K3, K7\(^0\)—K9, the conditions K4a)—K6a) and K4b)—K6b) are equivalent. 2) If a category with involution \(R\) satisfies axioms K1—K9, then its subcategory \(G(R)\) is semi-abelian.

The proof is carried out analogously to that of D. Puppe \((^1)\) for the abelian case, relying here on the results of the preceding paragraph.

In a semi-abelian category \(\mathcal K\) one can, in the usual way, introduce correspondences as subobjects of direct products and consider the category of correspondences \(R(\mathcal K)\) (see \((^4)\)).

Theorem 5. 1) The category of correspondences \(R(\mathcal K)\) over a semi-abelian category \(\mathcal K\) satisfies axioms K1—K9. 2) Between semi-abelian categories and categories with involution satisfying axioms K1—K9 there is a one-to-one correspondence up to equivalence of categories.

In \((^4)\) it is shown that the category \(R(\mathcal K)\) satisfies axioms K1—K6 and that the category \(\mathcal K\) is isomorphic to the subcategory \(G(R(\mathcal K))\) of proper morphisms of the category \(R(\mathcal K)\). Verifying axioms K7\(^0\)—K9, we obtain the first assertion of the theorem, and, using the assertion from § 1, also the second.

Moscow State University
named after M. V. Lomonosov

Received
7 V 1969

CITED LITERATURE

1. D. Puppe, Math. Ann., 148, 1 (1962).
2. M. S. Tsalenko, Matem. sborn., 73, 564 (1967).
3. M. S. Burgin, Tr. Mosk. matem. obshch., 22 (1970) (in press).
4. D. A. Raikov, DAN, 188, No. 5 (1969).
5. B. Mitchell, Theory of Categories, N.Y.—London, 1965.
6. A. Grothendieck, On Some Questions of Homological Algebra, Moscow, 1961.