MATHEMATICS

I. E. Burmistrovich

EMBEDDING AN ADDITIVE CATEGORY IN A CATEGORY WITH DIRECT PRODUCTS

(Presented by Academician P. S. Aleksandrov on 22 II 1960)

In this note the category of matrices over a given additive category \(K\) is introduced, and it is proved that it is the minimal extension of the category \(K\) in which direct products exist for arbitrary finite sets of objects*. From this there follows, in particular, a theorem on the endomorphism ring of an abelian group decomposed into a direct sum of a finite number of subgroups (see \((^4)\)).

Definition 1. We shall call a category additive if addition is defined in each set \(H(a,b)\), with respect to which it is an abelian semigroup with zero, and this addition is connected with multiplication by the distributive laws.

Definition 2. An object \(g\) of an additive category is called the direct product of the objects \(a_i,\ i=1,\ldots,n\), if mappings \(\pi_i:g\to a_i\), called projections, and \(\sigma_i:a_i\to g\), called embeddings, are given, and moreover

\[ \sigma_i\pi_j= \begin{cases} \varepsilon_{a_i}, & \text{for } i=j,\\ \omega, & \text{for } i\ne j; \end{cases} \tag{1} \]

\[ \sum_{i=1}^{n}\pi_i\sigma_i=\varepsilon_g. \tag{2} \]

We shall denote the direct product by

\[ g=a_1\times a_2\times \cdots \times a_n \quad (\pi_i,\sigma_i). \]

Definition 3. Let an additive category \(\overline{K}\) be given, and in it a full** subcategory \(K\). We shall say that \(\overline{K}\) is a completion of \(K\) if the direct product of any two objects of \(\overline{K}\) exists and every object of \(\overline{K}\) is the direct product of a finite number of objects of \(K\).

Definition 4. The skeleton of a category \(K\) is the full subcategory \(S(K)\) generated by objects chosen one from each class of equivalent objects. Categories with isomorphic skeletons are called coextensive.

For what follows we need to construct the category of matrices \(K^M\) over the additive category \(K\). As objects of \(K^M\) we take all possible finite ordered sets of objects from \(K\), not necessarily distinct. Let \(A=(a_1,\ldots,a_m)\) and \(B=(b_1,\ldots,b_n)\) be two such sets. By a mapping \(h:A\to B\)

* For definitions and notation from category theory see \((^{1-3})\).

** That is, a subcategory containing, together with any two objects \(a,b\), the entire set \(H(a,b)\).

we shall mean any matrix in which, at the intersection of the \(i\)-th row and the \(j\)-th column, there stands a mapping \(\alpha_{ij}: a_i \to b_j,\ \alpha_{ij}\in K;\ i=1,\ldots,m;\ j=1,\ldots,n\).

Let us call the pair of collections \((a_1,\ldots,a_m;\ b_1,\ldots,b_n)\) the type of the matrix \(h\).

Since the category \(K\) is additive, these matrices can be added and multiplied according to the rules usual for addition and multiplication of rectangular matrices.

It is easy to see that \(K^M\) will be an additive category; moreover, if in the category \(K\) all \(H(a,b)\) were abelian groups under addition, then the same is true in \(K^M\). For an object \(A=(a_1,\ldots,a_m)\) of the category \(K^M\), the identity mapping is the matrix

\[ \varepsilon_A= \begin{pmatrix} \varepsilon_{a_1} & 0\\ & \varepsilon_{a_2}\\ & & \ddots\\ 0 & & \varepsilon_{a_m} \end{pmatrix}. \]

It is obvious that \((K^M)^M \simeq K^M\).

A special case of the category \(K^M\) is the collection of all rectangular matrices over an associative ring with identity—the latter is an additive category with one object.

Lemma. The object \(C=(a_1,\ldots,a_m,b_1,\ldots,b_n)\) of the category \(K^M\) is the direct product of the objects \(A=(a_1,\ldots,a_m)\) and \(B=(b_1,\ldots,b_n)\).

Indeed, the projections are the matrices

\[ \pi_1= \begin{pmatrix} \varepsilon_{a_1} & 0\\ & \ddots\\ 0 & \varepsilon_{a_m}\\ & 0 \end{pmatrix} \qquad (m+n\ \text{rows},\ m\ \text{columns}), \]

\[ \pi_2= \begin{pmatrix} 0\\ \varepsilon_{b_1} & 0\\ & \ddots\\ 0 & \varepsilon_{b_n} \end{pmatrix} \qquad (m+n\ \text{rows},\ n\ \text{columns}), \]

and the embeddings are the matrices

\[ \sigma_1= \begin{pmatrix} \varepsilon_{a_1} & 0\\ & \ddots & 0\\ 0 & \varepsilon_{a_m} \end{pmatrix} \qquad (m\ \text{rows},\ m+n\ \text{columns}), \]

\[ \sigma_2= \begin{pmatrix} \varepsilon_{b_1} & 0\\ 0 & \ddots\\ & 0 & \varepsilon_{b_n} \end{pmatrix} \qquad (n\ \text{rows},\ m+n\ \text{columns}). \]

The matrices for the projections and embeddings are obtained, respectively, from the matrix \(\varepsilon_C\) if one “cuts” it in the vertical or, respectively, in the horizontal direction.

Properties (1) and (2) are obvious.

Theorem. The category \(K^M\) is a completion of the additive category \(K\). This completion is unique up to coessentiality.

The first assertion of the theorem follows immediately from the preceding lemma, since every object \(A=(a_1,\ldots,a_m)\) of the category \(K^M\) will be the direct product of the objects \(a_1,\ldots,a_m\) of the category \(K\).

Let now an additive category \(K\) and some completion \(\overline K\) of it be given. It is necessary to show that the skeletons \(S(K^M)\) and \(S(\overline K)\) of the categories \(K^M\) and \(\overline K\) are isomorphic.

For this we shall need the following

Lemma. There exists an additive covariant functor

\[ F:\ K^M \longrightarrow S(\overline{K}), \]

which maps \(K^M\) onto the whole category \(S(\overline{K})\) and has the property that the full inverse image of every object of \(S(\overline{K})\) is a class of equivalent objects in \(K^M\).

We define the functor \(F\) as follows. To an object \(A=(a_1,\ldots,a_m)\) of \(K^M\) we assign the object \(F(A)=a=a_1\times\cdots\times a_m\) \((\pi_i,\sigma_i)\) of \(S(\overline{K})\). Here it is not assumed that \(a_i,\pi_i,\sigma_i\in S(\overline{K})\), but the object \(a=a_1\times\cdots\times a_m\) in \(S(\overline{K})\), by assumption, exists and is unique. To a matrix \((\alpha_{ij})\) of type \((a_1,\ldots,a_m;\ b_1,\ldots,b_n)\) we assign the mapping

\[ F((\alpha_{ij}))=\sum_{i,j}\pi_i\alpha_{ij}\sigma'_j:\ a\longrightarrow b, \]

where

\[ a=a_1\times\cdots\times a_m \quad (\pi_i,\sigma_i), \]

\[ b=b_1\times\cdots\times b_n \quad (\pi'_j,\sigma'_j). \]

It is easy to see that this functor satisfies the requirements of the lemma. Therefore the functor

\[ F_1:\ S(K^M)\longrightarrow S(\overline{K}), \]

induced by the functor \(F\), gives an isomorphic mapping of the skeleton \(S(K^M)\) onto the skeleton \(S(\overline{K})\).

Moscow State University
named after M. V. Lomonosov

Received
16 II 1960

REFERENCES

¹ S. Eilenberg, S. MacLane, Trans. Am. Math. Soc., 58, 231 (1945).
² J. R. Isbell, Canad. J. Math., 9, 563 (1957).
³ A. G. Kurosh, Tr. Mosk. matem. obshch., 8, 391 (1959).
⁴ A. G. Kurosh, Group Theory, Moscow, 1953, p. 140.