MATHEMATICS

A. PULTR, Z. HEDRLIN

ON REPRESENTING SMALL CATEGORIES

(Presented by Academician P. S. Novikov on 1 VII 1964)

1. Notation and definitions. The letters (X, Y, Z) (possibly with indices) always denote sets; the cardinality of a set (X) is denoted by (|X|). A cardinality will be called attainable if it is smaller than the first unattainable (in the strong sense) cardinality.

If (R \subset X \times X), then the pair ((R, X)) (and sometimes also (R) itself) is called a relation on (X); the relation ((R, X)) is called reflexive, symmetric, etc., if (R) has the corresponding property in the usual sense. Instead of ((x, y) \in R) one sometimes writes (xRy). A mapping (f : X \to Y), carrying (X) into a subset of (Y), is understood as a triple ((\varphi, X, Y)), where (\varphi \subset Y \times X) satisfies the well-known conditions; for brevity, sometimes no distinction is made in notation between (f) and

[
\varphi={(f(x),x)\mid x\in X}.
]

If there are relations ((R, X)), ((S, Y)) and a mapping (f : X \to Y), then (f) is called (RS)-admissible if (xRx') implies (f(x)Sf(x')), i.e. if (f\circ R\subset S\circ f) in the sense of composition of relations, and strongly admissible if (f\circ R=S\circ f).

If ((R, X)) is a relation, then (C(R, X)) and, respectively, (C^{}(R, X)) denotes the semigroup of all (RR)-admissible (respectively strongly admissible) mappings; obviously, this is a semigroup with identity. The relation ((R, X)) is called invertible* if (C(R, X)) is a group with identity.

By (\mathfrak R) (respectively (\mathfrak R^{})) is denoted the category whose objects are relations ((R, X)), and whose morphisms from ((R, X)) to ((S, Y)) are the (RS)-admissible (respectively strongly (RS)-admissible) mappings; (\mathfrak R) may also be regarded as the category of directed graphs, with the morphisms being mappings that carry edges into edges. By (\mathfrak R_a) is denoted the full subcategory of (\mathfrak R) whose objects are the antireflexive ((R, X)); by (\mathfrak R_t), the full subcategory of (\mathfrak R^{}) whose objects are transitive antireflexive antisymmetric relations. If (A) is a set, then by (A\mathfrak R) is denoted the following category: the objects are families of relations of the form ({(R_a, X);\, a\in A}); the morphisms from ({(R_a, X);\, a\in A}) to ({(S_a, Y);\, a\in A}) are those (f : X \to Y) which are (R_aS_a)-admissible for every (a\in A). Obviously, if (|A|=1), then (A\mathfrak R) is isomorphic to (\mathfrak R). By (\mathfrak S_0(l)) will be denoted the category of (T_0)-topological spaces, whose morphisms are taken to be local homeomorphisms (onto a space).

Let (\mathfrak K) be a category. If the class (\mathfrak K') of its morphisms is a set, then (\mathfrak K) will be called a small category; the cardinality of (\mathfrak K') will be denoted by (|\mathfrak K|).

Let (\mathfrak K) and (\Omega) be categories. We shall say that the category (\Omega) is representable in (\mathfrak K) if it is isomorphic to a full subcategory of (\mathfrak K).

Finally, let us introduce abbreviated notation for the following assertions (in which (\mathfrak a) is a cardinality): (\mathcal F(\mathfrak a)): there exists an invertible ((R, X)) such that (|X|\ge \mathfrak a); (\mathcal M(\mathfrak a)): for every family of relations

({(R_aX): a\in A}), (|A|\leq \mathfrak a), there exists a relation ((R,Y)) such that the semigroups (C(R,Y)) and (\bigcap{C(R_a,X)\mid a\in A}) are isomorphic; (\mathfrak K(\mathfrak a)): if (|A|\leq \mathfrak a), then (A\mathfrak R) is representable in (\mathfrak R).

2. Basic constructions.

For a given relation ((R,X)) or family of relations ({(R_a,X)}), relations (R^{(1)}, R^{(2)}, R^{(3)}) are constructed (in an essentially unique way).

1) Let ((R,X)) be a relation on the set (X). Put (X^{(1)}=X\cup R\cup U), (U={u_1^1,u_2^1,u_1^2,u_2^2,u_3^2}). On the set (X^{(1)}) define the relation (R^{(1)}) as follows: (u_1^1R^{(1)}u_2^1); (u_i^2R^{(1)}u_{i+1}^2), (i=1,2); (u_2^1R^{(1)}u_1^1); (u_3^2R^{(1)}u_1^2); for every (x\in X) and (i=1,2), (u_1^iR^{(1)}x); for every ((x,y)\in R), (xR^{(1)}(x,y)), ((x,y)R^{(1)}y).

2) Let ((S_i,Y_i)) ((i=1,2)) be relations, (A) a set, (\varphi_i:A\to Y_i) ((i=1,2)) mappings such that there exist (y_i^0\in Y_i\setminus\varphi_i(A)); let (j_1,\ldots,j_4) be natural numbers, (U_k={u_k(1),u_k(2),\ldots,u_k(j_k)}), (U=U_1\cup U_2\cup U_3\cup U_4). Suppose ({(R_aX),\,a\in A}) is a family of relations on (X). Then we put (Z={(x,x',a)\mid xR_ax',\ x,x'\in X}), (X^{(2)}=X\cup Y_1\cup Y_2\cup Z\cup U), and define on the set (X^{(2)}) the relation (R^{(2)}) as follows:

[
\begin{gathered}
u_k(i)R^{(2)}u_k(i+1),\qquad k=1,2,3,4,\quad i=1,2,\ldots,j_k-1;\
u_k(j_k)R^{(2)}u_k(1),\qquad k=1,2,3,4;\
u_k(1)R^{(2)}y_i,\qquad y_i\in Y_i,\quad i=1,2,\quad k=i,i+2;
\end{gathered}
]

for every (x\in X), (i=1,2), (y_i^0R^{(2)}x); for every ((x,x',a)\in Z), (i=1,2), (\varphi_i(a)R^{(2)}(x,x',a)); for every ((x,x',a)\in Z), (xR^{(2)}(x,x',a)) and ((x,x',a)R^{(2)}x'); for (y_i,y_i'\in Y_i), (y_iR^{(2)}y_i') holds if and only if (y_iS_iy_i').

3) Let ((R,X)) be a relation. Let (V={(x,y,i)\mid i=1,2;\ xRy}). Put (X^{(3)}=X\cup V) and define (R^{(3)}) on the set (X^{(3)}) as follows: ((x,y,1)R^{(3)}x), ((x,y,1)R^{(3)}y), ((x,y,1)R^3(x,y,2)), ((x,y,2)R^{(3)}y).

Proposition 1. If ((R,X)) is antireflexive, then (C(R,X)) is isomorphic to (C(R^{(1)},X^{(1)})); moreover, ((R^{(1)},X^{(1)})) has no cycles of odd length not divisible by 3.

A cycle of length (m) for a relation ((S,Y)) is a sequence (x_1,x_2,\ldots,x_n) such that (x_iSx_{i+1}), (x_nSx_1).

Proposition 2. Let ((S_1,Y_1)), ((S_2,Y_2)) be discretized relations, and let there be distinct prime numbers (j_1,j_2,j_3,j_4) such that ((S_i,Y_i)) has no cycle of length (j_k) ((i=1,2;\ k=1,\ldots,4)). Let (\varphi_i) ((i=1,2)) be a one-to-one mapping of the set (A) into (Y_i), with (\varphi_i[A]\ne Y_i). Suppose a family of relations ({(R_a,X),\,a\in A}) is given such that for some (a) we have (R_a={(x,x)\mid x\in X}). Then, for (X^{(2)},R^{(2)}) defined above, the subsemigroup (C(R^{(2)},X^{(2)})) is isomorphic to (\bigcap{C(R_a,X)\mid a\in A}).

Proposition 3. If ((R,X)) is antireflexive, then the semigroup (C[R,X]) is isomorphic to (C_*[R^{(3)},X^{(3)}]).

3.

The main result of the paper is the following assertion:

Theorem 1. If (\mathfrak a) is an attainable cardinal, (\mathfrak K) a small category, (|\mathfrak K|\leq\mathfrak a), then (\mathfrak K) is representable in each of the categories (\mathfrak R), (\mathfrak R_), (\mathfrak R_a), (\mathfrak R_{f}), (\mathfrak T_0(l)).

More generally, the following holds:

Theorem 2. Suppose that for the cardinal (\mathfrak a) the condition (\mathcal F(\mathfrak a)) holds. Then every small category (\mathfrak K) such that (|\mathfrak K|\leq\mathfrak a) is representable in (\mathfrak R_a), (\mathfrak R_t^*), (\mathfrak T_0(l)).

These theorems follow from the following assertions:

Theorem 3. Every small category (\mathfrak K) is representable in the category (A\mathfrak R), and as (A) one may take the set of morphisms of the category (\mathfrak K).

Proof. Denote by (K) the set of objects of the category (\mathfrak K), and by (M(a,b)) the set of morphisms from the object (a) to the object (b). Put (T(a)=\bigl(\bigcup{M(b,a)\mid b\in K};\ {R_\alpha},\ \alpha\in\mathfrak K'\bigr)), where (\beta R_\alpha\gamma\Longleftrightarrow \beta=\gamma\circ\alpha); (T(\beta)={\gamma\mapsto\beta\circ\gamma}). It is easy to verify that this defines a mutu-

has a unique functor in (A\mathfrak R). It remains to show that for every morphism (f:T(a)\to T(b)) there exists (\varphi\in M(a,b)), (f=T(\varphi)). Let (\varepsilon) be the identity morphism of the object (b). Since (a=\varepsilon\circ a), it follows that (aR_\alpha\varepsilon) and, consequently, (f(a)R_{\alpha f}(\varepsilon)), or, by the definition of (R_\alpha), (f(a)=f(\varepsilon)\circ a). Hence (f=T(f(\varepsilon))).

Theorem 4. Let (\mathcal F(|A|)) hold. Then (A\mathfrak R) is representable in (\mathfrak R_a).

Proof. Let ((S_i,Y_i)), (i=1,2), be marked relations, (|Y_i|\ge |A|). We may assume (|Y_i|>1); then the (S_i) are antireflexive. According to Proposition 1, there are distinct primes (j_1,\ldots,j_4) such that (Y_i) has no cycles of length (j_k). We now carry out construction 2) for each object of the category (A\mathfrak R), each time with the same ((S_1,Y_1)), ((S_2,Y_2)). It can be shown that the morphisms of the category (A\mathfrak R) are put in one-to-one correspondence with the morphisms of the objects of the category (\mathfrak R) obtained in the indicated way; more precisely, every morphism from ({(R_a,X);\ a\in A}) to ({(\overline{R}_a,\overline{X});\ a\in A}) is extended uniquely to a morphism from ((R^{(2)},X^{(2)})) to ((\overline{R}^{(2)},\overline{X}^{(2)})), and conversely, every morphism from ((R^{(2)},X^{(2)})) to ((\overline{R}^{(2)},\overline{X}^{(2)})) induces, for each (a\in A), an (R_a\overline{R}_a)-admissible mapping.

Remark. As a consequence, we obtain the following equivalence of statements:

[
\mathcal F(a)\Longleftrightarrow \mathcal M(a)\Longleftrightarrow \mathcal K(a).
]

Theorem 5. The category (\mathfrak R_a) is representable in (\mathfrak R_t^).*

The proof is not difficult to give, using construction 3).

From Theorems 3 and 4, Theorem 2 follows directly. The assertion about the category (\mathfrak L_0(l)) can be obtained by means of a modification of the result on (\mathfrak R_t^*). Theorem 1 is a consequence of Theorem 2 and of the fact that, for attainable cardinal numbers, the assertion (\mathcal F(a)) holds.

Let us note that, according to a personal communication from P. Vopěnka, (\mathcal F(a)) holds even for some further cardinalities (of course, under the assumption of their existence).

Additional remarks. In particular, Theorems 1 and 2 can be applied to semigroups with identity. We obtain the assertion: every semigroup with identity and of attainable cardinality is isomorphic to some (C(R,X)) and to some (C^*(S,Y)).

By definition,
[
C(R,X)={f\mid f\circ R\subset R\circ f},\qquad
C^*(R,X)={f\mid f\circ R=R\circ f}.
]
In this connection, at first sight the question of representing semigroups in one of the following ways seems interesting (capital letters denote relations, and lowercase letters mappings):

A. ({f\mid f\circ g\subset g\circ f}), in other words, ({f\mid f\circ g=g\circ f}).

B. ({S\mid S\circ f\subset f\circ S}).

C. ({S\mid S\circ f=f\circ S}).

D. ({S\mid S\circ R\subset R\circ S}).

E. ({S\mid S\circ R=R\circ S}).

The answer, however, is negative, since even the permutation group of a set consisting of three elements cannot be represented in any of the ways A–E.

Charles University
Prague, Czechoslovakia

Received
24 IV 1964