UDC 519.48+519.54

MATHEMATICS

L. M. KISSINA, A. S. SHVARTS

ON THE QUESTION OF DESCRIPTION OF THE DUALITY FUNCTOR

(Presented by Academician P. S. Aleksandrov on 15 VI 1965)

The concept of duality of functors, first introduced by D. B. Fuks \((^{1})\) for functors in the category of topological spaces with a distinguished point, was subsequently extended \((^{2})\) to functors in a broad class of categories (in the so-called \(D\)-categories). In the present article, dual functors are described in a certain sense in a number of categories.

1. Let \(\mathcal T\) be a category consisting of topological spaces with a distinguished point. We shall consider functors acting from the category \(\mathcal T\) to the category \(\mathcal G\) of all topological spaces with a distinguished point (by morphisms in the categories \(\mathcal T\) and \(\mathcal G\) we mean continuous mappings taking the distinguished point to the distinguished point). We introduce notation: the set \(\operatorname{Hom}(X,Y)\) of all morphisms of a space \(X \in \mathcal G\) into a space \(Y \in \mathcal G\) with the compact-open topology will be denoted by \(H(X,Y)\), the same set with the topology of pointwise convergence by \(H_s(X,Y)\); the set of all (not necessarily continuous) mappings of \(X\) into \(Y\) taking the distinguished point to the distinguished point by \(\operatorname{Map}(X,Y)\). A functor \(F\) will be called continuous if the mapping defined by it
   \(F:\operatorname{Hom}(X,Y) \to \operatorname{Hom}(F(X),F(Y))\) is a continuous mapping \(H(X,Y)\) into \(H_s(F(X),F(Y))\). We shall assume that all functors under consideration satisfy the condition
   \(F(0_{X,Y}) = 0_{F(X),F(Y)}\) (by \(0_{X,Y}\) we denote the mapping of \(X\) into \(Y\) taking all of \(X\) to the distinguished point of the space \(Y\)). Define the functor
   \(\Sigma_A(X)= A \otimes X\) as the direct product \(A \times X\), factorized by the “coordinate cross” \(A \times 0 \cup 0 \times X\); here we assume that a fundamental system of the zero point of \(A \otimes X\) obtained from any pair of nonzero points \((a,x)\in A \times X\) consists of sets of the form \(U \otimes V\), where \(U\) is a neighborhood of the point \(a \in A\), \(V\) a neighborhood of the point \(x \in X\), while a fundamental system of the zero point of \(A \otimes X\), obtained from any pair \((a,0)\in A \times X\) or \((0,x)\in A \times X\), consists of sets of the form
   \(U \otimes X \cup A \otimes V\), where \(U\) is a neighborhood of the point \(O\) in \(A\), \(V\) a neighborhood of the point \(O\) in \(X\) (this definition of the topology in \(\Sigma_A(X)\) differs from that adopted by D. B. Fuks \((^{3})\), but in the case when \(A\) and \(X\) are bicompact, the two definitions coincide). To each functor \(F\) acting from \(\mathcal T\) to \(\mathcal G\), put in correspondence the dual functor \(DF\), acting from \(\mathcal G\) to the category of sets with a distinguished point; namely, for any \(A \in \mathcal G\) define the set \(DF(A)\) as the set \(\{F \to \Sigma_A\}\) of mappings of the functor \(F\) into the functor \(\Sigma_A\).

By \(E\) and \(P\) we shall denote, respectively, the line and the plane (the distinguished point is the origin). We shall assume that the category \(\mathcal T\) considered by us consists of completely regular spaces and contains the spaces \(E\) and \(P\).

Define the set \(D_{\mathcal R}F(A)\) as the set of mappings of the functor \(F\) into the functor \(\Sigma_A\), if these functors are considered only on the category \(\mathcal R \subset \mathcal T\), consisting of the two spaces \(E, P\) and the morphisms \(P \to E\).

Theorem 1. Let \(F\) be a continuous functor acting from the category \(\mathcal T\) to the category \(\mathcal G\). Then the natural mapping of the set

\(DF'(A)\) into the set \(D_{\mathcal R}F'(A)\) is a one-to-one mapping onto \(D_{\mathcal R}F'(A)\).

Proof. Denote by \(\overline X\) the set of morphisms of the space \(X\) into the line \(E\), and by \(A'\) the space \(A\otimes E\). Define the mapping
\[ i_X:A\otimes X\to \operatorname{Map}(\overline X,A'), \]
by assigning to each point \(a\otimes x\in A\otimes X\) the mapping \(\varphi:\overline X\to A'\) that takes a function \(\psi\in \overline X\) to the point \(a\otimes \psi(x)\in A\otimes E=A'\). From the complete regularity of the space \(X\) it follows that the mapping \(i_X\) is injective. For each morphism \(\mu:F(E)\to A\) we construct a mapping
\[ \widetilde\mu_X:F(X)\to \operatorname{Map}(\overline X,A'), \]
by setting \(\widetilde\mu_X(z)=\psi\), where \(\psi(\varphi)=\mu F(\varphi)z\) (here \(z\in F(X)\), \(\varphi\in \overline X\)). It is not hard to verify that for any \(\lambda\in DF'(A)\) we have
\[ i_X\lambda_X=(\widetilde{\lambda_E})_X; \]
using the injectivity of the mapping \(i_X\), we see that the mapping of functors \(\lambda\in DF'(A)\) is uniquely determined by the morphism \(\lambda_E:F(E)\to A\). A functor \(\widetilde\Sigma_A\) is naturally defined, for which
\[ \widetilde\Sigma_A(X)=\operatorname{Map}(\overline X,A') \]
(this functor acts from the category \(\mathcal T\) to the category of sets); the mappings constructed above
\[ i_X:A\otimes X\to \operatorname{Map}(\overline X,A') \]
and
\[ \widetilde\mu_X:F(X)\to \operatorname{Map}(\overline X,A') \]
generate mappings of functors (if the functors \(\widetilde\Sigma_A\) and \(F\) are also regarded as acting in the category of sets).

Lemma 1. If the condition
\[ \widetilde\mu_XF(X)\subset i_X(A\otimes X) \]
is satisfied for \(X=P\), then it is satisfied for every space \(X\in\mathcal T\).

Let \(z\in F(X)\), \(v=\widetilde\mu_X z\in \operatorname{Map}(\overline X,A')\). We shall show that, under the hypotheses of the lemma, there is a point \(a\otimes x\in A\otimes X\) for which \(i_X(a\otimes x)=v\). Take a morphism \(\sigma:X\to P\) (such a morphism is, evidently, determined by a pair of morphisms \(\psi_1,\psi_2\in\overline X\)) and consider the commutative diagram
\[ \widetilde\mu_PF(\sigma)=\widetilde\Sigma_A(\sigma)\widetilde\mu_X. \]
By the condition of the lemma there is a point
\[ a_\sigma\otimes\tau_\sigma\in A\otimes P \]
such that
\[ \widetilde\Sigma_A(\sigma)(v)=\widetilde\mu_PF(\sigma)z=i_P(a_\sigma\otimes\tau_\sigma). \]
Unraveling the definitions of the mappings \(\widetilde\Sigma_A(\sigma)\) and \(i_P\), we arrive at the following assertion, which we shall call assertion (*) below. Let \(\psi_1,\psi_2\in\overline X\); let \(\lambda(t_1,t_2)\) be a continuous numerical function of two numerical variables, and let \(l_\lambda\) be the function from \(\overline X\) defined by the relation
\[ l_\lambda(x)=\lambda(\psi_1(x),\psi_2(x)). \]
Then there exist points \(\tau=(\tau_1,\tau_2)\in P\) and \(a\in A\) (depending on \(v,\psi_1,\psi_2\)) such that, for every function \(\lambda\),
\[ v(l_\lambda)=a\otimes \lambda(\tau_1,\tau_2). \]

Using assertion (), we first come to the conclusion that if
\[ v(\psi_1)=a_1\otimes v_1\ne0,\qquad v(\psi_2)=a_2\otimes v_2\ne0, \]
then \(a_1=a_2\) (it is enough to consider the functions \(\lambda_1(t_1,t_2)=t_1\) and \(\lambda_2(t_1,t_2)=t_2\)). In other words, the mapping
\[ v:\overline X\to A' \]
has the form
\[ v(\varphi)=a_0\otimes \nu(\varphi), \]
where \(a_0\) is a fixed point of \(A\), and \(\nu(\varphi)\) is a numerical functional on \(\overline X\). It turns out that \(\nu(\varphi)\) is a multiplicative linear functional on the space of functions \(\overline X\). Indeed, applying assertion () to three functions—an arbitrary function \(\lambda(t_1,t_2)\) and the functions \(\lambda_1(t_1,t_2)=t_1\), \(\lambda_2(t_1,t_2)=t_2\)—we obtain that, for any functions \(\psi_1,\psi_2\in\overline X\) and any function \(\lambda(t_1,t_2)\), the relation
\[ \nu(\lambda(\psi_1,\psi_2))=\lambda(\nu(\psi_1),\nu(\psi_2)) \]
holds. Specializing this relation to the functions
\[ \lambda(t_1,t_2)=t_1t_2 \]
and
\[ \lambda(t_1,t_2)=\alpha t_1+\beta t_2, \]
we obtain the multiplicativity and linearity of the functional \(\nu\). Next, we endow the space of functions \(\overline X\) with the compact-open topology and note that, from the continuity condition imposed by us on the functor \(F\), the continuity of the functional \(\nu(\varphi)\) follows. It is known ([4], p. 28) that if the space \(X\) is completely regular and the space \(\overline X\) of continuous functions on \(X\) is endowed with the compact-open topology, then every continuous multiplicative linear functional \(\nu\) on \(\overline X\) is determined by some point \(x_0\in X\) by the formula
\[ \nu(\varphi)=\varphi(x_0). \]
Thus we have
\[ v(\varphi)=a_0\otimes \varphi(x_0), \]
i.e.
\[ v=i_X(a_0\otimes x_0). \]
The assertion of the lemma is proved.

Let us now consider the mapping of the set \(DF(A)\) into the set \(D_{\mathcal R}F'(A)\) appearing in the assertion of the theorem. The one-to-one property

of this mapping follows from the possibility, mentioned above, of reconstructing the mapping of functors \(\lambda:F\to\Sigma_A\) from the mapping \(\lambda_E:F(E)\to\Sigma_A(E)\). It remains for us, therefore, to prove that the mapping \(DF(A)\) into \(D_{\mathscr R}F(A)\) is onto. For this, first note that if an element \(D_{\mathscr R}F(A)\) is determined by a pair of morphisms \(\mu:F(E)\to A\otimes E\) and \(\rho:F(P)\to A\otimes P\), then one may assert that \(\mu_P=i_P\rho\), i.e. Lemma 1 is applicable to the morphism \(\mu\). By Lemma 1, \(\mu_XF(X)\subset i_X(A\otimes X)\) for every \(X\in\mathscr T\), and hence one may define the mapping \(\lambda_X=i_X^{-1}\mu_XF(X)\to A\otimes X\), which, however, a priori need not be a morphism. It turns out, however, that from the compatibility of the mappings \(\lambda_X\) and the continuity of the mapping \(\lambda_E\) one can derive the continuity of all the mappings \(\lambda_X\).

Remark. Theorem 1 may be formulated differently, as a theorem on the independence of the dual functor from the choice of the category with respect to which the dual functor is defined.

If one assumes that the category \(\mathscr T\) contains \(T_0\)-spaces with a finite number of points, then one can prove an analogue of Theorem 1 in which the spaces \(E\) and \(P\) are replaced by still simpler spaces (\(T_0\)-spaces with 2, 3, and 4 points). Consider the category \(\mathscr T\), consisting of \(T_0\)-spaces with a distinguished point \(0\) and satisfying the following conditions: a) the distinguished point of each space \(X\in\mathscr T\) is closed; b) if \(L\) is such a subset of a space \(X\in\mathscr T\) that the system \(V_L\) of open sets containing at least one point of \(L\) is centered, then the intersection of all sets of the system \(V_L\) is nonempty. Suppose also that the category \(\mathscr T\) contains, in particular, the following spaces: the connected two-point space \(I\) (the space with 2 points \(0\) and \(1\), the first of which is closed and the second open); the space with 4 points \(Q=\{0;\,1;\,2;\,3\}\) with open sets \(\{1,3\}\), \(\{2,3\}\), \(\{1,2,3\}\), \(\{3\}\); the space with 3 points \(T=\{0,1,2\}\) with open sets \(\{1,2\}\), \(\{0,1\}\), \(\{1\}\).

Let \(F\) be a functor acting from the category \(\mathscr T\) to the category of all topological spaces \(\mathscr C\). Denote by \(\mathscr L\) the subcategory of the category \(\mathscr T\) consisting of the three spaces \(I,Q,T\) and all morphisms \(Q\to I\), \(T\to I\). Introduce the set \(D_{\mathscr L}F(A)\) of mappings of the functor \(F\), considered only on the category \(\mathscr L\), into the functor \(\Sigma_A\).

Theorem 2. If the functor \(F\) is continuous, then the natural mapping of the set \(DF(A)\) into the set \(D_{\mathscr L}F(A)\) is a one-to-one mapping onto \(D_{\mathscr L}F(A)\).

The proof of Theorem 2 is to a considerable extent analogous to the proof of Theorem 1. Here the role of the line \(E^1\) in the definition of the spaces \(\bar X\) and \(A'\) is played by the connected two-point space \(I\); the space \(\bar X\) is naturally realized as the collection of open sets in \(X\setminus 0\). The role of the assertion used above from \((^4)\) is played by

Lemma 2. Let \(\varphi\in\operatorname{Hom}(\bar X,A')\). In order that the morphism \(\varphi\) belong to \(i_X(A\otimes X)\), it is sufficient that for every pair of sets \(M_1,M_2\in\bar X\) one of the following three conditions be fulfilled:

\[ \begin{aligned} &\text{either }1)\quad \varphi(M_1\cup M_2)=\varphi(M_1)=\varphi(M_2)=\varphi(M_1\cap M_2);\\ &\text{or }2)\quad \varphi(M_1\cup M_2)=\varphi(M_1);\quad \varphi(M_2)=\varphi(M_1\cap M_2)=0;\\ &\text{or }3)\quad \varphi(M_1\cup M_2)=\varphi(M_2);\quad \varphi(M_1)=\varphi(M_1\cap M_2)=0. \end{aligned} \]

The proof of the lemma proceeds according to the following plan. We shall call a set \(M\in\bar X\) nonzero if \(\varphi(M)\ne0\); we shall call a point \(x\in X\) nonzero if every set \(M\in\bar X\) containing it is nonzero. First of all we prove by contradiction that every nonzero set contains a nonzero point. Next, applying condition b), imposed on the category \(\mathscr T\), to the set \(L\) of all nonzero points, we ascertain that there exists a nonzero point \(x_0\) contained in every nonzero set. To complete the proof, we observe that \(\varphi=i_X(x_0\otimes a)\), where \(a\) is the common value of \(\varphi\) on all nonzero sets.

1. Consider a \(D\)-category \(\mathscr K\) satisfying the following conditions: 1) in the category \(\mathscr K\) the tensor product coincides with the direct product (i.e., there exists an isomorphism of functors of two variables \(X\otimes Y\) and \(X\times Y\)); 2) for an arbitrary spectrum \(\{X_\lambda\}\) of objects of the category \(\mathscr K\), from the existence of the inductive limit \(\varinjlim \hat X_\lambda\) of the sets \(\hat X_\lambda\) there follows the existence of the limit \(\varinjlim X_\lambda\); 3) in the category \(\mathscr K\) there is an object \(J\) such that for every object \(X\in\mathscr K\) the set of morphisms of the object \(X\) into the object \(J\) is nonempty.

Theorem 3. For any functor \(F\) in a category satisfying the requirements listed above, the dual functor \(DF\) is a composition \(\Sigma_A\Omega_B\) of the functors \(\Sigma_A\) and \(\Omega_B\).

Proof. By definition,

\[ DF(X)=H(F,\Sigma_X)=\varprojlim \{N_\lambda,\Pi_\lambda^{\lambda'}\}_{\lambda\in\Lambda}, \]

where \(\Lambda\) is the set of morphisms of the category \(\mathscr K\),

\[ N_\lambda=H(F(Y),X\otimes Z), \quad \text{if } \lambda\in \operatorname{Hom}(Y,Z). \]

Let us note that

\[ N_\lambda=H(F(Y),X)\times H(F(Y);Z) \]

(this follows from the coincidence of the tensor product with the direct product and from the relation, valid in any \(D\)-category,

\[ H(A,B\times C)=H(A,B)\times H(A,C)). \]

Therefore \(DF(X)\) also decomposes into a product of two factors; it is not difficult to verify that one of them is isomorphic to \(DF(I)\), where \(I\) is the integral object, and the other to \(H(F,T_X)\), where \(T_X\) is the functor assigning to each object of the category the fixed object \(X\), and to each morphism the identity morphism. To complete the proof it remains only to verify that \(H(F,T_X)=\Omega_B(X)\), where \(B\) is the inductive limit of the spectrum \(\{F(Y),R_Y^{Y'}\}_{Y\in K}\), in which \(R_Y^{Y'}\) is the set of morphisms of the type \(F(d)\); the existence of the limit \(B\) follows from conditions 2 and 3.

The conditions of Theorem 3 are satisfied by the categories of sets, functionally Hausdorff \(k\)-spaces (3), and partially ordered sets (as morphisms one takes, respectively, arbitrary mappings, continuous mappings, and mappings preserving order). In the categories just listed, with the aid of Theorem 3 it can be shown that the stock of reflexive functors is exhausted by functors of the form \(\Sigma_A\) and \(\Omega_A\) (and, consequently, the relation \(D^3=D\) is not satisfied).

Received
15 VI 1965

CITED LITERATURE

1. D. B. Fuks, DAN, 141, No. 4, 818 (1961).
2. A. S. Shvarts, DAN, 148, No. 2, 288 (1963).
3. D. B. Fuks, Matem. sborn., 62, 2, 160 (1963).
4. E. Michael, Locally Multiplicatively—Convex Topological Algebras, Providence, 1952.