V. V. KUZNETSOV

DUALITY OF FUNCTORS IN THE CATEGORY OF SETS WITH A DISTINGUISHED POINT

(Presented by Academician P. S. Aleksandrov, 12 VI 1964)

We consider the category \(K\), whose objects are sets in each of which a certain point is fixed (we shall denote it by \(0\)), and the set \(\operatorname{Hom}(X,Y)\) of morphisms of an object \(X\) into an object \(Y\) is the set of mappings of \(X\) into \(Y\) taking \(0\) to \(0\). Speaking of a functor in \(K\), we shall regard it as covariant and acting from \(K\) to \(K\). The equality sign between two functors will denote their isomorphism. The category \(K\) is naturally turned into a \(D\)-category \((^1)\), and thereby a duality of functors acting in this category is defined. The functors \(H(X,Y)\) and \(X \otimes Y\), which occur in the definition of a \(D\)-category, are constructed as follows: by \(H(X,Y)\) is denoted the set of morphisms of \(X\) into \(Y\), with the zero mapping being taken as the distinguished point; by \(X \otimes Y\) is denoted the set obtained from the direct product \(X \times Y\) by identifying into one point \(0\) the sets \(X \times 0 \cup 0 \times Y\) (here \(X,Y \in K\); the values of the functors \(H(X,Y)\) and \(X \otimes Y\) on morphisms are defined in the natural way). The functors \(H(A,X)\) and \(A \otimes X\), where \(A\) is a fixed object, are denoted respectively by \(\Omega_A\) and \(\Sigma_A\).

The main results are Theorems 2 and 3. A proposition analogous to Theorem 3 for the case of the category of topological spaces had already been stated by D. B. Fuks \((^2)\). However, it still remains unproved.

Let in \(K\) there be given a certain family of objects \(M_\lambda,\ \lambda \in L\). Considering the direct product \(\prod_{\lambda \in L} M_\lambda\) as an object of \(K\), we take as the zero point in it the point with zero components. The set \(U\), obtained from the union of all \(M_\lambda\) by identifying into one zero point the zeros of the sets \(M_\lambda\), will be called the bouquet of objects of the given family and denoted

\[ U=\bigvee_{\lambda \in L} M_\lambda . \]

For a given family of functors \(P_\sigma,\ \sigma \in S\), the functors \(\prod_{\sigma \in S} P_\sigma\) and \(\bigvee_{\sigma \in S} P_\sigma\) are naturally defined. We shall call a functor \(H\) the union of the functors of the given family if for every \(X \in K\) we have

\[ H(X)=\bigcup_{\sigma \in S} P_\sigma(X) \]

and for every \(\sigma \in S\) the functor \(P_\sigma\) is a subfunctor of the functor \(H\).

Let \(F=\prod_{\sigma \in S} P_\sigma\). Then one can prove that

\[ DF=\bigvee_{\sigma \in S} DP_\sigma \]

(the proof completely coincides with the proof given by D. B. Fuks \((^3)\) of the analogous fact for topological spaces). This proposition means that if \(\varphi_\sigma\) is the natural mapping \(P_\sigma \to \Sigma_B\), and \(f\) is an arbitrary mapping \(F \to \Sigma_B\), then \(f\varphi_\sigma\) is nonzero for at most one \(\sigma \in S\); on the other hand, for any \(\sigma_0 \in S\) and \(\psi: P_{\sigma_0} \to \Sigma_B\) there exists a unique \(f: F \to \Sigma_B\) such that \(f\varphi_{\sigma_0}=\psi\), while for \(\sigma \ne \sigma_0\) the mapping \(f\varphi_\sigma\) is zero.

Let \(L\) and \(S\) be two sets, and suppose that to each \(\lambda \in L\) there is assigned a set \(S_\lambda \subset S\), so that \(S=\bigcup_{\lambda\in L} S_\lambda\), while to each \(\sigma\in S\) there corresponds some functor \(G_\sigma\). We shall consider the functor
\[ F=\bigcup_{\lambda\in L} F_\lambda, \]
where
\[ F_\lambda=\prod_{\sigma\in S_\lambda} G_\sigma . \]
Here, for every \(X\in K\), in \(F_{\lambda'}(X)\) and \(F_{\lambda''}(X)\) we identify those two points \(\{g'_\sigma\}\), \(\sigma\in S_{\lambda'}\), and \(\{g''_\sigma\}\), \(\sigma\in S_{\lambda''}\), for which, if \(\sigma\in S_{\lambda'}\cap S_{\lambda''}\), we have \(g'_\sigma=g''_\sigma\), while, if \(\sigma\notin S_{\lambda''}\) and \(\sigma\notin S_{\lambda'}\), respectively, \(g'_\sigma=0\) and \(g''_\sigma=0\). It turns out that the functor \(DF\) can be obtained by means of a construction of the same kind, the role of the sets \(S\) and \(S_\lambda\) being played respectively by the set \(S\) and by the collections \(s\) of points of the set \(S\) which meet each \(S_\lambda\) in not more than one point, and the role of the set \(L\) by the family \(R\) of all such collections. Namely, we have

Lemma 1. The functor \(DF\) is isomorphic to the functor
\[ T=\bigcup_{s\in R}\prod_{\sigma\in S} DG_\sigma . \]

Let \(f:F\to\Sigma_B\) be an arbitrary point of \(DF(B)\), and let \(f_\lambda\) be the mapping \(F_\lambda\to\Sigma_B\) induced by the mapping \(f\); let \(\psi_B\) be the mapping of the functor \(G_\sigma\subset F_\lambda\) (\(\sigma\in S_\lambda\)) into \(\Sigma_B\) induced by \(f_\lambda\) (it is not hard to see that in the case where \(\sigma\in S_{\lambda'}\) and \(\sigma\in S_{\lambda''}\), the mappings \(\varphi_{\lambda'}\) and \(\varphi_{\lambda''}\) give rise to one and the same mapping \(\psi_\sigma\)). Then the set \(s\) of those indices \(\sigma\) for which \(\psi_\sigma\ne0\) belongs to the family \(R\). Indeed, if there existed \(\sigma',\sigma''\in s\) and \(\lambda\in L\) such that \(\sigma',\sigma''\in S_\lambda\), then the mapping \(\varphi_\lambda\) would induce nonzero mappings \(\psi_{\sigma'}\) and \(\psi_{\sigma''}\) for two factors \(G_{\sigma'}\) and \(G_{\sigma''}\) of the direct product \(F_\lambda\), which is impossible.

Thus the collection \(\bar f=\{\psi_\sigma\}\), \(\sigma\in s\), may be regarded as a point of \(T(B)\). The mapping
\[ \chi_B:DF(B)\to T(B), \]
under which \(f\mapsto\bar f\), is obviously a monomorphism. We shall show that \(\chi_B\) is also an epimorphism.

Let \(s\in R\), and let
\[ f\in \prod_{\sigma\in S} DG_\sigma(B) \]
be a collection of mappings \(\psi_\sigma:G_\sigma\to\Sigma_B\). For each \(\lambda\in L\) define a mapping \(\varphi_\lambda:F_\lambda\to\Sigma_B\) by the following rule: if \(S_\lambda\) contains a point \(\sigma\in s\), then \(\varphi_\lambda\) is defined by the mapping \(\psi_B\), so that \(\varphi_\lambda\theta_\sigma=\psi_\sigma\), where \(\theta_\sigma\) is the natural mapping \(G_\sigma\to F_\lambda\); if, however, the set \(S_\lambda\cap s\) is empty, then the mapping \(\varphi_\lambda\) is zero. It is obvious that, for any \(\lambda',\lambda''\in L\) and any \(X\in K\), the mappings \((\varphi_{\lambda'})_X\) and \((\varphi_{\lambda''})_X\) coincide on the set \(F_{\lambda'}(X)\cap F_{\lambda''}(X)\), and this means that the mappings \(\varphi_\lambda\) define a mapping \(\bar f:F\to\Sigma_B\). Clearly \(\chi_B(\bar f)=\bar f\). The functoriality of the mappings \(\chi_B\) is easily verified. The lemma is proved.

Choose in \(K\) an object \(M\), and specify in it some family \(\mathfrak M\) of subobjects \(M_\lambda\), \(\lambda\in L\). Define the functor \(\Omega_M^{\mathfrak M}\) as a subfunctor of the functor \(\Omega_M\), for which \(\Omega_M^{\mathfrak M}(X)\) is the set of those mappings \(f:M\to X\) such that \(f(M_\lambda)=0\) for at least one \(\lambda\in L\). Associate to every \(\Lambda\subset L\) the set \(N_\Lambda\subset M\), putting
\[ N_\Lambda=\left(\bigcap_{\lambda\in\Lambda} M_\lambda\setminus \bigcup_{\lambda\in L\setminus \Lambda} M_\lambda\right)\cup 0, \]
if \(\Lambda\) is nonempty, and
\[ N_\Lambda=\left(M\setminus \bigcup_{\lambda\in L} M_\lambda\right)\cup 0, \]
if \(\Lambda\) is empty. Obviously,
\[ M/M_\lambda=\bigvee_{\lambda\in L,\ \lambda\notin\Lambda} N_\Lambda . \]
Then we have a series of isomorphisms:
\[ \Omega_M^{\mathfrak M} = \bigcup_{\lambda\in L}\Omega_{M/M_\lambda} = \bigcup_{\lambda\in L}\prod_{\Lambda\subset L-\lambda}\Omega_{N_\Lambda}. \]
We see that the functor \(\Omega_M^{\mathfrak M}\) belongs to the class of functors whose duals were found in Lemma 1.

Using Lemma 1 and taking into account that \(D\Omega_A=\Sigma_A\), \(D\Sigma_A=\Omega_A\), one can verify the validity of the following three assertions.

Lemma 2.
\[ D\Omega_M^{\mathfrak M} = \bigcup_{s\in S}\prod_{\Lambda\in s}\Sigma_{N_\Lambda\setminus\Lambda}, \]
where \(S\) is the family of all collections \(s\) of pairwise disjoint nonempty subsets \(\Lambda\subset L\).

Lemma 3.
\[ DD\Omega^{\mathfrak M}_{M}=\bigcup_{r\in R}\prod_{\Lambda\in r}\Omega_{N_L\setminus\Lambda}, \]
where \(R\) is the family of all sets \(r\) of pairwise intersecting nonempty \(\Lambda\subset L\).

Theorem 1. Every functor of the form \(D\Omega^{\mathfrak M}_{M}\) is reflexive.

Let now \(F\) be some functor. Denote by \(I\) the two-point object of \(K\), and for each \(\alpha\in DF(I)\) construct in \(A=F(I)\) an equivalence \(\lambda_\alpha\) as the least one for which, for any \(X\in K\) and \(y\in F(X)\), if \(\mu',\mu''\in \operatorname{Hom}(X,I)\) are such that
\[ \mu'\alpha_X(y)=\mu''\alpha_X(y)\ne 0, \]
then
\[ F(\mu')y\sim F(\mu'')y; \]
and if \(\mu\in \operatorname{Hom}(X,I)\) is such that \(\mu\alpha_X(y)=0\), then \(F(\mu)y\) is equivalent to \(0\). We shall denote the quotient set of the set \(A\) by the equivalence \(\lambda_\alpha\) (in which the class of points equivalent to zero is distinguished) by \(A_\alpha\), and the identification map \(A\to A_\alpha\) by \(\pi_\alpha\). We also note that, for each \(B\in K\), there is a morphism \(\varepsilon_B:B\to I\) such that \(\varepsilon_B(b)\ne 0\) for any nonzero point \(b\in B\), and, for any \(\varphi\in DF(B)\), a map \(\alpha^\varphi\in DF(I)\) defined by the formula
\[ \alpha^\varphi=DF(\varepsilon_B)\varphi . \]

Lemma 4. Let \(\varphi\in DF(B)\). If
\[ a'\sim a''\pmod{\lambda_{\alpha^\varphi}}, \]
then
\[ \varphi_I(a')=\varphi_I(a''). \]
Conversely, if for some \(\alpha\in DF(I)\) and \(\psi:A\to B\), from
\[ a'\sim a''\pmod{\lambda_\varphi} \]
it follows that
\[ \psi(a')=\psi(a''), \]
then there exists a \(\varphi\in DF(B)\) such that \(\psi=\varphi_I\).

It suffices to prove the first part of the lemma for the case when there exist \(X\in K\), \(y\in F(X)\), and \(\mu',\mu''\in \operatorname{Hom}(X,Y)\) such that
\[ \mu'(\alpha_X^\varphi(y))=\mu''(\alpha_X^\varphi(y))\ne 0 \]
and
\[ F(\mu')y=a',\qquad F(\mu'')y=a''. \]
But then
\[ \varphi_I(a')=\Sigma_B(\mu')\varphi_X(y),\qquad \varphi_I(a'')=\Sigma_B(\mu'')\varphi_X(y), \]
and, since for some \(b\in B\) we have
\[ \varphi_X(y)=\alpha_X^\varphi(y)\otimes b, \]
it follows that
\[ \varphi_I(a')=\varphi_I(a'')=b. \]

Being under the hypotheses of the second part of the lemma, for any \(X\in K\) and \(y\in F(X)\) put
\[ \varphi_X(y)=0 \]
if
\[ \alpha_X(y)=0; \]
but if
\[ \alpha_X(y)\ne 0, \]
then
\[ \varphi_X(y)=\alpha_X(y)\otimes b, \]
where \(b=F(\mu)y\) for such a \(\mu:X\to I\) that
\[ \mu(\alpha_X(y))\ne 0. \]
Let us verify that the morphisms
\[ \varphi_X:F(X)\to B\otimes X \]
define a map of the functor \(F\) into the functor \(\Sigma_B\). Suppose that for some
\[ \nu:X'\to X'' \]
we have
\[ F(\nu)y'=y''. \]
Compare the points
\[ x'\otimes b'=\Sigma_B(\nu)\varphi_{X'}(y') \]
and
\[ x''\otimes b''=\varphi_{X''}(y''). \]
Since
\[ \nu(\alpha_{X'}(y'))=\alpha_{X''}(y''), \]
we have
\[ x'=x''=\alpha_{X''}(y''), \]
and if
\[ \alpha_{X''}(y'')\ne 0, \]
then there exist \(\mu':X'\to I\) and \(\mu'':X''\to I\) such that
\[ \mu'(\alpha_{X'}(y'))=\mu''(\alpha_{X''}(y''))\ne 0 \]
and
\[ F(\mu')y'=F(\mu'')y'', \]
and therefore
\[ b'=b''. \]

Theorem 2. For every functor \(F\), the functor \(DF\) has the form \(\Omega^{\mathfrak M}_{M}\).

We shall exhibit an isomorphism of the functors \(DF\) and \(\Omega^{\mathfrak M}_{M}\) for the set \(M\) obtained from the bouquet
\[ \bigvee_{\alpha\in DF(I)} A_\alpha \]
by identifying, for any \(\alpha',\alpha''\in DF(I)\), points \(a'\in A_{\alpha'}\) and \(a''\in A_{\alpha''}\) if
\[ \pi_{\alpha'}^{-1}(a')=\pi_{\alpha''}^{-1}(a''), \]
and for the family \(\mathfrak M\) of sets \(M_\alpha\), \(\alpha\in DF(I)\), defined by the formula
\[ M_\alpha=(M\setminus \varkappa(A_\alpha))\cup 0, \]
where \(\varkappa\) is the natural map
\[ \bigvee_{\alpha\in DF(I)} A_\alpha\to M. \]

For each \(B\in K\) construct a map
\[ \chi_B:DF(B)\to \Omega^{\mathfrak M}_{M}(B) \]
by assigning to each \(\varphi\in DF(B)\) the map \(\omega:M\to B\) that is zero on the set \(M_{\alpha^\varphi}\) and equal to
\[ \theta_\varphi\varkappa^{-1} \]
on the set
\[ M\setminus M_{\alpha^\varphi}, \]
where \(\theta_\varphi\) is the map
\[ A_{\alpha^\varphi}\to B \]
induced by the map \(\varphi_I\), so that
\[ \varphi_I=\theta_\varphi\pi_{\alpha^\varphi} \]
(the existence of \(\theta_\varphi\) is ensured by Lemma 4).

For any \(B\), the map \(\chi_B\) is a monomorphism. Indeed, suppose that for distinct \(\varphi',\varphi''\in DF(B)\) we have
\[ \chi_B(\varphi')=\chi_B(\varphi'')=\omega . \]
This means that
\[ \omega(M_{\alpha^{\varphi'}})=\omega(M_{\alpha^{\varphi''}})=0, \]
i.e., for all \(a'\in A_{\alpha^{\varphi'}}\) for which
\[ \varkappa(a')\notin \varkappa(A_{\alpha^{\varphi''}}), \]
we have
\[ \omega(\varkappa(a'))=0. \]
Hence the map \(\varphi'_I\) differs from the zero map only at those nonzero points \(a\in A\) for which
\[ \varkappa(\pi_{\alpha^{\varphi'}}(a'))=\varkappa(\pi_{\alpha^{\varphi''}}(a'')); \]
it is easy to conclude that
\[ \varphi'_I=\varphi''_I. \]
But it is not difficult to verify directly that, for distinct \(\varphi',\varphi''\in DF(B)\), the maps \(\varphi'_I\) and \(\varphi''_I\) are distinct.

On the other hand, let us show that for every \(B\) the mapping \(\chi_B\) is also an epimorphism. For this, for an arbitrary \(\omega \in \Omega_M^{\mathfrak M}(B)\) choose \(\alpha\) for which \(\omega(M_\alpha)=0\), and, by means of the mapping \(\psi=\omega\chi\pi_\alpha\), which obviously satisfies the condition of the second part of Lemma 4, construct \(\varphi:F\to\Sigma_B\) for which \(\varphi_I=\psi\). We shall verify that \(\chi_B(\varphi)=\omega\). Denoting by \(A_0\) the set of those points \(a\in A\) for which \(\omega(\chi(\pi_\alpha(a)))=0\), we note that if \(a\in A\), \(X\in K\), \(y\in F(X)\), and \(\mu:X\to I\) are such that \(F(\mu)y=a\), then \(\alpha_X(y)=\alpha_x(y)\ne 0\) when \(a\notin A_0\), and \(\alpha_X^\varphi(y)=0\) for \(a\in A_0\). Hence it is easy to conclude that \(\pi_\alpha^{-1}(0)=A_0\) and that on the set \(A\setminus A_0\) the equivalences \(\lambda_a\) and \(\lambda_{a\varphi}\) coincide. But then \(M_{\alpha\varphi}=M_\alpha\setminus \chi(\pi_\alpha(A_0))\). Comparing the mappings \(\omega\) and \(\omega'=\chi_B(\varphi)\), we shall have \(\omega'(M_{\alpha\varphi})=\omega(M_{\alpha\varphi})=0\) and \(\omega'(M\setminus M_{\alpha\varphi})=\theta_\varphi(\chi^{-1}(M\setminus M_{\alpha\varphi}))=\omega(M\setminus M_{\alpha\varphi})\), i.e. \(\omega'=\omega\).

Thus the mappings \(\chi_B:DF(B)\to\Omega_M^{\mathfrak M}(B)\) are isomorphisms. It is easy to verify the functoriality of these mappings. The theorem is proved.

Theorem 3. For every functor \(F\) in the category \(K\), the functor \(DF\) is reflexive.

From Theorems 1 and 2 it follows immediately that \(DDF\) is reflexive for an arbitrary functor \(F\). It remains to refer to (3), where D. B. Fuks proved that from the reflexivity of \(DDF\) follows the reflexivity of the functor \(DF\). (D. B. Fuks’s proof was given for the category of topological spaces, but in substance it does not use the specifics of this category.)

I take this opportunity to express my gratitude to A. S. Schwarz for advice and suggestions.

REFERENCES

1. A. S. Schwarz, DAN, 148, No. 2, 288 (1963).
2. D. B. Fuks, DAN, 141, No. 4, 818 (1961).
3. D. B. Fuks, Dissertation, M. V. Lomonosov Moscow State University, 1963.