THE HOCHSCHILD-MITCHELL DIMENSION OF THE SET OF REALS IS EQUAL TO 3 A. A. Khusainov UDC 513.83 The present article is devoted to the Hochschild-Mitchell dimension [1] of totally ordered subsets of the real line. We prove that the Hochschild-Mitchell dimension of every subset which has cardinality 2o ~ and lies in the set N of reals equals 3 independently of the continuum hypothesis. This answers the question of finding the dimension of the set of reals which was raised by Mitchell in the articles [1, 2]. w1. Preliminary Facts from the Cohomology Theory of Small Categories Let C be a small category. Denote by CAb the category of functors from the category C to the category Ab of abelian groups. The projective dimension p.d. G of a functor G : C ---* Ab is the greatest integer n for which the functor Ext"(G,-) : CAb ~ Ab is nontrivial in the sense that there is a functor F E CAb such that Extn(G, F) r 0. If no such n exists then the projective dimension of the functor G is declared infinite. The category CAb has sufficiently many projectives and injectives, the bifunctor CAb(-, =) is right and left balanced; therefore, the projective dimension p.d. G equals the least length of all projective resolutions of G E CAb. For the sake of exemplification, we consider the functor C(c, -) = h c : C ---, Ens, to the category Ens of sets, associating with each object d E C the set C(c, d) of morphisms c --~ d of C and with each morphism (a: dl ~ d2) E Mor (C), the map C(c,a): C(c, dl) ~ C(c, dz) that takes/~ 6 C(c, dl)into the composition a o fl E C(c, d2). Look at the composition Lh c of the functor h ~ : C -+ Ens and the functor L : Ens ~ Ab that associates with each set E the abelian group LE with basis E and with a map between sets, the homomorphism of free abelian groups which extends the map. It is easy to see that the functor Lh c is a projective in the category CAb; in consequence, p.d. Lh c = O. Given a small category C, denote by colimq C 9 CAb ~ Ab, q > 0, the left derived functors of the colimit functor colim c 9 CAb -* Ab and denote by lim~: 9 CAb ~ Ab the right derived functors of the limit functor limc : CAb ~ Ab. Let S : C --~ D be a functor between small categories. Define the functor (-)S : DAb ~ CAb that associates with each functor F E DAb the composition FS and with a natural transformation r/ : F --* G E Mor (DAb), the natural transformation 7/*S : FS ~ GS that is defined by (~*S)e = ~?s(c)" F(S(c)) ~ G(S(c)). The functor (-)S has a left adjoint functor Lan S " CAb -~ DAb and a right adjoint functor Rans : CAb ~ DAb [3]. The values of LanSF and RansF on the objects d E D are calculated as follows [3]. The fiber S/d of the functor S : C ~ D over an object d 6 D is thought of as the category whose objects are the pairs (c,a) such that c E C and (a : S(c) --~ d) E Mor (D), and the set of morphisms (c,(~) ~ (c',c~'), where c' E C and o/ E C(S(c'),d), between any two of these pairs is constituted by the triples (/3 E C(c,c'), a E D(S(c),d), a' 6 D(S(c'),d)) of morphisms meeting the relation ~' o S(/3) = c~. Define the forgetful functor of a fiber Qd : S/d ~ C as the functor that associates with an object (c,c~) 6 S/d the object c 6 C and with a morphism (fl, c~,~') E Mor(S/d), the morphism E Mor (C). Then, for every functor F 9C --* Ab, the isomorphism LanSF(d) ~- colimS/'~FQd holds [3]. Moreover, the left derived functors Lanq S : CAb --* DAb of the functor Lan s can be Komsomol~sk-na-Amure. Translated from Sibirski~Matematicheski~ Zhurnal, Voh 34, No. 4, pp. 217-227, July-August, 1993. Original article submitted August 28, 1991. 786 0037-4466/93/3404-0786 $12.50 C) 1993 Plenum Publishing Corporation