The Brauer group of a closed category

Abstract

The Brauer group of a finitely bicomplete closed category is defined. This group gives known Brauer groups for the appropriate choices of the closed category. There is a Brauer group functor from the category of commutative monoids in the closed category to the category of Abelian groups. For a finitely bicomplete closed category V [8, p. 180] we define the Brauer group of V, B(V). This paper depends heavily on the calculus of Morita contexts given in [7]. If V is the category of left-modules over a commutative ring R, we have B(V) is the Brauer group of R as defined in [2]; if V is the category of sheaves of modules over a commutative ringed space (X, CU), we have B(V) is the Brauer group of (X, if) as defined in [1]; if V is the category of Z/2Z graded R-modules M with R a commutative ring concentrated in degree zero and with symmetric homomorphism c: M OM'--4M'?M defined by c(m (m M) =(1)m m (m r(mX m) for homogeneous elements m e M and m' E M' of respective degrees (m and 8m', then B(V) is the Brauer-Wall group of R as defined in [4]. We give necessary conditions on a functor T from one closed category V to another V' to insure that T induces a homomorphism from B(V) to B(V'). As a corollary we see that B induces a functor from the category of commutative monoids of V [8, p. 167] to the category of Abelian groups. We recall that there is a one-to-one correspondence between monoids A in V and one-object V-categories [A]. The opposite monoid AO corresponds to the V-category [A]? and the enveloping monoid A' corresponds to the V-category [A] 0 [A]0. We will denote the functor category V[ A] by Presented to the Society, January 25, 1973 under the title Morita contexts and the Brauer group of a closed category; received by the editors July 20, 1973 and, in revised form, February 26, 1974. AMS (MOS) subject classifications (1970). Primary 18D15, 18D35; Secondary 13A20, 16A16.