S. Verwulgen
In the sequel S is always a set. Let X
be a category. Fix a codomain IÎ|X|and let K = [
¾,
I]X be the contravariant hom functor. If
X has products then K has a left adjoint: define
FS = IS, then the map h: S
®
KFS: s |®
prs is universal for K. Hence K factors through the (Eilenberg-Moore) algebra's ([4]) of the associated monad. See [7] for a detailed investigation of monads and their
algebra's.
First we fix the codomain to be R and we apply K to the seminormed spaces with linear contractions. It is shown that the spaces of probability charges are the algebra's of the monad induced by
K. This gives rise to a strong representation of B(S, F
), the set of all bounded charges on a field
of subsets of S, as a dual space. A space of probability charges is an abstraction of the algebraic structure on B(S,
F ). A comprehensive account on charges can be found in [1].
The category of locally convex approach spaces lcApVec is studied in [2]. A locally convex approach space is a vector space X together with a saturated ideal
Mx in the lattice of seminorms on
X. For example ]R is considered as such an object by putting MR
= {h:
R®
R | h is a seminorm and h
<| ¾
|} (in the same way the category of seminormed spaces is a full subcategory of
lcApVec). So we can fix R as a codomain and apply K on lcApVec. We show that
AC, the category of absolutely convex modules, is the category of
algebra's of the monad induced by K applied on lcApVec. We show that the category of spaces of probability charges is a full subcategory of
AC. Exactly in this weaker sense, the representation of B(S,
F) as a dual space is understood in the classical way, that is, as an isomorphism between Banach
spaces. Information on absolutely convex modules can be found in [5].
[1] K.P.S. Bhaskara Rao and M. Bhaskara Rao: "Theory of charges", Academic Press, 1983
[2] R. Lowen and S. Verwulgen and M. Sioen: "Locally convex approach vector spaces", to appear
[3] R. Lowen and S. Verwulgen: "Approach Vector Spaces", Rocky Mountain J. of Math., submitted
[4] S. Mc Laine: "Cateories for the Working Mathematician", Springer 1998 (second edition)
[5] D. Pumplun and H. Rohrl: "Banach Spaces and Totally Convex Spaces I", Communications in algebra, 12(8) pp 953--1019 (1984)
[6] D. Pumplun and H. Rohrl: "Banach Spaces and Totally Convex Spaces II", Communications in algebra, 13(5) pp. 1047--1113 (1985)
[7] D. Pumplun: "Eilenberg--Moore algebra's revisited", Seminarberichte, FB Manthematic und Informatic Fernuniversit\"at, Nr. 29 pp 57--144 (1988)