§1. The category of sets in vector spaces
1.1. Basic concepts. Let k be a field. We introduce the category SV^ of sets
in k-vector spaces as follows and refer to [43] for notions of category theory. The
objects of SVfc are the pairs (R, X) where X is a k-vector space, and R C X is
a subset which spans X and contains the zero vector. To have a typographical
distinction between the elements of R and those of X, the former will usually be
denoted by Greek letters a, /?,..., and the latter by x, y, z,....
The morphisms f: (R, X) (5, Y) are the ^-linear maps f: X —Y such that
f(R) C S. Hence / is an isomorphism in SV& if and only if / is a vector space
isomorphism mapping R onto 5. Clearly, the pair 0 = ({0}, {0}) is a zero object of
SVfc.
There are two forgetful functors S and V from SV^ to the category Set*
of pointed sets and the category Vec/c of &-vector spaces, respectively, given by
S(R,X) = R and V(R,X) = X on objects, and S(/) = f\R and V(f) = f on
morphisms, respectively. Here the base point of the pointed set R is defined to be
the null vector. We will use the notation
R*
:
=R\{Q}
for the set of non-zero elements of R. Thus R {0} U
Rx.
Clearly V is faithful and so is § because, due to the requirement that R span
X, a linear map on X is uniquely determined by its restriction to R. It is easy
to see that V has a right adjoint which assigns to any vector space X the pair
(X, X) G SVfc. Also, § has a left adjoint £, which assigns to any S G Set* the
following object. Denote by 0 the base point of S and let, as above,
Sx
= 5 \ { 0 } .
Then £(S) is the pair ({0} U {es : s G
Sx},fc(s
)), i.e., the free /c-vector space on
5
X
and its canonical basis {es : s G
Sx}
together with the null vector 0. For a
morphism / : S T of pointed sets, the induced morphism £(/ ) maps es to £f(s)-
The adjunction condition
SVfc0G(S), (R,X)) ^ SeU(S,S(R,X)) = Set*(S,R)
is clear from the universal property of the free vector space on a set. As a conse-
quence, § commutes with limits and V commutes with colimits. This can also be
seen in the following lemmas and propositions.
We next investigate some further basic properties of the category SV^.
1.2.
LEMMA.
Let f: (R,X) (S,Y) be a morphism of SVfc.
(a) f is a monomorphism ^= §(/) is a monomorphism, i.e., f\R: R 5 is
injective.
(b) / is an epimorphism ^=^ V(/) is an epimorphism, i.e., f:X Y is
surjective.
(c) SVfc admits finite direct products and arbitrary coproducts, given by
6
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