§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

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