  The dimension algebra of graded groups is introduced. With the help of known
geometric results of extension theory that algebra induces all known results of
the cohomological dimension theory. Elements of the algebra are equivalence
classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations
of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and
$L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric
products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in
AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \linebreak 2. For pointed
compact spaces $X$ and $Y$, $\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))$
if and only if $X$ and $Y$ are of the same dimension type (i.e.,
$\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of
Hurewicz Theorem in extension theory becomes
$\dim(\pi_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept
of cohomological dimension $\dim_A(X)$ of a pointed compact space $X$ with
respect to a graded group $A$ is introduced. It turns out $\dim_A(X)\leq 0$ iff
$\dim_{A(n)}(X)\leq n$ for all $n\in\Z$. If $A$ and $B$ are two positive graded
groups, then $\dim(A)=\dim(B)$ if and only if $\dim_A(X)=\dim_B(X)$ for all
compact $X$.
