A notion of dimension is a notion of “size” of objects. There are various variations of what exactly this means, applicaple in various contexts, that tend to agree when they jointly apply.
The dimension of a vector space is the cardinality of any set such that is free (as a vector space) on . (It is a theorem, equivalent to the axiom of choice, that every vector space has a unique dimension.) For modules over rings that are not fields (for which the theorem above does not hold, neither existence nor uniqueness) the term used is rank.
A topological space has (Lebesgue) dimension less than if every open cover of has a refinement such that every element of belongs to fewer than elements of . (Then has dimension if it has dimension less than but not less than .) By negative thinking, this makes sense for ; precisely the empty space has dimension , and precisely the point (of course) has dimension .
See at length of an object.
The dimension of a (finite dimensional) vector space over a field is equivalently the trace of the identity morphism in the symmetric monoidal category Vect (which is a linear map , canonically identified with an element in )
This definition subsumes standard notions of Euler characteristic and hence may also be thought of as generalizing that notion.
For the dimension in symmetric monoidal categories see the references at Euler characteristic.
A general abstract (∞,1)-topos theoretic discusssion of notions of homotopy/cohomology/covering dimension is in section 7.2 of