nLab
dimension

Idea

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.

Of spaces

There are many notions of dimension of spaces. What they all have in common is that the cartesian space ${ℝ}^n$ has dimension $n$.

• The dimension of a vector space $V$ is the cardinality of any set $X$ such that $V$ is free (as a vector space) on $X$. (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.

  The trace of a vector space coincides with its categorical trace in the symmetric monoidal category Vect of vector spaces.

• A manifold is of dimension $n$ if it is locally isomorphic to the Cartesian space ${ℝ}^n$. A complex manifold is of complex dimension $n$ if it is locally isomorphic to ${ℂ}^n$, hence has (real) dimension $2n$.

• A topological space $X$ has (Lebesgue) dimension less than $n$ if every open cover $U$ of $X$ has a refinement $V$ such that every element of $X$ belongs to fewer than $n+1$ elements of $V$. (Then $X$ has dimension $n$ if it has dimension less than $n+1$ but not less than $n$.) By negative thinking, this makes sense for $n\ge -1$; precisely the empty space has dimension $-1$, and precisely the point (of course) has dimension $0$.

• A metric space has a Hausdorff dimension which may be any non-negative real number.

Of objects in a symmetric monoidal category: Euler characteristic

The dimension of a (finite dimensional) vector space $V$ over a field $k$ is equivalently the trace of the identity morphism in the symmetric monoidal category Vect (which is a linear map $k\to k$, canonically identified with an element in $k$)

$$\mathrm{tr}(V):=\mathrm{tr}({\mathrm{Id}}_V):k\to V\otimes V^{*}\stackrel{\simeq }{\to }{V}^{*}\otimes V\to k.$$tr(V) := tr(Id_V) : k \to V \otimes V^* \stackrel{\simeq}{\to} V^* \otimes V \to k \,.

Therefore it makes sense for any symmetric monoidal category $C$ and every dualizable object $V$ to call $\mathrm{tr}({\mathrm{Id}}_V):1\to 1$ the categorical trace of $V$.

This definition subsumes standard notions of Euler characteristic and hence may also be thought of as generalizing that notion.

Of objects in an $(\mathrm{\infty },1)$-topos

The following notions of dimension capture aspects of the concept for objects in a topos or more generally an (∞,1)-topos:

• homotopy dimension

• cohomological dimension

• covering dimension

• Heyting dimension

Properties

• global dimension theorem

References

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

• Jacob Lurie, Higher Topos Theory