Dimension

# 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 $\mathbb{R}^n$ 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 $\mathbb{R}^n$. A complex manifold is of complex dimension $n$ if it is locally isomorphic to $\mathbb{C}^n$, hence has (real) dimension $2 n$.

• A topological space $X$ has (Lebesgue) covering 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 $n \geq -1$; precisely the empty space has dimension $-1$, and precisely the point (of course) has dimension $0$.

• A CW-complex has dimension of a CW-complex, this being the largest $n$ for which there are nontrivial $n$-cells.

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

• A space in noncommutative geometry (spectral geometry via spectral triples) may have a notion of KO-dimension.

### Of objects in an abelian category: Length

See at length of an object.

### 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 $k$)

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

Therefore it makes sense for any symmetric monoidal category $C$ and every dualizable object $V$ to call $tr(Id_V) : 1 \to 1$ 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 $(\infty,1)$-topos

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

## Properties

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