nLab
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 n\mathbb{R}^n has dimension nn.

Of objects in a symmetric monoidal category: Euler characteristic

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

tr(V):=tr(Id V):kVV *V *Vk. 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 CC and every dualizable object VV to call tr(Id V):11tr(Id_V) : 1 \to 1 the categorical trace of VV.

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

Of objects in an (,1)(\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

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

Revised on March 20, 2014 07:41:29 by Urs Schreiber (89.204.154.77)