Is there a topos-theoretic notion of "dimension"?

It seems like almost any "topological" phenomenon has a generalization to toposes. For instance, in Site Characterizations for Geometeric Properties of Toposes, Olivia Caramello shows how we can compute topos theoretic analogues of discrete spaces, locally connected spaces, etc.

The notion of Lebesgue Covering Dimension seems like something which might be amenable to generalization to topoi, since there is already a lot of machinery around understanding (generalizations of) open covers and their refinements. However I can't find any references to any kind of topos theoretic "dimension", either on the nlab, googling or searching the arxiv for things like "topos dimension", or in any of the standard references (Borceux Vol 3, Maclane and Moerdijk, Goldblatt's book, Johnstone's "Topos Theory", as well as The Elephant™).

Of course, following Colin McLarty in the talk Nevertheless One Should Learn the Language of Topos (available here), we expect a topos to behave like a geometric object. If $X$ is a space, then we can identify $X$ with the terminal object of $\mathsf{Sh}(X)$, so we can think of a topos as being sheaves on some idealized geometric object represented by the terminal object.

Obviously when studying a geometric object, its "dimension" is frequently an invariant of much interest, so it surprises me that it's this hard to find anybody talking about a topos theoretic analogue. The question then:

Is there a well studied notion of topos dimension? Or if not, have there been some beginning steps in this direction? Where can I read about it? Has it been used for anything?

If there isn't a pre-existing notion of dimension, is there a reason why? Maybe some of the obvious ideas (like lebesgue covering dimension) don't actually work, or aren't particularly useful? I would still be interested in this case, particularly if there are some examples of things that have been tried and are known to have failed.

Thanks in advance! ^_^

Section 7.2 of Lurie's Higher Topos Theory explores various notions of dimension for toposes, including the homotopy dimension and the cohomology dimension.

The former is a generalization of the Lebesgue covering dimension in a sense that for any paracompact space its covering dimension coincides with the homotopy dimension of its ∞-topos of sheaves.