Q:

In what sense is extensivity a minimal requirement on an opfibration to conform with a notion of "family"?

The following is an excerpt from Lawvere's Some thoughts on the future of category theory.

To clarify the above considerations, generalize to distributive
  categories and seek philosophical guidance. Even though the determination
  of which maps are epimorphisms is the more profound
  question studied with Grothendieck topologies, it takes place within a
  topos of the following kind. Call a small category $\mathsf C$ "extensive" if it
  has finite coproducts which yield an equivalence $\mathsf C/A+B = \mathsf C/A \times \mathsf C/B$
  and $\mathsf C/ \mathbf{0} = \mathbf{1}$ (this seems a minimum requirement on an op-fibration to
  conform with the notion of "family" and with Grassmann's
  "combinatorics of continuous magnitudes"); for example. "the" homotopy
  category or the category of spaces of dimension at most 4.

In what sense does extensivity make the codomain opfibration conform with the notion of family? Why is it worthy of being called minimal?
What is Grassman's "combinatorics of continuous magnitudes"? Where can I read about it? How and why does extensivity furnish/oil it?

A:

The only way to know for sure is to ask Lawvere. You could try mailing him.  
For (1), I guess what he had in mind is the following.  
A family $(X_i)_{i \in I}$ of sets indexed by a set $I$ is essentially the same thing as a single set $X$ together with a map $\pi: X \to I$.  The idea here is that $X = \coprod_{i \in I} X_i$ and that $\pi$ sends an element $x \in X_i$ to $i$.  Once you've also done some thinking about maps between families, you conclude that the category of $I$-indexed families of sets is equivalent to the slice category $\mathbf{Set}/I$.
Now, for any two sets $I$ and $J$, a family indexed over $I + J$ should consist of a family indexed over $I$ together with a family indexed over $J$.  That's a theorem if we're talking about families of sets:
$$
\mathbf{Set}/(I + J) \simeq \mathbf{Set}/I \times \mathbf{Set}/J.
$$
But if we're attempting to generalize the notion of family to categories $\mathbf{C}$ other than $\mathbf{Set}$ (as Lawvere is doing), then this identity becomes a sensible axiom.  Thus, he requires that
$$
\mathbf{C}/(A + B) \simeq \mathbf{C}/A \times \mathbf{C}/B
$$
for all $A, B \in \mathbf{C}$.
Similarly, but more trivially, it's a theorem that there's exactly one family of sets indexed over the empty set:
$$
\mathbf{Set}/\emptyset \simeq \mathbf{1}.
$$
Hence Lawvere imposes the axiom
$$
\mathbf{C}/0 \simeq \mathbf{1}
$$
on $\mathbf{C}$.
That's an explanation of why those two axioms on $\mathbf{C}$ are reasonable if you want to be able to talk about "families" in $\mathbf{C}$ and have them behave at all like families of sets.  And though one can imagine imposing further axioms on $\mathbf{C}$ to make families in $\mathbf{C}$ behave even more like families of sets, you don't have to: hence, "minimal".
I have no idea about the Grassmann thing.  A short web search suggests that when Lawvere puts the words "combinatorics of continuous magnitudes" in quotation marks, he's not actually quoting anyone, but indicating that he's coining a term.  It's a shame he doesn't give a reference.

