### Topos Theory Can Make You a Predicativist

#### Posted by Mike Shulman

Recall that predicative mathematics is mathematics which rejects “impredicative definitions,” and especially power sets. (Power sets are “impredicative” because you can say things like “let $X$ be the intersection of all blah subsets of $S$” and proceed to prove that $X$ is itself blah; thus $X$ has been defined “impredicatively” in terms of a collection of which it is a member.) So it might seem odd to claim that topos theory can make you a predicativist, since the basic ingredient in the definition of an elementary topos is a power object.

However, I mean instead to refer to Grothendieck topos theory. This is usually regarded as a sub-field of elementary topos theory, since every Grothendieck topos is an elementary topos. But actually, that’s really only true *if* $Set$ is itself an elementary topos! So if we were predicativists, we would instead say that $Set$ is some kind of pretopos, and we would conclude instead that any category of sheaves is a pretopos of the same kind. Moreover, we could do lots of ordinary Grothendieck-topos-theory with these “Grothendieck pretopoi” (modulo the usual sorts of difficulties that come up in doing any sort of mathematics predicatively). So “Grothendieck topos theory” is, aside from the confusion of names, fully compatible with predicativism.

But how can Grothendieck topos theory *make* you a predicativist?

To be precise, it’s not really ordinary Grothendieck 1-topos theory that I’m referring to, but higher topos theory and lower topos theory. Just as a 1-topos (I’m going to drop the “Grothendieck” from now on), or (1,1)-topos, is a category of sheaves of sets on a site, and sets are 0-groupoids, so an (n,1)-topos is an $(n,1)$-category of sheaves of $(n-1)$-groupoids on an $(n,1)$-site. It’s established by now, due to the work of many people (but Jacob Lurie wrote a big book about it), that the theory of $(n,1)$-topoi makes good sense for any $0\le n\le \infty$, and mirrors the behavior of 1-topos theory in many ways (and improves on it in others).

In particular, that is so for $n=0$. A (0,1)-topos is a (0,1)-category of sheaves of (-1)-groupoids on a (0,1)-site. There’s a nice description of this process at (0,1)-site; sheaves in this case are often called *ideals*.

However, when one starts to think about $(n,1)$-topoi for arbitrary $n$, one is struck by some curiously distinctive features of the case $n=0$. Most of these arise from the fact that:

- a (0,1)-topos, unlike an $(n,1)$-topos, is an (essentially)
*small*category. As a poset, it can be identified with a frame or a locale, depending on which direction one takes the morphisms as going.

And of course, this fact is a consequence of the power set axiom for $Set$. So if you think about these things a lot, you might start to wonder (as I have), *why* should the case $n=0$ be different from all other values of $n$? We’ve learned not to expect the $(n+1)$-category of all small $n$-categories to be itself a small $(n+1)$-category for any value of $n\ge 0$; why should we expect it to be so for $n=-1$?

By the way, the analogue for higher $n$ of the impredicative definition “let $X$ be the intersection of all blah subsets of $S$” would be “let $X$ be the limit of the diagram of all blah presheaves on $S$.” We know that a presheaf category is complete, but that only means that it has *small* limits, so if we don’t know that there are only set-many blah presheaves, that latter “definition” of $X$ is illegitimate. For instance, “let $X$ be the product of all (set-valued) presheaves on $S$” is certainly illegitimate! So why should we expect to be able to say something analogous like “let $X$ be the intersection of all subsets of $S$”? (Since a subset of $S$ is just a (-1)-groupoid-valued presheaf on $S$.)

So that’s half of how topos theory can make you a predicativist: you might start wondering why we believe that the case $n=0$ is special. The other half is that topos theory (and, really, category theory in general) can show you that often *it doesn’t really matter* whether or not $n=0$ is special. When you first encounter predicativism, you might be shocked at the idea of not having a power set and thus not being able to do any of the familiar things with power sets. (I know I was.) But the negative-thinking approach suggests that that should be *no more or less shocking* than the idea of not having a category of all sets.

But, of course, we *do* have a category of all sets and we work with it all the time; it’s just a *large* category, which is itself a proper class instead of a set. Likewise, predicatively any set has a power *class*. In particular, when doing topos theory, we can work with frames and locales in the same way as before, except that they will no longer necessarily be *small* (0,1)-categories. Rather, they will just be assumed to have a small set of generators, just like we do for Grothendieck $(n,1)$-topoi for all other values of $n$. Yes, certainly, some things will get a bit more tedious, but they won’t come completely crashing down around our ears. This is, I believe, essentially what predicative mathematicians do under the name of formal topology.

## Re: Topos Theory Can Make You a Predicativist

Interesting. Simple as it is now that you say it, I had not been aware of this connection.

So what precisely do we do to set us up in the predicative context in which $(0,1)$-sheaf toposes do show size behaviour analogous to $(n,1)$-sheaf toposes for higher $n$?

We

discard the power set axiom from the definition of $SET$;

and from the definition of Grothendieck universes.

Anything else we need/want to fine tune about our foundations?