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Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of $\mathcal{U}$-small sets (which is also $\mathcal{V}$-small). Suppose that $\mathcal{E}$ is a category which can obtained as a left exact localization of $$Set^{C^{op}}.$$ $\mathcal{E}$ is not (usually) a Grothendieck topos in either universe, but in many ways, it behaves as if it is. It can fail to be locally presentable however.

My question: Exactly which characteristic properties of topoi (e.g. Giraud's axioms, being Cartesian closed...) hold for $\mathcal{E}$ and which do not?

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up vote 6 down vote accepted

By definition, a $\mathbf{U}$-pretopos is a category $\mathcal{C}$ that satisfies Giraud's axioms except for the existence of topological generators, i.e.

  • $\mathcal{C}$ has finite limits,
  • $\mathcal{C}$ is a $\mathbf{U}$-extensive category, i.e. $\mathcal{C}$ has coproducts for $\mathbf{U}$-small families of objects, and coproducts are disjoint and stable under pullback, and
  • $\mathcal{C}$ is an effective regular (= Barr exact) category, i.e. $\mathcal{C}$ has quotients for equivalence relations, and these are effective and stable under pullback.

Let $\mathbf{Set}$ be the category of $\mathbf{U}$-sets. It is clear that $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is always a $\mathbf{U}$-pretopos, regardless of whether $\mathcal{A}$ is $\mathbf{U}$-small or not. And because the definition of $\mathbf{U}$-pretopos involves only colimits and finite limits, any "left exact localisation" (= reflective subcategory with finite-limit-preserving reflector) of a $\mathbf{U}$-pretopos is again a $\mathbf{U}$-pretopos.

By Giraud's theorem, the following are equivalent for a $\mathbf{U}$-pretopos $\mathcal{E}$:

  • $\mathcal{E}$ is a Grothendieck $\mathbf{U}$-topos.
  • $\mathcal{E}$ is a locally presentable $\mathbf{U}$-category.
  • $\mathcal{E}$ has a $\mathbf{U}$-small separating family.

Thus any $\mathbf{U}$-pretopos that fails to be a Grothendieck $\mathbf{U}$-topos is necessarily not locally presentable. This can happen even if it is locally $\mathbf{U}$-small: for instance, take $\mathcal{A}$ to be a one-object groupoid that is $\mathbf{U}$-small and consider $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, this is even an elementary topos!)

Now, if $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is locally $\mathbf{U}$-small, then it is cartesian closed (using the essentially same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). And if $\mathcal{E}$ is a reflective subcategory of $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ with a reflector that preserves finite products, then $\mathcal{E}$ is cartesian closed and inherits exponential objects from $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, $\mathcal{E}$ is an exponential ideal in this case.)

On the other hand, if $\mathcal{A}$ has the property that each slice category $\mathcal{A}_{/ a}$ is essentially $\mathbf{U}$-small (e.g. $\mathcal{A} = \mathbf{Ord}$), then $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ has a subobject classifier (using essentially the same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). This is not a necessary condition: cf. the example where $\mathcal{A}$ is a "big group".

Finally, let me remind you of an important theorem in the theory of elementary toposes: if $\mathcal{F}$ is an elementary topos and $\mathcal{E}$ is a reflective subcategory of $\mathcal{F}$ with finite-limit-preserving reflector, then $\mathcal{E}$ is also an elementary topos.

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Thanks Zhen for the nice answer. Could you perhaps recommend a good reference? –  David Carchedi Nov 28 '13 at 22:59
    
By the way, in light of Todd's answere here: mathoverflow.net/questions/24540/… it seems $[\mathcal{A}^{op},\mathbf{Set}]$ is only locally small if $\mathcal{A}$ is essentially small. –  David Carchedi Nov 28 '13 at 23:04
    
That's certainly not true. As I said, $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is locally small when $\mathcal{A}$ is a "big group". Indeed, the hom-sets are even subsets of the hom-sets of the underlying sets! –  Zhen Lin Nov 28 '13 at 23:07
    
The Elephant has a few things to say about pretoposes scattered here and there, but you can probably find more information by looking up lextensive and effective regular categories separately. –  Zhen Lin Nov 28 '13 at 23:09
    
Ah- well, I suppose one needs the caveat that $\mathcal{A}$ is also locally small, which a big group is not. –  David Carchedi Nov 28 '13 at 23:09
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If one is not comfortable with Grothendieck universes, then a good heuristic is to replace $\mathcal{U}$-small set by "finite set", and $\mathcal{V}$-small set by "set" (or even in some cases by "countable set"), try to answer analogical questions in this setting, and then transfer answers back to the original setting (this is possible because $\aleph_0$ behaves almost like a strongly inaccessible cardinal).

Here are some trivial observations for finite sets:

  • $\mathbf{FinSet}^{\aleph_0}$ does not have finite generating family,

  • $\mathbf{FinSet}^{N}$, where $N$ is the free monoid $\langle 0, +1\rangle$ on a single generator $0$, is not cartesian closed, neither it has a subobject classifier

  • Every category of the form $\mathbf{FinSet}^{\mathbb{C}^{op}}$ has finite limits and colimits (constructed pointwise); therefore every left exact localization of $\mathbf{FinSet}^{\mathbb{C}^{op}}$ has finite colimits (the same is true for finite limits, but for limits this is not actually a useful heuristic).

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Thanks, this is a nice way of thinking :) –  David Carchedi Nov 28 '13 at 23:19
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