The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a topos. Is that the only way there is?

Well, I suppose there's another way: the category of topological spaces and local homeomorphisms is locally cartesian closed. So one can get a similar effect by throwing out morphisms. But in this example the category is not even accessible.

Let's stipulate that the category should also not be a poset.

After all, a topos is just a locally cartesian closed category with a subobject classifier. Is the subobject classifier merely a representability condition, or does it have additional exactness implications beyond local cartesian closure?

  • $\begingroup$ Isn't a locale locally cartesian closed? $\endgroup$ – Marc Hoyois Feb 28 '18 at 2:46
  • $\begingroup$ Well, a locale is a (1,0)-topos, right? Maybe I should ask my category to be large. $\endgroup$ – Tim Campion Feb 28 '18 at 2:53
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    $\begingroup$ Right. If you allow ∞-categories a standard example would be motivic spaces, but I don't have another 1-categorical example at hand. $\endgroup$ – Marc Hoyois Feb 28 '18 at 2:54
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    $\begingroup$ It's not easy: one can show that the free (discrete) abelian group on the punctured affine line is not a loop space, see Remark 3.1.7 in arxiv.org/pdf/1711.05248.pdf. $\endgroup$ – Marc Hoyois Feb 28 '18 at 3:04
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    $\begingroup$ Ok... so the upshot is that the statement "every 0-truncated group object is the loop space of something [its bar construction?]" is an exactness condition that holds in $\infty$-toposes but not in locally cartesian closed $\infty$-categories. That's really neat! I'm still curious if there are similar conditions that hold in the 1-category case. $\endgroup$ – Tim Campion Feb 28 '18 at 3:12

Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form a topos is the one-point topological space: the category of separated presheaf on it is the category of sets with an initial object $0$ freely added. This category doesn't have disjoint coproducts, since $X\times_{X\sqcup Y}Y\simeq \emptyset $ is not initial. More generally, this should work for any site with an initial object that is covered by the empty sieve.

For completeness, let me add the examples discussed in the comments, which are of a different nature: a locale, a.k.a. a $0$-topos, is presentable and locally cartesian closed, but not a topos. This is an instance of the more general fact that every $n$-topos is presentable and locally cartesian closed, but not an $m$-topos for $m>n$.

Finally, if $k$ is a field, then the ∞-category of motivic spaces over $k$, i.e., $\mathbb A^1$-invariant Nisnevich sheaves on smooth $k$-schemes, is presentable and locally cartesian closed, but is known not to be an ∞-topos: the $0$-truncated group object $a_{Nis}(\mathbb Z[\mathbb A^1-0])$ is an $\mathbb A^1$-invariant Nisnevich sheaf, but it is not equivalent to the loops on its bar construction (so the simplicial colimit in the bar construction is not van Kampen). In light of this, it seems likely that the 1-category of $\mathbb A^1$-invariant Nisnevich sheaves of sets on smooth $k$-schemes (which is locally cartesian closed) is not a topos, but I don't know.

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    $\begingroup$ The category of diffeological spaces would be an example, then. $\endgroup$ – David Roberts Feb 28 '18 at 5:53
  • $\begingroup$ Chalk this up as another one in the category of "I should have realized this"! I suppose it's not a total loss -- I've just added this class of examples to the nlab article on locally cartesian closed categories. $\endgroup$ – Tim Campion Feb 28 '18 at 6:02
  • $\begingroup$ @MarcHoyois this begs the question -- is the unstable motivic category an $\infty$-quasitopos? i.e. is it the separated presheaves on a site? $\endgroup$ – Tim Campion Feb 28 '18 at 6:04
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    $\begingroup$ Just a minor correction: not every Grothendieck quasitopos is "the category of separated presheaves on a site"; in general they consist of presheaves on some small category C that are separated for one topology on C and a sheaf for another topology on C. In other words, the separated objects for a Lawvere-Tierney topology on a Grothendieck (not necessarily presheaf) topos. $\endgroup$ – Mike Shulman Feb 28 '18 at 20:28
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    $\begingroup$ Also, a locale is actually a particular case of a Grothendieck quasitopos: it is the category of sheaves on itself that are separated for the maximal topology. $\endgroup$ – Mike Shulman Feb 28 '18 at 20:45

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