### A crash course in topos theory

10 Mar 2014 09:15 pmI ended up spending this weekend learning a lot about topos theory. Why? I have no idea. But for whatever reason, this happened. Like many areas of category theory, the biggest hurdle is figuring out the terminologyâ€” in particular, figuring out what terminology you actually need to know now vs the terminology you can put off learning until later.

So, with that in mind, I threw together a quick sketch of topos/logos theory. I should emphasize that this is only a *very quick* sketch. It shows when one kind of category is also another kind of category, but for the most part[1] it doesn't say what the additional characteristics of the subkind are (unlike in my other terminology maps). One major thing missing from this sketch is a notation for when one kind of category is exactly the intersection of two other kinds (e.g., a pretopos is *precisely* an extensive exact category). Once I figure out how to show that in PSTricks, I'll post a new version. As before, the dashed lines indicate things you get for free (e.g., every pretopos is coherent for free). The dotted line from Heyting categories to well-powered geometric categories is meant to indicate that, technically, all WPG categories are also Heyting categories, but the Heyting structure is not preserved by geometric functors and therefore should not be "relied on" when reasoning about WPG categories as a whole. And finally, the table at the bottom shows what operators exist in the internal logic of these various categories, as well as what structure the subobject posets have.

Despite being far less polished than my other maps, hopefully it'll give you enough to go on so that you can read through the pages at n-lab in the correct order.

[1] For the arcs which are explained, the following abbreviations are used: "So.C." = subobject classifier; "AMC" = axiom of multiple choice; "NNO" = natural numbers object.; "LCCC" = locally cartesian closed.