A crash course in topos theory

[winterkoninkje]

I 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 terminology— 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.

Comments

[xuenay.livejournal.com]

Unrelated to this post, but I just thought that I'd mention that the topic of you randomly came up on an IRC channel mostly containing Finnish CS/math majors but also some Americans (and one Russian). Which was seriously weird.

22:11 <@arcatan> does wren work with you, btw?
22:11 <@lucidian> who is wren?
22:11 <@arcatan> nwf clearly knows her
22:11 <@ksotala> I think I might know wren
22:12 <@ksotala> Oddly enough
22:12 <@lucidian> nwf did tell me about a wren who once worked here.
22:12 <@arcatan> right
22:13 <@ksotala> Are we talking about the person behind http://winterkoninkje.dreamwidth.org/ ?
22:13 <@arcatan> yeah
22:13 <@ksotala> Huh, the world is tiny
22:13 <@lucidian> this is all new to me, but if it's the wren who used to work here, then yeah, nwf knows her
22:14 <@ksotala> This is seriously weird
22:14 <@ksotala> Like to me wren was this random person who I met because she was reading some transhumanist LiveJournal community where I posted waaay back
22:15 <@Lor> I recognize the name from haskell lists waaay back.
22:15 <@arcatan> i know her from haskell, too
22:15 <@lucidian> (hmm maybe I should not be speaking on nwf's behalf, explaining that he's not ordinary American, and saying who he knows.)
22:15 <@lucidian> (hopefully he'll return soon.)
22:15 <@arcatan> :P
22:15 <@ksotala> (nwf is gone, but you telepathically know that he knows wren?)
22:17 <@ksotala> (Oh, you said that he mentioned her earlier)
22:17 <@lucidian> I have heard the name "wren" come up in tales of former lab members
22:18 <@lucidian> I don't know the details of wren's affiliation with our lab though.
22:19 <@ksotala> I need to send wren a copy of this conversation, obv without specifying who any of you are
22:19 <@ksotala> To maximize the confusion

[winterkoninkje]

Heh :) The world must be especially small for me, cuz stuff like this happens pretty regularly

So, yeah, I was at JHU/CLSP for a couple years. For my masters I was working with nwf and Jason on Dyna 2. Afterwards I had a research position with ccb, Zhifei, et al. working on Joshua.