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Today I had the following question: "Is the category of continuous functors a cartesian closed subcategory of ${\bf Cat}\ $?" Suppose I want to find an existing reference for this claim. In that case,

I need to find the result in the literature somehow!

And for me, that's always hard. I look through some of my category theory books, I look on nlab, I check out wikipedia, and I can't find what I'm looking for. Does that mean the result is not in those places? Absolutely not. I usually take it to mean that I don't have an efficient search strategy. Maybe the result is sitting right there, disguised as "Any lextensive category with strongly orthogonal orbits is quintessentially monoidal", or something that I'm simply not able to "see". Or maybe it's there in plain site, but I just missed it. Why can't we avoid this stupid human error?

The present MO question really has three parts, in ascending order of coolness but descending order of concreteness.

  1. Does anyone know a reference for my category theory claim in quotes above?

  2. Does anyone know of a nice strategy for triangulation in math research? I'd like some way to search for "continuous functors" and "cartesian closed", but wouldn't you know it -- doing that in google returns useless results. How can one use books, the web, etc. to perform the kind of research I'm talking about: finding what's known about your question? E.g. suppose the answer is in a work of Johnstone or Kelly. What techniques would I use to realize that fact, given that I don't know it to begin with?

  3. I'd like to hear ideas about the proper structure for the world of mathematical theorems. I'm not looking for an answer such as "It's ${\bf Prop}$, the category of propositions," or anything so simple. Instead, I'm looking for a "strongly searchable" structure in which to store mathematical literature. By strongly searchable, I mean a structure that enables the kind of triangulation I discuss above in 2. This is of course an open-ended question, but perhaps someone has a good idea.

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Could you make this claim more precise? I'm not at all sure what it is you're claiming, but if you mean the category of small categories and functors that preserve all limits that exist in the domain category, then the claim doesn't look at all right to me: this isn't cartesian closed. For example, there is no doubt that the terminal object in $Cat$ is also terminal in this category. Now if $B^C$ is the putative exponential in this cartesian closed category, we should have that continuous functors $1 \to B^C$ correspond to continuous functors $B \to C$. But there is at most one continuous ... –  Todd Trimble Mar 13 '13 at 1:06
(sorry, I meant continuous functors $C \to B$ in the preceding sentence) but there is at most one continuous $1 \to B^C$, namely one which takes the object of $1$ to the terminal object of $B^C$ (if one exists). [There are some other issues here, such as the fact the if we really want to work with $Cat$ or this subcategory 1-categorically, then we should really be working with chosen limits.] If this is how you meant the claim, the problem could be in part that searches failed because the claim is false. –  Todd Trimble Mar 13 '13 at 1:10
Oh boy, you're right -- this was off-base; no wonder I was having trouble both with proving it and with finding it proven in the literature :-P Thanks Todd. –  David Spivak Mar 13 '13 at 2:16
So do you expect that ${\bf Prop}$ is a triangulated category? –  Noam D. Elkies Mar 13 '13 at 4:36
Normally I get reasonable results (far from perfect, but also far from useless) from Google. One needs to use some tricks though: for instance: "sentence", -word, filetype:pdf. –  Federico Poloni Mar 13 '13 at 12:16
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4 Answers

This risks being a useless answer, but the correct method to find references is (drum roll)...

...asking someone who knows more; perhaps by email. Even in the age of MathSciNet, Google, and MO, networking with experts is the way to go. There is someone out there who knows, or knows someone who knows, or gives you a hint to an obscure reference that may or may not have the answer. Plus you get to learn other (un)related math bits and you get to network with very knowledgeable people.

Addendum (in reply to David's comment): My point is that you cannot encode an automated database of mathematical theorems because you do not know how the search query will look like. What happens more often to me is that I find a structure in the setting I study, and notice that some property must hold. When I ask, the answer is something like "That looks like Laramie's quintionic permafrost algebras, but not quite. Your formula is equivalent to Zygyljnski's Platypus Lemma, but the indices are different."

A computer system smart enough to notice that what I describe is related to permafrost algebras would be smart enough to prove theorems of its own. I do not see that coming in the near future.

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But this is a pathetic method. For question 3, I'm not looking for what's avaiable now, I'm looking for what should exist. Looking at the trend in technology, there is clearly some teleological point we're heading toward in which information is better organized. What's the right organization for math? –  David Spivak Mar 13 '13 at 2:00
David, could you please switch to using terms like "20th century" or "uninclusive" instead of "pathetic" on this site? –  S. Carnahan Mar 13 '13 at 2:22
@David, you are rapidly converging to a discussion, and it should most likely happen somewhere else. –  Mariano Suárez-Alvarez Mar 13 '13 at 2:54
I apologize for my wording above. I think I was embarrassed by being wrong in my quoted category theory claim and took it out on the method suggested by Rodrigo. By "pathetic method" I meant "method that should quickly become outdated, especially if we have the courage to think about this problem in earnest". Thanks Scott and others. –  David Spivak Mar 13 '13 at 13:23
@David: Triangulation is in general a harder problem than you seem to give it credit for. It is closely related to the problem, given a proposition P and a set of axioms A, is there a proof of P from A of length at most n? This kind of problem is in general NP-complete. So it's a little optimistic to think that we can completely solve the problem just by building the right widget. –  Timothy Chow Mar 13 '13 at 14:20
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MathOverflow itself serves as an excellent reference tool for this sort of triangulation. When a future researcher googles the topic of lextensive categories with strongly orthogonal orbits (perhaps wondering about their quintessential monoidality) she will find a negative answer in Todd Trimble's comment, as well as your reformulation of her question as "Is the category of continuous functors a cartesian closed subcategory of $\operatorname{Cat}$?"

The actual semantic indexing that's going on is a complex mix of expert networking, StackExchange's software, and Google's search index. How these three work together in a formal, structured way is anybody's guess. Start recording some data and see what structures they elucidate.

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I find myself doing searches like this a lot. In the pre-google days I imagine my method would be basically depth first search through the references, but I don't claim this is optimal. Anyway, I know David knows what DFS is, but I'll describe what I mean in detail anyway because it's the basis for what I really do. First, I'll try to remember where I've read something like this before or start googling till I find something in the ballpark of the result I want. If that paper gives references to other sources for background material (e.g. textbooks, etc) then I'll go look them up. I keep tracing back the chain of references till I am pretty sure the result is not there. Then I go up the chain of papers till I find another avenue which seems fruitful and check it out. Because I keep digital copies of almost all papers and books it's easy to search within them for the keywords. The benefit of doing it this way is that if the terminology has changed at some point in history, and if the papers I'm tracing back along are well-written, then I'll be made aware of which keywords to search for in older documents.

Since we live in a world with google, I often supplement this process above by adding keywords or phrases to my original google search to pare down the number of hits. I never thought of it like this before, but this is the triangulation part of the search. The idea is that if a paper is going to contain the result I want then it will also probably contain this phrase. If I know it should contain the definition of a foobar then I might add the phrase "a foobar is" or "is called a foobar" to the googling. I usually add enough phrases, one step at a time, till the number of results is down to between 5 and 30 hits.

Obviously, there are lots of choices being made in which phrases to add, so I have to estimate probabilities on the fly to guess at which choice is going to be most likely to pare down the results without throwing away the paper that contains the reference I want. I guess this would be the hardest part for a machine to emulate. I make my estimates based on all the papers I've read and the way in which authors usually write. Computers can do things like this nowadays, e.g. the algorithms used to summarize the news. I keep multiple tabs open with different google search results so I can try several of the most likely searches. The tabs usually have some overlap but also some unique results. In the end I don't always get the original place the result appeared, but I can find a place where it's there as a lemma or something. And if I really care about the original place then I can follow the references in that paper and usually find it, or copy the verbiage of that lemma into google to see where the author got it from.

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In addition to moving downward the reference tree, moving upward using citation search is also very helpful. So is moving horizontally (after finding one vaguely relevant paper by some author, scanning other papers of that author to find a more relevant paper, or even better, a survey). Gravitating towards papers that have a high prior probability of containing a good survey of results and "speaks your language" (e.g. if you recognise the author as someone who does a good job in this regard) also cuts down the search time significantly. –  Terry Tao Mar 13 '13 at 16:12
+1 to TT for mentioning upward movement in the tree - I assumed everyone does this but if they don't and can then they should –  Yemon Choi Mar 13 '13 at 16:39
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I do not know much about categories, but I interpret the question as a general question: how to find out whether some mathematical fact is known/true. Here I strongly agree with Rodrigo: ask an expert. Or become an expert yourself:-)

It is still a VERY long way (fortunately!) until Google, Mathscinet and other automatic tools will replace a a real expert. MO is actually a very interesting tool which allows you to address a question to very large number of people. You may be lucky: one of those people might be a good expert, s/he may notice your question and answer it. I have a pretty high rate of success with asking questions on MO :-)

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