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Suppose I want to establish a theory of the category $C$ (vector spaces or whatever), but what I really have is $D$, some precisely known category. This is to say, I know all the axioms of $D$, but I only have an intuition for $C$ and I want to develop a theory of $C$. I would normally have diagrams in $C$ by mapping cpos or Domains into $C$. But instead, I want to do it with $D$. What I do is define the largest category, $J$, of Domains in $D$. I do this by defining a dcpo with objects as elements in $D$ and relations as arrows in $D$. Then any functor will map the domains in $J$ to diagrams in $C$.

It seems like I am just inserting a category $D$ in the normal diagram functor $J \rightarrow C$ resulting in $J \rightarrow D \rightarrow C$ which seems to miss the point of the exercise. The point of the exercise, I think, is to try to do a lot of category theory when you have to live in some category $D$.

We start by saying that we "have access to" all diagrams in $D$. Further, we say that we have access to none of the morphisms in any other category. So if we want to talk about a category $C$, it will have to be in terms of diagrams in $D$. Next, we intuit the existence of a category $C$ (I am using this restricted language to reflect the notion that we do don't have access to $C$). Next, we consider endofuntors of $D$, but we really see them as diagrams in $D$ indexed by the domains we constructed in $D$ by $J$. These endofunctors are meant to mimic functors from $D$ to $C$. We are pretending to have access to $C$, by attempting a construction of $C$ in $D$.

Sorry that this is so unclear, especially the idea of having an "intuition of C" and "attempting a construction of". I think that this is an expression of a Topos, and so I have some questions. Firstly, what kind of minimum structure do we need in $D$ to really start doing some work? Second, if we really want to say that we only have access to $D$, then we cannot present $D$ as a set of morphisms and a set of objects because that would imply we are actually in SET, not $D$. Is there any way to start working only in $D$? This goes back to the first question (although thinking about this too much is a bit of a morass).

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closed as not a real question by Todd Trimble, Andreas Blass, Dan Petersen, David Roberts, Johannes Ebert Apr 13 '11 at 8:30


Ben, I find your question a confusing. Are you trying to approximate things in C by diagrams of things in the 'subcategory' D? If that is a bit like what you are trying to do, look at some of the ideas of Categorical Shape Theory. This is related to ideas of generating a full clone of operations from a functor but is also sort of trying to probe the unknown things in C by known things. But my main point is, can you try to clarify the question a bit. (Even breakng it up into paragraphs would help a little.) –  Tim Porter Apr 12 '11 at 20:33
I accidently posted the comment as an answer... which it was not. Sorry if this confuses! –  Tim Porter Apr 12 '11 at 20:35
Tim, Thank you for your questions. I made some small changes. I hope they clarify things. The important concept is "access to". So, if I say I only have access to D, then any discussion (diagram) about C, is actually a diagram in D. –  Ben Sprott Apr 12 '11 at 21:01
Please clarify the question a lot, not just a bit. Can you give an example of what you're trying to do? –  Andrej Bauer Apr 12 '11 at 21:07
Sorry, but I vote to close this as being "not a real question". I encourage you to think very deeply about what you want, and how to turn it into a precise question. –  Todd Trimble Apr 12 '11 at 21:56

2 Answers 2

I am guessing that your $C$ is an internal category in $D$ because I do not understand your question very well. Anyhow, in order to be able to speak sensibly about internal categories in $D$ you should at least require that $D$ have finite limits (otherwise you cannot even express composition of morphisms in an internal category). And having finite limits already carries you a long way. You certainly don't need $D$ to be a topos just to get started, but you may need more conditions on $D$ once you try to do fancy things. It would help if you could give an example of what you are after.

Also, please do not confuse a category $D$ with its presentation. Just because someone presented $D$ to you in terms of sets, that does not prevent us from taking $D$ as the ambient category inside of which we want to do whatever we want to do. In general a category can have many presentations, but so what?

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Andrej, I just want to respond to your final question. I am thinking of realism in the face of quantum theory as in hardy paradox or Kochen-Specker, and asynchronous systems where the set of all processes cannot be agreed on. It is my contention that a presentation of the theory of categories in SET, could be used to translate any, good relational theory of physics, back into one where the realism problem shows up again and we start talking about vectors that exist. Rather, I am exploring the notion that, in physics, we can only approximate a category that can support this kind of language. –  Ben Sprott Apr 13 '11 at 9:58

The question is confusing, but looking at the linked paper that was added in a comment I think that there may be a connection between this and some work that I did in a paper with Jon Gratus (see papers at http://drops.dagstuhl.de/portals/index.php?semnr=04351 but also the published version 'A Spatial View of Information, Theoretical Computer Science 365 (2006) 206-215.)

My advice would be not to impose too much structure on your categories to start with. Look how the probes using D are related. (Sort of philosophically you could take the view that C is there only as much as it can be probed, so you have to relate the probes.) My comment about Shape Theory is also one you might follow up. (Perhaps chat to Rick Blute as you are in the right place to contact him.)

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Tim, Thanks for the link. As you can see, I am avoiding geometry for now in the hopes that it will magically appear later. This kind of hope is strengthened by work like Jamie Vicary where he finds space-like categories in a symmetric monoidal one (see arXiv:0706.0711v2). However, your paper is very interesting and touches on geometry in a way I will want to think more about. –  Ben Sprott Apr 13 '11 at 9:59
