In my experience, category theory is very successful at providing powerful machinery to reason about large objects or objects unrestricted in size, for example (logical) models (via accessible categories) or "nice" topological spaces (via simplicial sets). However I haven't seen many impressive applications of category theory to categories of finite objects, for example finite graphs or more generally finite models, whose behaviors can be radically different from the corresponding objects of unrestricted size. I certainly haven't read widely enough, but I want to ask anyway: is this finite/infinite divide a conception or misconception? If the former, are there any intuitive explanations? If the latter, what are some convincing counterexamples (for example, applications that provided major insight in what graph theorists would be interested in)?
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4$\begingroup$ Fraïssé limits should be mentioned, I guess golem.ph.utexas.edu/category/2009/11/fraisse_limits.html $\endgroup$– David Roberts ♦Commented Aug 10, 2015 at 12:50
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4$\begingroup$ It may (or may not) be helpful to note that "small" (finite?) objects are often characterised by being compact in their category. E.g. a compact $k$-vector space is the same as a finite-dimensional space, or (at the other end of the range of concreteness) a compact object in the stable homotopy category is the same as the natural generalisation of a finite CW complex. This way of embedding small (finite) objects into categories of huge objects is actually very useful, c/f Brown representability. $\endgroup$– Tom BachmannCommented Aug 10, 2015 at 13:43
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2$\begingroup$ I don't really understand the question. You might be claiming that category theory doesn't say much about finite graphs, to be concrete, that combinatorialists might care about, but I think category theory doesn't say much about infinite graphs that combinatorialists might care about either. I think finitude is a red herring here and that your question is about something else. $\endgroup$– Qiaochu YuanCommented Aug 10, 2015 at 17:35
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1$\begingroup$ On the other hand, why there hasn't been more interaction between category theory and say graph theory (the kind that might appeal to Erdos) is IMO a rather interesting question. The Robertson-Seymour theorem on graph minors being quasi-well-ordered is an intensely interesting structural result that I would love to see category theorists pay attention to. $\endgroup$– Todd TrimbleCommented Aug 10, 2015 at 18:36
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2$\begingroup$ @QiaochuYuan that's part of the question, whether this finitude is a red herring. $\endgroup$– SorcererofDMCommented Aug 10, 2015 at 19:50
2 Answers
I would say you could make good headway on this by looking over some of the research projects of Tom Leinster, Mark Meckes, and Simon Willerton (and others I may be forgetting), centering on various notions of size/magnitude for graphs, finite metric spaces, and various other structures. You could start by perusing the entries under "Size" on Leinster's webpage. Many of the entries with link to "discussions" refer to discussions taking place at the $n$-Category Café over a period of years.
I don't exactly understand the question, but I think you should familiarise yourself with the notion of a fully dualisable object in a (possibly higher) symmetric monoidal category. For example, a vector space is fully dualisable iff it is finite dimensional. In the bicategory of algebras and bimodules, the separable algebras are fully dualisable. This notion often captures desirable finiteness conditions.
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$\begingroup$ Dualizability requires a monoidal structure to be well-defined, and with respect to either the product or the coproduct it's uninteresting (exercise). The OP explicitly gives the example of finite graphs, which dualizability doesn't capture. As I said in the comments, I think finiteness is a red herring here; the details of the OP's question seem to be about something else. $\endgroup$ Commented Aug 20, 2015 at 16:17