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Suppose that a small category $C$ has sets of objects and arrows that carry the structure of, say, Polish spaces, in some appropriately compatible way. For example $C$ might be an internal category in some category of Polish spaces (with continuous morphisms? Borel morphisms?).

Then it might make sense to study categorical limits that only hold generically (according to some choosen ideal or $\sigma$-ideal).

For example, even if $C$ does not have products, a priori one might still hope for an almost product $A \times_{\rm almost} B$ for objects $A$ and $B$ such that for, say, merely a co-meager set of triples $(C,f:C\rightarrow A,g:C\rightarrow B)$ there exists a unique map from $C$ to $A \times_{\rm almost} B$ so that the appropriate diagram commutes.

For now I'm only asking for: trenchant examples and/or pointers to the literature where anything along these lines arises.

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Doesn't this make the most sense in the category of Polish spaces with maps up to almost everywhere equivalence? –  Will Sawin Oct 24 '12 at 21:02
@Will I thought about that, but I phrased the question the way I did because I thought it would be more attractive to more people if $C$ still constituted an old-fashioned garden-variety category, albeit with some extra structure. Make $C$ internal to a non-concrete category like yours and it gets hard to talk about the source and target of specific arrows, etc. So my motivation, to repair or relax the condition of the existence of limits and colimits for specific diagrams needs reframing. If I understand you?? –  David Feldman Oct 24 '12 at 21:59

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