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.
