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In Robert Paré's 2018 presentation "Double Categories: The best thing since slice categories", the slides non-exhaustively give the following listing of interesting situations where double categories arise:

  • External/internal
  • Total/partial (restriction categories; e.g. Pfn & Set)
  • Deterministic/stochastic
  • Classical/quantum
  • Linear/smooth
  • Classical/intuitionistic
  • Lax/oplax (MonCat)
  • Strong/weak
  • Horizontal/vertical (categories of quintets, categories of commuting squares)

I didn't attend the presentation, so I don't know the examples that were used; what are typical examples for each of these situations? I felt that this was small enough for one single question, particularly since somebody might have Paré's original examples, but won't complain if it needs to be split up.

I already know a family of examples for "total/partial"; every restriction category (nLab) can be rearranged into a double category. Also, for "lax/oplax", I gather that the 2-category MonCat of monoidal categories and lax monoidal functors can be turned into a double category with oplax monoidal functors as the other kind of morphism. The slides give examples of generic horizontal-and-vertical-arrow constructions too.

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