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Seven Dogmas of Category Theory

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Seven Dogmas of Category Theory

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Joseph Goguen gave seven dogmas in his paper A Categorical Manifesto.

  1. To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it.
  2. To any natural construction on structures of one species, yielding structures of another species, there corresponds a functor from the category of the first species to the category of the second.
  3. To each natural translation from a construction F : A -> B to a construction GA -> Bthere corresponds a natural transformation F => G.
  4. A diagram D in a category C can be seen as a system of constraints, and then a limitof D represents all possible solutions of the system.
  5. To any canonical construction from one species of structure to another corresponds anadjuction between the corresponding categories.
  6. Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a super-widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected.
  7. Given a species of structure C, then a species of structure obtained by “decorating” or “enriching” that of C corresponds to a comma category under C (or under a functor from C).

Although category theory is all about general patterns, it’s hard to learn what the general patterns of category theory are. The list above is the best high-level description of category theory I’ve seen.

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