# Seven Dogmas of Category Theory

# Seven Dogmas of Category Theory

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

- To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it.
- 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.
- To each natural translation from a construction
*F*:*A*->*B*to a construction*G*:*A*->*B*there corresponds a natural transformation*F*=>*G*. - 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. - To any canonical construction from one species of structure to another corresponds anadjuction between the corresponding categories.
- 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.
- 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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