Conceptual Mathematics A First Introduction to Categories

Much mathematical struggle (e.g. ‘solving equations’) aims to partially invert a given transformation. In particular, in the case of a functor Φ from one category to another, Kan (1958) noticed that there is sometimes a uniquely determined functor in the opposite direction that, while not actually inverting Φ, is the ‘best approximate inverse’ in either a left- or a right-handed sense. The given functor is typically so obvious that one might not have mentioned it, whereas its resulting adjoint functor is a construction bristling with content that moves mathematics forward.

The uniqueness theorem for adjoints permits taking chosen cases of their existence as axioms. This unification guides the advance of homotopy theory, homological algebra and axiomatic set theory, as well as logic, informatics, and dynamics.

Roughly speaking these reverse functors may adjoin more action, as in the free iteration Φ!(X) of initial data on X, or the chaotic observation Φ*(X) of quantities in X, where Φ strips some of the ‘activity’ from objects Y in its domain. When Φ is instead the full inclusion of constant attributes into variable ones, then the reverse functors effect an ‘averaging’, as in the existential quantification Φ!(X) and the universal quantification Φ*(X) of a predicate X. Exponentiation of spaces satisfies the exponential law that Φ*()L is right adjoint to Φ=()×L. Similarly, implication L ⇒ () of predicates is right adjoint to ()&L.