Adjoint functors are powerful tools in category theory, connecting different mathematical structures. They pair up functors that work together, like free and forgetful functors in algebra, or product and exponential functors in cartesian closed categories.

These functors solve universal problems, construct important objects, and reveal deep connections. From free groups to tensor products, adjunctions pop up everywhere, linking seemingly disparate areas of math and providing a unified framework for understanding categorical structures.

Adjoint Functors and Their Applications

Adjoint functors in categories

• Free-forgetful adjunction pairs functors constructing free objects and stripping away structure
  • Free functor builds objects with minimal structure (free groups, free modules)
  • Forgetful functor removes algebraic structure retaining underlying set
• Group-Set adjunction connects groups and sets
  • Free functor generates free group from set elements as generators
  • Forgetful functor views group solely as collection of elements
• Vector space-Set adjunction relates vector spaces to sets
  • Free functor creates vector space with set as basis
  • Forgetful functor considers only underlying set of vectors
• Abelianization functor maps groups to abelian groups preserving commutativity
• Tensor-Hom adjunction in module categories connects tensor products and homomorphisms

Adjunctions for universal mapping

• Universal property of adjunctions establishes natural bijection between morphism sets
• Adjoint functors solve universal problems by constructing objects with specific properties
  • Initial objects serve as sources for unique morphisms to all objects
  • Terminal objects receive unique morphisms from all objects
  • Products combine objects while coproducts offer choices between objects
• Free group construction universally embeds set into group category
• Tensor product universally linearizes bilinear maps between modules
• Adjoint functor theorem provides conditions for existence of adjoints using solution sets

Exponential objects and adjunctions

• Cartesian closed categories possess internal hom-objects and terminal objects
• Exponential objects act as right adjoints to product functors
  • $(-) \times Y \dashv (-)^Y$ formalizes currying in category theory
• Curry-Howard-Lambek correspondence connects logic, computation, and category theory
  • Types correspond to objects, functions to morphisms, propositions to types
• Product-exponential adjunction establishes isomorphism between hom-sets
  • $\text{Hom}(X \times Y, Z) \cong \text{Hom}(X, Z^Y)$ enables currying and function evaluation
• Power set functor right adjoins to singleton functor mapping elements to one-element sets

Adjunctions in categorical constructions

• Limits and colimits universally construct objects from diagrams in categories
• Adjoint functor pairs for limits and colimits connect constant and (co)limit functors
  • $\Delta \dashv \lim$ relates constant diagrams to limit objects
  • $\text{colim} \dashv \Delta$ connects colimit objects to constant diagrams
• Kan extensions generalize limits and colimits as adjoint functors
  • Left Kan extensions act as best approximations to left adjoints
  • Right Kan extensions approximate right adjoints
• Monads arise from adjunctions describing algebraic structures
  • Eilenberg-Moore construction yields category of algebras for a monad
  • Kleisli construction produces category of free algebras for a monad
• Galois connections form order-theoretic adjunctions between partially ordered sets
• Adjoint functors in algebraic theories connect free algebras to underlying sets
  • Free algebra functor generates algebras from sets of generators
  • Forgetful functor views algebras as sets forgetting operations