Adjoint Functors and Equivalences

The adjunction bijection

Two functors F : C → D and G : D → C are [[adjoint-functor|adjoint]], written F ⊣ G (F left adjoint, G right adjoint), when there is a bijection, natural in both variables, Hom_D(F(A), B) ≅ Hom_C(A, G(B)). Read it as: a map out of the “free” object F(A) is the same data as a map into the “underlying” object G(B). The universal properties of guide 3 are exactly the statement that certain functors have adjoints.

Free group F : Set -> Grp  is LEFT adjoint to forgetful U : Grp -> Set.

  Adjunction:   Hom_Grp( F(S), H )  ~=  Hom_Set( S, U(H) ).

Unwound: a group homomorphism out of the free group on S is
the SAME thing as a plain function from S to the elements of H.
That is the universal property of a free group -- 'specify a
homomorphism by sending the generators anywhere you like.'

Tensor-Hom adjunction (R commutative, fix module M):

  Hom_R( A (x) M , B )  ~=  Hom_R( A , Hom_R(M, B) ).

So  - (x) M  is LEFT adjoint to  Hom_R(M, -).
This is exactly the universal property of the
[[tensor-product|tensor product]]: bilinear maps A x M -> B
<-> linear maps A (x) M -> B.

Unit and counit (the structural data):
  unit    eta : id_C => G F   (e.g. S -> U F(S), x |-> x as a word)
  counit  eps : F G => id_D   (e.g. F U(H) -> H, evaluate the word)
satisfying the triangle identities
  (eps F)(F eta) = id_F,   (G eps)(eta G) = id_G.

Adjoints preserve, and the exactness clue

Adjunctions are not just bookkeeping — they have teeth. Left adjoints preserve colimits and right adjoints preserve limits (the slogan: LAPC, RAPL). This single theorem explains, for example, why the tensor product distributes over direct sums (⊗ is a left adjoint, ⊕ is a colimit) and why Hom(M, −) sends products to products. It also explains failures: ⊗ is a left adjoint, so it need not preserve the limit that is a kernel — which is precisely why ⊗ is only right exact and Tor appears to measure the defect.

Equivalence of categories

Isomorphism of categories (F and G with FG = id, GF = id on the nose) is too rigid to be useful. The right notion is [[equivalence-of-categories|equivalence]]: functors F : C → D, G : D → C with natural isomorphisms GF ≅ id_C and FG ≅ id_D. Equivalent categories are “the same for all categorical purposes” even when their objects are wildly different sets. Equivalently, F is an equivalence iff it is fully faithful (a bijection on each Hom-set) and essentially surjective (every object of D is isomorphic to some F(A)).

1. Finite-dimensional Vect_k is equivalent to the category of natural numbers with n × m matrices as morphisms n → m. Linear algebra “is” matrix algebra — that is an equivalence, not an isomorphism.
2. An adjunction F ⊣ G upgrades to an equivalence exactly when its unit and counit are isomorphisms. So equivalence is a strong, symmetric special case of adjunction.