Adjunctions

The previous post looked at adjoints in the context of linear algebra. This post will look at adjoints in the context of category theory.

The adjoint of a linear operator T between inner product spaces V and Wis the linear operatorT* such that for allv inV and allw inW,

Tv, w_W = v, T*w_V.

The subscripts are important because the left side is an inner product inW and the right side is an inner product inV. Think of the inner products as pairings. We pair together elements from two vector spaces. An adjunction in category theory will pair together objects in two categories.

Now suppose we have two categories,C andD. An adjunction betweenC andD is a pair of functors (F,G) with

F:D C

and

G:C D

such that

Hom_C(Fd, c) Hom_D(d, Gc)

for every object d in Dand c in C. Here Hom" means the set of morphisms from one object to another. The subscript reminds us what category the morphisms are in. The left hand side is a set of morphisms in C and the right hand side is a set of morphisms inD.

Note the typographical similarity between

Tv, w_W = v, T*w_V

and

Hom_C(Fd, c) Hom_D(d, Gc).

The functorsF andG are analogous to the operators T andT*. Sets of morphisms in different categories are analogous to inner products in different spaces.

Fine print

There's a lot more going on in the definition of adjoint than is explicit above. In particular, the symbol " is doing a lot of heavy lifting. It's not saying that two sets are equal. They can't be equal because they're sets of different things, sets of morphisms in different categories. It means that there is a bijection between the two sets, and the bijection is natural in d andc." This in turn means that for every fixedd inD, there is a natural isomorphism between Hom_C(Fd, -) and Hom_D(d, G-), and for every fixedc inC there is a natural isomorphism between Hom_C(F-, c) Hom_D(-, Gc). More on this here.

Direction

Note that there is a direction to adjunctions. In the case of linear operators T* is the adjoint ofT; it is not necessarily the case thatT is the adjoint ofT*. In Hilbert spaces,T** = T, but in Banach spaces the two operators might be different.

Similarly,F is the left adjoint of G, and G is the right adjoint of F. This is writtenF G, which implies that direction matters. Adjunctions arefrom one categoryto another. It's possibleto have F G F, but that's unusual.

Note also that in the case of an adjunction from a categoryC to a categoryD, theleft adjoint goes fromD toC, and the right adjoint goes fromC toD. This is the opposite of what you might expect, and many online sources get it wrong.

Hand-wavy examples

Left and right adjoints are inverses in some abstract sense. Very often in an adjunction pair (F,G) the functorF adds structure and the functionG takes it away.F is free" in some sense, andG is forgetful" in some corresponding sense.

For example, letC be the category of topological spaces and continuous functions, andD the space of sets and functions. LetF be the functor that takes sets and gives them the discrete topology, making all functions continuous functions. And letG be the functor that takes a topological space and forgets" its topology, regarding the space simply as a set and regarding its continuous functions simply as functions. Then F G.

Not every adjunction pair follows the free/forgetful pattern, but most examples you'll find in references do. As noted above, sometimes F G F, which could not happen if you always lose structure as you move down the chain of adjunctions.

Other metaphors for left and right adjoints are that left adjoints approximate something from below" (i.e. with less structure) and right adjoints approximate from above" (i.e. with more structure). Some say left adjoints are optimistic" and right adjoints are pessimistic." (This reminds me of last weekend's post about kernels and cokernels: kernels are degrees of freedom and cokernels are constraints.)

Related posts

• Topological Abelian groups
• Category theory without categories
• Universal properties

The post Adjunctions first appeared on John D. Cook.