Construction of Kan Extensions

Here, we will introduce Kan extensions, and give an explicit construction thereof. The data we will work with consists of two small categories, and , along with an arbitrary category that has small colimits. We will fix a functor . Note first that we have a functor . This functor has a left adjoint, called the (left) Kan extension along , denoted . We will explicitly define and show that it has the desired property.

For that, we will need a few additional notations -- for , let denote the comma category consisting of pairs and denote by the forgetful projection functor .

So, for a functor , define on objects as follows:

Theorem: This assignment rule on objects actually extends to a functor.

Proof: We need to give a a well-defined behavior on maps. For maps, take a map in . We need to assign a canonical map . Note that we have a limit cone that comes with a family of maps

for every . At the same time, we have a limit cone and family of maps

for every . But observe that, given an , we can produce . Then we can form the co-cone:

If this is an honest co-cone, then we are guaranteed a unique map , the desired value for . Let's verify that the above is actually a co-cone. Take a map in , which is actually a map such that . Note that (similarly for ) and . So we just the following triangle to commute:

But notice that since , its also true that , thus is also permissible as a map in . Then applying the fact that (2) is a cone to this map produces exactly the commuting triangle (4). So, (3) really is a co-cone, which makes well-defined.

It's fairly straightforward to see that is itself a functor . To see how it behaves on a map (natural transformation) , observe that for each , induces a natural transformation between diagrams . Such a natural transformation induces a map between the limits , so let have components .

Now, let's show the functor is actually the left adjoint.

Theorem:

\bigg( (F(A) \xrightarrow{f} B) \xrightarrow{P_B} A \xrightarrow{X} X(A) \bigg) \xrightarrow{\varepsilon_A} Y(F(A)) \xrightarrow{Y(f)} Y(B)

\bigg( (F(A) \xrightarrow{f} B) \xrightarrow{PB} A \xrightarrow{X} X(A) \bigg) \xrightarrow{p{A,1{FA}}} (\Lan_F X)(F(A)) \xrightarrow{\eta{F(A)}} Y(F(A)) \xrightarrow{Y(f)} Y(B)

\bigg( (F(A) \xrightarrow{f} B) \xrightarrow{P_B} A \xrightarrow{X} X(A) \bigg) \xrightarrow{\varepsilon_A} Y(F(A)) \xrightarrow{Y(f)} Y(F(A))

(note the contravariance due to the Yoneda embedding). Evaluating at a single , we have:

Without further ado, we have:

Using the commuting square from the universal property of , we can rewrite , and we continue:

So and the diagram commutes.

Our final task is to show naturality in . Fix an . We must show is naturally isomorphic to , which is to say this commutes:

Evaluating at an , we have:

And we proceed:

So and the diagram commutes.

So, is well-defined and left-adjoint to , making it the left Kan extension along .