§ First over functions (fake category fluff)

Given a function $g: C \rightarrow E$ and an embedding $j: C \rightarrow D$, then the Right Kan extension of $g$ along $j$, denoted $g/j$ is a new function $g/j: D \rightarrow E$. So we are extending the domain of $g$, along the extender $j$. Informally, we write the new function as $g/j$ because:

C---|   \-g---*j         ||         |v         vD--g/j--->E

foo(j(c)) = g(c)foo = (g/j)(c)g/j(j(c)) = g(c)

§ Next over preorders (real category stuff)

The kan extension provides us a bijection, for any function f: D -> E

C---|   \-g---*j         ||         |v         vD--g/j--->ED------f->Ehom(f.j, g) ~= hom(f, g/j)

That is, if we have some way to make congruent f.j with g, then we can "split" the congruence to have f.j congruent with (g/j).j. Cancelling j, we can have f congruent with g/j.

Consider a preorder with a ordering ≤. Equip with a monoidal structure <>, which is a monotone map with a neutral element. (For example, integers with multiplication and ≤).

The bijection of hom-sets is equivalent to saying

m*k <= n iff m <= n/k

(how? I have no idea; I gotta work this out!)

§ Question: What happens in the context of vector spaces?

Since linear algebra is probably the nicest thing we have in math, I really want to understand what happens in the linear algebra case. I don't really understand how to make the correct version of a kan extension inside $Vect$, though. A kan extension seems to fundamentally be a 2 categorical construct, than a 1 categorical construct.

• Art and dan explain an old trick