Let be a diagram in a category . Also, for any objects in , let and denote cones over .
A cone morphism is a natural transformation such that the diagram
commutes. Note that naturality of any such implies that for all , , so that for some in . The single component itself is often referred to as the cone morphism.
An equivalent definition of a cone morphism says that all component diagrams
The following discussion took place at the component diagram above:
Eric: What is the “component free” way to say that?
Finn Lawler: I think the category of cones over is the comma category , so that a morphism should be just a natural transformation such that . That gives your condition in components, I think.
Eric: Thanks Finn! I’m still learning all this, so it’ll take me some time to absorb what you said. It sounds good though :) Either way, it sounds like some potentially good additional content.
Finn Lawler: I should point out that a natural transformation is very nearly exactly the same thing as a morphism (it’s in each component, which you’ll see if you draw ‘s naturality square, so it’s for some ). Now look at the triangle above, write instead of and erase the s and you have the morphism in the comma category.
I hope this helps. If it’s done the opposite, apologies. I’ve a habit of trying the one and accomplishing the other.
Eric: Hmm. I’m probably confused, but when I draw the naturality square for , I get
for every .
Eric: I think I got it. My diagram is correct, except we have and we want this to be . I made that more explicit in the definition above by adding “whose component is .”
The full blown diagram looks like