I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book.

Let us begin with some notation: Let $C$ be a category, and $J$ be an index category, let $F: J \to C$ functor and let $\Delta : C \to C^J$ the diagonal functor (associate to each $c \in \text{Obj}(C)$ the constant functor $J \to F$ and to each map $f \in \text{Arr}(C)$ the natural transformation $\Delta f$ which has the same value $f$ at each object $i$ of $J$ (page 67 of CWM).

I cannot understand the one he gave for the limit (pag. 68):

a limit for a functor F is a universal arrow $\langle r,v \rangle$ from $\Delta$ to $F$.

But according to the definition of universal arrow (pag. 55), $\Delta$ should be an object of $C$ (because $F \colon J \to C$), but instead is an object of $(C^J)^C$. Can someone explain this discrepancy? Am I missing something?

The definition (as universal arrow) he gave for the colimit (pag. 67) was fairly clear, because the functor $F$ is indeed an object of $C^J$, hence the definition of universal arrow from $F$ (seen as an object) to $\Delta$ makes perfectly sense


The point is that the concept of universal arrow in Mac Lane's book is used in two directions: you have universal arrow from a functor to an object but also universal arrow from an object to a functor.

In the case of colimit an universal arrow is a natural transformation $u \colon F \to \Delta(r)$ where $F \in C^J$, $r \in C$ and $\Delta \colon C \to C^J$. For limit the situation is dual: an universal arrow is a natural transformation $u \colon \Delta(r) \to F$ where $F \in C^J$, $r \in C$ and (again) $\Delta \colon C \to C^J$.

The difference between these cases is that in the colimit case universality means that the arrow $u$ is initial between all the natural transformations $F \to \Delta(-)$ while in the limit case the arrow $u$ is terminal between all the natural transformation of type $\Delta(-) \to F$.

Hope this helps.

| cite | improve this answer | |
  • $\begingroup$ Dear Giorgio, thanks for the answer, I'll try writing down the definition of the dual of the universal arrow. Just a clarification, according to my notes a colimit is made of an element $r \in C$ and a natural transformation from $F to \Delta(r)$, while you suggest the contrary. Am I wrong? (I'm thinking of a CW-complex $X$ seen as a colimit of a sequence of sub complexes, and the inclusions goes into $X$, and not from $X$) $\endgroup$ – Luigi M Mar 26 '15 at 16:34
  • 1
    $\begingroup$ @LuigiM forgive me, in the first part of the answer I've got confused... I've edited to correct the mistake. :) $\endgroup$ – Giorgio Mossa Mar 26 '15 at 16:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.