Taking limits is functorial

It turns out that in that happy situation where a category $\mathcal{C}$ has $\mathcal{J}$ -limits for some small category $\mathcal{J}$ , the assignment of a limit $\varprojlim_\mathcal{J}F$ to a functor $F$ is itself a functor from $\mathcal{C}^\mathcal{J}$ to $\mathcal{C}$ . That is, given functors $F$ and $G$ from $\mathcal{C}$ to $\mathcal{J}$ and a natural transformation $\eta:F\rightarrow G$ we get an arrow $\varprojlim_\mathcal{J}\eta:\varprojlim_\mathcal{J}F\rightarrow\varprojlim_\mathcal{J}G$ , and composition of natural transformations goes to composition of the limit arrows.

Since the proof of this is so similar to how we established limits in categories of functors, I’ll just refer you back to that and suggest this as practice with those techniques.

About these ads

June 24, 2007 - Posted by John Armstrong | Category theory

2 Comments »

Is there a good reference for this result? I can’t seem to find it in my category theory books including Mac Lane. Thanks!

Comment by bfitzpat22 | February 27, 2012 | Reply
I don’t recall offhand, though I’d have guessed CWM. If it’s not in there, I may have just recognized it and thought it worth mentioning separately.

Comment by John Armstrong | February 27, 2012 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects
Archives

June 2007

M T W T F S S

« May Jul »

1 2 3

4 5 6 7 8 9 10

11 12 13 14 15 16 17

18 19 20 21 22 23 24

25 26 27 28 29 30
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider