Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Here's something that I'd like to use in my thesis.. but Im feeling too lazy to write a proof of it, I feel pretty sure this is correct though. I have a feeling that this can be found in a book on category theory. So maybe someone can point me to a reference (I have only used Adamék, Herrlich and Strecker so far).

Conjecture: Short Statement: pullback of inverse limits is the inverse limit of pullbacks

Long Statment: Let $I$ be a directed set and let $\mathbf D,\mathbf E: I \rightarrow \mathcal C$ be two diagrams to a complete category. Let $C$ be an object in $\mathcal C$ and suppose that we have natural sinks $\mathbf Di \rightarrow C$ and $\mathbf Ei\rightarrow C$ (Mac Lane calls these "cones"). Let $A$ and $B$ be the inverse limit of $\mathbf D$ and $\mathbf E$ respectively. We get another diagram that goes to the pullback namely $\mathbf D \times_C \mathbf E : I \rightarrow \mathcal C \times_C \mathcal C$. The claim is that the inverse limit of $\mathbf D \times_C \mathbf E$ is actually the fiber product (or pullback, however you want to call it) $A \times_C B$ (where $A\rightarrow C$ and $B\rightarrow C$ are canonical map that results from the inverse limit and the natural sinks).

share|improve this question
    
I am having difficulty displaying the sinks above using LaTeX. They dont seem to appear correctly.. I wanted to write sinks $\{\mathbf Di \rightarrow C\}$ etc. but they dont appear. –  Jose Capco Jun 18 '10 at 22:05
    
I meant $\{\mathbf Di \rightarrow C\}_I$ –  Jose Capco Jun 18 '10 at 22:06

1 Answer 1

up vote 3 down vote accepted

Yes -- filtered/directed colimits commute with finite limits. See Mac Lane, Categories for the Working Mathematician, theorem IX.2.1.

Edit: Oh, I thought you meant colimits, but it seems you meant limits. But limits always commute with each other (CWM IX.2).

share|improve this answer
    
ah yes pullbacks considered as limits then you make a bifunctor out of the whole thing. Still I needed the reference. Thanks for that. –  Jose Capco Jun 18 '10 at 22:29

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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