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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).

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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
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).

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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

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