# Inverse limit in the category of sets

This is in reference to page 153 of the notes found here http://www.math.lsa.umich.edu/~hochster/614F10/614.pdf

In the second paragraph, the author is considering inverse limit of a family of subsets of a certain set where the defining maps of the inverse systems are inclusions. I am not sure why the inverse limit is the intersection of the subsets.

For e.g. consider a set $X$ and four subsets $X_1,X_2,X_3,X_4$ s.t. $X_1\subset X_2$ and $X_3\subseteq X_4$, but $X_2\cap X_4=\emptyset$. Now, by the general construction of the inverse limit in the category of sets, we look at the elements of $X_1\times X_2\times X_3\times X_4$ s.t. any element in the second coordinate is also in the first coordinate and any element in the fourth coordinate is also in the third. So, the inverse limit is not empty, but the intersection is empty. What am I missing here?

• To follow-up on Jonas's point: when we talk about an "inverse limit", we always assume either that the index set is directed and the morphisms "go the other way", so that if $i\leq j$ then you have a map $X_j\to X_i$, or else that the index set is inversely directed. When we talk about more general index sets, we refer to these constructions as "limits", not as "inverse limits". – Arturo Magidin Jan 24 '11 at 3:15
• @Arturo: This is not common practice. – Martin Brandenburg Mar 7 '12 at 11:24
• @Martin: Sorry... it's been over a month. What is not standard practice? That "inverse limits" refer to limits over directed index sets/inversely directed sets, as opposed to arbitrary sets? – Arturo Magidin Mar 7 '12 at 17:08
• Heh; "over a month" should be "over a year and a month..." – Arturo Magidin Mar 7 '12 at 21:07
• I think it is important to note that in the example in the linked notes the family is ordered by reverse inclusion, i.e. $i \leq j$ iff $X_i \supseteq M_j$. Then the maps $f_{ji} : X_j \to X_i$ are the inclusion maps from $X_j \to X_i$ (using the first of Arturo's conventions). I think when viewed this way it is in fact a standard inverse limit. The notes would have been clearer if they had emphasised that this relies on reverse inclusion as opposed to inclusion. – Bill Apr 7 '16 at 11:31

In those notes it is assumed that the system is directed. In your example that means that one of the $X_i$ must be a subset of $X_2\cap X_4=\emptyset$.