# Is there any way to define morphisms between filters in order to get a category, one which its opposit category would be the category of ideals?

It's well known that filters and ideals are dual. I would like to see how to express this fact "Categorically". I would be very thankful if someone could help me with that.

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That's not the sense in which filters and ideals are dual. Rather, a filter is a certain kind of subcategory of a poset (thought of as a category), and if you take the opposite of the poset, then filters become ideals. – Zhen Lin Oct 30 '12 at 22:21
There is such a thing as the category of filters, but as Zhen Lin points out, its dual has nothing to do with ideals. Its objects are pairs $(X,F)$ where $X$ is a set and $F$ is a filter on $X$. A morphism between objects $(X',F')$ and $(X,F)$ is a function $f:X'\rightarrow X$ with the property that $f^{-1}(A)\in F'$ whenever $A\in F$. – Shawn Henry Nov 1 '12 at 20:02

Having said that, the answer to your question could turn out to be a positive one, I'm just not sure. The thing is that there are many ways to define the morphisms in a category whose objects are all pairs $(X,\mathcal F)$ where $X$ is a set and $\mathcal F$ is a filter on it. There seems to be quite a lot of flexibility on the choice of morphisms, so maybe one that suits your needs exists.