Inverse Limit as a Functor, is this correct?

Question

Suppose we have a homomorphism $\psi: R/I^n\to S/J^n$.

Does it follow from the fact that inverse limit is a functor that it extends to a homomorphism

$$\widetilde{\psi}: \varprojlim R/I^n\to \varprojlim S/J^n$$?

Thanks a lot.

Answer 1

A few things:

• The inverse limit is a covariant functor from [the category of directed systems in $C$] to $C$.

• The category of directed systems in $C$ has a slick description in terms of the opposite category of some category describing the indexing, which is what the Wikipedia article you're linking is doing. But you should not think the inverse limit itself is a contravariant functor; it is covariant (as the page explicitly says).

• Your notation is a little ambiguous. To get a morphism of directed systems, we don't just want one $\psi$, but for each $n$ we want a $\psi_n: R/I^n \to S/J^n$. Further, we require that these commute with the natural quotient maps $R/I^n \to R/I^{n-1}$ and $S/J^n \to S/J^{n-1}$. It's only with this data that we get a morphism of inverse limits (EDIT: or rather, it's only with this data that we get a morphism of inverse limits by directly appealing to functoriality -- see the comments).