## Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. III)

**Point of Post: **This is a continuation of this post.

*The Category of Inverse Systems and the Functor*

We have discussed the category of directed systems and the associated direct limit functor, and so we now move in a parallel direction to define the category of inverse systems and the associated inverse limit functor.

We suppose now, similar to the case of , that we have fixed some preordered set (from hereon out denoted ) and some ring . We then define the category of inverse systems in over to be such that the objects of the category are inverse systems of left -modules over and whose morphisms are sets of morphisms such that following square commutes whenever

It is completely routine to verify, and in fact is almost isomorphic to the case of directed systems, that this, in fact, a category.

We’d now like to define the inverse limit functor . We already have an object-map defined by and so we must only construct, given a morphism a corresponding morphism . We proceed as we did for the case of directed systems by picking out limit cones and for and respectively. We then note that for each we have maps , and by assumption that is a morphism we check that

Thus, by the definition of the inverse limit we are guaranteed a unique morphism such that . We define to be this .

So, to check that is, in fact, a functor we must merely check that it respects identities and compositions. To check that it respects identities we merely note that if we take each to be the identity then our functional equation for says that for every . But, since also satisfies this equation, we may conclude by the definition of the inverse limit that . Similarly, to check that respects compositions we suppose that we have a third inverse system , a morphism , and that has a limit cone . We see then that for each

and by definition of the inverse limit we may then conclude that . Thus, really is a functor.

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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[…] pair of indexed sets of objects/arrows. It’s not then hard to check that the category of inverse systems of modules over is the same thing as . We can thus define, in general, the category of inverse systems in over […]

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