Are inverse limits just special cases of limits?

Question

I have a quick question

Are inverse limits as they are defined for inverse systems just a special case of a limit? I am not really proficient in category theory. I guess for a given inverse system we can always define a diagram (=Functor?) which looks exactly alike the inverse system. Then the limit of this diagram is exactly the inverse limit of the inverse system?

In particular I want to know why inverse limits are unique up to isomorphism. I found several sources proving that limits are unique up to isomorphism but none which prove the fact for inverse limits, so I guess it is rather trivial. I see quite the correspondence between the two definitions, which is how I came up with this question, however my foundations on category theory don't allow me to come up with a proof myself. If my assumption is true it would obviously follow very quickly that inverse limits are also unique up to isomorphism.

I would also be very thankful on any sources explaining the very basics of category theory all around limits and inverse limits suitable for a beginner like me.

Answer 1

Inverse limits (or projective limits) are the same thing as limits. Or, as some authors might prefer, restrict inverse limits to refer to limits over special kinds of diagrams (cofiltered or codirected).

Dually, direct limits (or inductive limits) are the same thing as colimits. Or, as some coauthors might prefer, restrict direct limits to refer to colimits over special kinds of diagrams (filtered or directed).

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Some other remarks:

• Yes, a diagram of shape $J$ in a category $\mathcal C$ is a functor $J \to \mathcal C$. Here $J$ is regarded as a small category. If, e.g. in the diagram, a triangle does not commute, then the diagram shape $J$ should break the triangle apart to four or more objects, that happens to be mapped by the functor to the same object in $\mathcal C$.
• The proof of this should be straightforward if you are familiar with the definitions. So it might help if you draw lots of pictures and explicitly write out all the definitions until you are satisfied with yourself.