# nLab derived limit functor

If $A$ is a category and $I$ a small category then we can consider the category of functors $A^I$, whose objects are the functors from $I$ to $A$. If $A$ admits limits of shape $I$, then the limit $lim = lim_I:A^I\to A$ is a functor which is right adjoint to the constant diagram functor hence it commutes with limits, and in particular it is left exact.

If $A$ has some notion of homotopy theory then usually $A^I$ has the notion as well (e.g. if we work with model category structures or say Abelian categories) hence we can then form right derived functors of $lim$ which are usually denoted $lim^n$ and called the derived limit functors. As usual we can also assemble them into the total right derived functor $\mathbb{R}lim$.

An important example is where $A$ is a category of modules over a commutative ring, then one has $Ch(A)$ and any $M:I\to A$ can be thought of as $M:I\to Ch(A)$ by thinking of each $M(i)$ as being a chain complex concentrated in dimension 0.

These include homotopy limit, lim^1 and Milnor sequences and cohomology of small categories?, this latter in the case of coefficients in a category of modules. This is a special case of the more general Baues-Wirsching cohomology.

A classic text with links to the theory of modules is

• C. U. Jensen, 1972, Les foncteurs dérivés de Lim et leurs applications en théorie de modules, volume 254 of Springer Lecture Notes in Maths.

A proof that derived limit functors give invariants of a corresponding pro-obect can be found in

• John Duskin, Pro-objects (after Verdier), Sém. Heidelberg- Strasbourg1966 -67, Exposé 6, I.R.M.A.Strasbourg.

Some results on the vanishing of ‘derived limits’ are in

• Barbara Osofsky?, The subscript of $\aleph_n$ projective dimension, and the vanishing of $lim^{(n)}$, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8 - 26.