nLab Mittag-Leffler condition

Idea

An inverse sequence (sequential diagram) $G_\bullet$ of groups is said to satisfy the Mittag-Leffler condition if the images of groups from far down the sequence do not get smaller.

This is a condition used to assure the vanishing of the first derived functor of the limit-functor, $\underset{\longleftarrow}{lim}^1 G_\bullet$. See at lim^1 and Milnor sequences.

This is relevant for the preservation of exactness when applying limiting processes to exact sequences. _

Definition

An inverse sequence of groups consists of some groups $G_n$ indexed by the natural numbers and between them group homomorphisms: if $m \gt n$, there is a homomorphism $p^m_n : G_m \to G_n$ and if $l\gt m \gt n$, $p^m_n p^l_m= p^l_n$, so that we really just need the $p^{n+1}_n$s to define everything.

An inverse system $G = \{G_n,p^m_n\}$ is said to satisfy the Mittag-Leffler property (or condition) if

for any $n$, there is an $n' \gt n$ such that for any $n'' \gt n'$,

$$p^{n''}_n(G_{n''}) = p^{n'}_n(G_{n'}).$$

Discussion

This is the ‘classical’ form of the condition. It can also be applied in any category where images make sense.

An inverse sequence is a special type of pro-object.

Any Mittag-Leffler pro-object is known to be essentially epimorphic in the sense that it is isomorphic to a pro-object whose connecting morphisms are epis, that is to a strict pro-object.

Literature

Related $n$Lab entries include movable pro-object.

Mittag-Leffler property of pro-objects in the category of pointed sets and in the category of (nonabelian) groups is studied in Ch.II, Sec. 6.2 of

• S. Mardešić, J. Segal, Shape theory, North Holland 1982