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.