- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

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. _

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 $n'' \gt n'$,

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

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.

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

Last revised on September 15, 2019 at 14:52:15. See the history of this page for a list of all contributions to it.