Editing Inverse limit (section)

=== General definition ===

The inverse limit can be defined abstractly in an arbitrary [[category (mathematics)|category]] by means of a [[universal property]]. Let <math display=inline> (X_i, f_{ij})</math> be an inverse system of objects and [[morphism]]s  in a category ''C'' (same definition as above). The '''inverse limit''' of this system is an object ''X'' in ''C'' together with morphisms {{pi}}<sub>''i''</sub>: ''X'' → ''X''<sub>''i''</sub> (called ''projections'') satisfying {{pi}}<sub>''i''</sub> = <math>f_{ij}</math> ∘ {{pi}}<sub>''j''</sub> for all ''i'' ≤ ''j''. The pair (''X'', {{pi}}<sub>''i''</sub>)  must be universal in the sense that for any other such pair (''Y'', ψ<sub>''i''</sub>) there exists a unique morphism ''u'': ''Y'' → ''X'' such that the diagram

<div style="text-align: center;">[[File:InverseLimit-01.svg|175px|class=skin-invert]]</div>

[[commutative diagram|commutes]] for all ''i'' ≤ ''j''. The inverse limit is often denoted
:<math>X = \varprojlim X_i</math>
with the inverse system <math display=inline>(X_i, f_{ij})</math> and the canonical projections <math>\pi_i</math> being understood.

In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits ''X'' and ''X''' of an inverse system, there exists a ''unique'' [[isomorphism]] ''X''&prime; → ''X'' commuting with the projection maps.

Inverse systems and inverse limits in a category ''C'' admit an alternative description in terms of [[functor]]s. Any partially ordered set ''I'' can be considered as a [[small category]] where the morphisms consist of arrows ''i'' → ''j'' [[if and only if]] ''i'' ≤ ''j''. An inverse system is then just a [[contravariant functor]] ''I'' → ''C''. Let <math>C^{I^\mathrm{op}}</math> be the category of these functors (with [[natural transformation]]s as morphisms). An object ''X'' of ''C'' can be considered a trivial inverse system, where all objects are equal to ''X'' and all arrow are the identity of ''X''. This defines a "trivial functor" from ''C'' to <math>C^{I^\mathrm{op}}.</math> The inverse limit, if it exists, is defined as a [[right adjoint]] of this trivial functor.