An 'inverse system' in a categoryC is a functor from a cofiltered categoryI to C. An inverse system is sometimes called a pro-object in C. Look up category in Wiktionary, the free dictionary. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, filtered categories generalize the notion of directed set. ...
The category of inverse systems
Pro-objects in C form a category pro-C. Two inverse systems F:I C and G:J C determine a functor Iop x JSets, namely the functor HomC(F(i),G(j)). The set of homomorphisms between F and G in pro-C is defined to be the colimit of this functor in the first variable, followed by the limit in the second variable.
If C has all inverse limits, then the limit defines a functor pro-CC. In practice, e.g. if C is a category of algebraic or topological objects, this functor is not an equivalence of categories. In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...
Direct systems/Ind-objects
An ind-object in C is a pro-object in Cop. The category of ind-objects is written ind-C.
Examples
If C is the category of finite groups, then pro-C is equivalent to the category of profinite groups and continuous homomorphisms between them.
If C is the category of finitely generated groups, then ind-C is equivalent to the category of all groups.
In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ...
In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects.
Inverse limits can be defined in any category, but we will initially only consider inverse limits of groups.
Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit.
Inverse marking is a special morphological mark on the verb in the 31 and 32 configurations; some languages may also apply the same mark to 3Obv3Prox and/or to one of the 12 or 21 categories.
Viewed from this perspective, the "semantic" inverse can be seen to be primary, and to the extent that some "pragmatic" constructions can be shown to be connected to the classic inverse pattern, it is probable that they represent extension of an originally deictic pattern, rather than the reverse.
The phenomena of inverse marking and split ergative case marking demonstrate that the unmarked transitive configuration is one in which the onset of natural attention flow corresponds to the most natural viewpoint; inverse marking can be seen as reflecting a conflict between natural viewpoint and natural attention flow.
C and G:J
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