![]() If the local Kan extension of every single functor exists for some given p : C → C ′ p\colon C\to C' and D D, then these local Kan extensions fit together to define a functor which is the global Kan extension. It is also a special case of the fact discussed at adjoint functor that an adjoint functor can fail to exist completely, but may still be partially defined. This is a generalization of the fact that a particular diagram of shape C C can have a limit even if not every such diagram does. There is also a local definition of “the Kan extension of a given functor F F along p p” which can exist even if the entire functor defined above does not. P ! F p_! F represents the cocones under F F: this means by definition that p ! F = lim → F p_! F = \lim_\to F is the colimit of F F. P * F p_* F corepresents the cones over F F: this means by definition that p * F = lim ← F p_* F = \lim_\leftarrow F is the limit over F F Ordinary or weak Kan extensions Global Kan extensions This distinction is even more important in enriched category theory. It is certainly true that most Kan extensions which arise in practice are pointwise. Some authors (such as Kelly) assert that only pointwise Kan extensions deserve the name “Kan extension,” and use the term as “weak Kan extension” for a functor equipped with a universal natural transformation. If the pointwise version exists, then it coincides with the “ordinary” or “weak” version, but the former may exist without the pointwise version existing. These define the value of an extended functor on each object (each “point”) by a weighted (co)limit.įurthermore, a pointwise Kan extension can be “absolute”. Which define extensions of single functors only, which may exist even if not every functor has an extension. Which define extensions of all possible functors of given domain and codomain (if all of them indeed exist) Here we (have to) distinguish further between These define the extension of an entire functor, by an adjointness relation. ![]() We (have to) distinguish the following cases: In good cases they all exist and all coincide, but in some cases only some of these will actually exist. There are various slight variants of the definition of Kan extension. Product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum Preserved limit, reflected limit, created limit
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