nLab
c-Reedy category

c-Reedy categories

Idea

A c-Reedy category is a generalization of a Reedy category, even more general than a generalized Reedy category, in which the level morphisms need not even be invertible.

Definition

A c-Reedy category is a category CC equipped with an ordinal-valued degree function on its objects, and subcategories C\overset{\leftrightarrow}{C}, C\overset{\to}{C}, and C\overset{\leftarrow}{C} containing all the objects, such that

  • CCC\overset{\leftrightarrow}{C} \subseteq \overset{\to}{C}\cap \overset{\leftarrow}{C}.
  • Every morphism in C\overset{\leftrightarrow}{C} is level (i.e. its domain and codomain have the same degree).
  • Every morphism in CC\overset{\to}{C}\setminus\overset{\leftrightarrow}{C} strictly raises degree, and every morphism in CC\overset{\leftarrow}{C}\setminus\overset{\leftrightarrow}{C} strictly lowers degree.
  • Every morphism ff factors as ff\overset{\to}{f} \overset{\leftarrow}{f}, where fC\overset{\to}{f} \in \overset{\to}{C} and fC\overset{\leftarrow}{f}\in\overset{\leftarrow}{C}, and the category of such factorizations with connecting maps in C\overset{\leftrightarrow}{C} is connected.
  • For any xx and any degree δ<deg(x)\delta\lt\deg(x), the functor C(x,):C =δSet\overset{\leftarrow}{C}(x,-):\overset{\leftrightarrow}{C}_{=\delta} \to \Set is a coproduct of retracts of representables, where C =δ\overset{\leftrightarrow}{C}_{=\delta} denotes the full subcategory of C\overset{\leftrightarrow}{C} on objects of degree δ\delta.

References

Last revised on October 6, 2015 at 18:49:13. See the history of this page for a list of all contributions to it.