Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
The relative version of the notion of (∞,1)-limit.
For an (∞,1)-functor between (∞,1)-categories which is presented by an inner fibration of quasi-categories (which we denote by the same symbols), and for
a cocone diagram in over the -shaped diagram
then is an -colimiting cocone if the canonical map
(from the co-slice quasi-category) is an acyclic Kan fibration of simplicial sets.
If is the terminal category and is the unique functor, then an -colimit is the same thing as a colimit in the usual sense.
If is the terminal category so that picks out an object and picks out an edge, an -colimit is precisely an -cocartesian lift.
If is a cocartesian fibration such that the fibers have all -shaped colimits and the reindexing functors preserve -shaped colimits for each morphism in , then for any extension of , the relative colimit is given by the functor which extends to the fiber (where is the cocone point) in a cocartesian way, and carries the cocone point of to the colimit of the induced diagram . (See Lurie, Cor 4.3.1.11 and its proof.)
Last revised on February 27, 2021 at 15:22:38. See the history of this page for a list of all contributions to it.