For an (∞,1)-category and an object, the over--category or slice -category is the -category whose objects are morphism in , whose morphisms are 2-morphisms in of the form
hence 1-morphisms as indicated together with a homotopy ; and generally whose n-morphisms are diagrams
in of the shape of the cocone under the -simplex that take the tip of the cocone to the object .
This is the generalization of the notion of over-category in ordinary category theory.
We give the definition in terms of the model of (∞,1)-categories in terms of quasi-categories.
Recall from join of quasi-categories that there are two different but quasi-categorically equivalent definitions of join, denoted and . Accordingly we have the following two different but quasi-categoricaly equivalent definitions of over/under quasi-categories.
Let be a quasi-category. let be any simplicial set and let
be an (∞,1)-functor – a morphism of simplicial sets.
The under-quasi-category is the simplicial set characterized by the property that for any other simplicial set there is a natural bijection of hom-sets
where is the canonical inclusion of into its join of simplicial sets with .
Similarly, the over quasi-category over is the simplicial set characterized by the property that
naturally in , where is the canonical inclusion map .
with denoting the other definition of join of quasi-categories (as described there)
has a right adjoint
and its image is another definition of the quasi-category under .
The first definition in terms of the the mapping property is due to Andre Joyal. Together with the discussion of the concrete realization it appears as HTT, prop 18.104.22.168. The second definition appears in HTT above prop. 22.214.171.124.
The simplicial sets and are indeed themselves again quasi-categories.
This appears as HTT, prop. 126.96.36.199
The two definitions yield equivalent results in that the canonical morphism
is an equivalence of quasi-categories.
This is HTT, prop. 188.8.131.52
From the formula
we see that
an object in the over quasi-category is a cone over ;.
For instance if then an object in is a 2-cell
a morphism in is a morphism of cones,
So we may think of the overcategory as the quasi-category of cones over . Accordingly we have that
Relation to over-1-categories
For (the nerve of) an ordinary category and , this construction coincides with the ordinary notion of overcategory in that there is a canonical isomorphism of simplicial sets
This appears as HTT, remark 184.108.40.206.
Functoriality of the slicing
If is a categorical equivalence then so is the induced morphism .
This appears as HTT, prop 220.127.116.11.
This is a special case of HTT, prop 18.104.22.168 and prop. 22.214.171.124.
Let be a map of small (∞,1)-categories, an -category, and . The induced (∞,1)-functor between slice -categories
is an equivalence of (∞,1)-categories for each diagram precisely if is an op-final (∞,1)-functor (hence if is final).
This is (Lurie, prop. 126.96.36.199).
Hom-spaces in a slice
For an (∞,1)-category and an object in and and two objects in , the hom-∞-groupoid is equivalent to the homotopy fiber of over the morphism : we have an (∞,1)-pullback diagram
This is HTT, prop. 188.8.131.52.
Limits and Colimits in a slice
The forgetful functor out of a slice (dependent sum) reflects (∞,1)-colimits:
Let be a diagram in the slice of an (∞,1)-category over an object . Then if the composite has an (∞,1)-colimit, then so does itself and the projection takes the latter to the former. Conversely, a cocone under is an (∞,1)-colimit of precisely if the composite is an -colimit of the projection of .
This appears as (Lurie, prop. 184.108.40.206).
On the other hand (∞,1)-limits in the slice are computed as limits over the diagram with the slice-cocone adjoined:
For an (∞,1)-category, a diagram, the comma category (the over--category if is the point) and a diagram in the comma category, then the (∞,1)-limit in coincides with the limit in .
For a proof see at (∞,1)-limit here.
Section 1.2.9 of