Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
The notion of final -functor (also called a cofinal -functor) is the generalization of the notion of final functor from category theory to (∞,1)-category-theory.
An (∞,1)-functor is final precisely if precomposition with preserves colimits:
if is final then for for any (∞,1)-functor we have
when either of these colimits exist.
(final morphism of simplicial set)
A morphism of simplicial sets is final if for every right fibration the induced morphism of simplicial sets
is a homotopy equivalence.
So in the overcategory a final morphism is an object such that morphisms out of it into any right fibration are the same as morphisms out of the terminal object into that right fibration.
This definition is originally due to Andre Joyal. It appears as HTT, def 188.8.131.52.
This is equivalent to the following definition, in terms of the model structure for right fibrations:
The morphism is final precisely if the terminal morphism in the overcategory is a weak equivalence in the model structure for right fibrations on .
(preservation of undercategories and colimits)
A morphism of simplicial sets is final precisely if for every quasicategory
- and for every morphism that exibits a colimit co-cone in , also is a colimit co-cone.
and equivalently precisely if
The following result is the -categorical analog of what is known as Quillen’s Theorem A.
(recognition theorem for final -functors)
A morphism of simplicial sets with a quasi-category is final precisely if for each object the comma-object is weakly contractible.
More explicitly, the comma object in question here is the pullback
where is the under quasi-category under .
The following says that up to equivalence, the cofinal maps of simplicial sets are the right anodyne morphisms
A map of simplicial sets is cofinal precisely if it factors as a right anodyne map followed by a trivial fibration.
This is (Lurie, cor. 184.108.40.206).
The inclusion of a terminal object is final.
By theorem 1 the inclusion of the point is final precisely if for all , the (∞,1)-categorical hom-space is contractible. This is the definition of being terminal.
On categories of simplices
For sSet a simplicial set, write for its category of elements, called here the category of simplices of the simplicial set:
an object of is a morphism of simplicial sets of the form for some (hence an -simplex of ) and a morphism is a commuting diagram
for the full subcategory on the non-degenerate simplices.
is a cofinal morphism of quasi-categories.
This appears as (Lurie, variant 220.127.116.11).
For every simplicial set there exists a cofinal map
This is (Lurie, prop. 18.104.22.168).
Section 4.1 of
Section 6 of