nLab
lax (∞,1)-colimit

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Limits and colimits

Contents

Idea

The refinement of the concept of lax colimits from category theory to (infinity,1)-category theory.

Properties

In the special case of functors f:C(,1)Catf : C \to (\infty,1)Cat, lax (co)limits can be given by the (∞,1)-end and coend.

laxcolim(f) cC(C c/) op×f(c) laxcolim(f) \simeq \int^{c \in C} (C_{c/})^{op} \times f(c)
oplaxcolim(f) cCC c/×f(c) oplaxcolim(f) \simeq \int^{c \in C} C_{c/} \times f(c)
laxlim(f) cCFun(C /c,f(c)) laxlim(f) \simeq \int_{c \in C} Fun(C_{/c}, f(c))
oplaxlim(f) cCFun((C /c) op,f(c)) oplaxlim(f) \simeq \int_{c \in C} Fun((C_{/c})^{op}, f(c))

where C /:C(,1)CatC_{/\bullet} : C \to (\infty,1) Cat and C /:C op(,1)CatC_{\bullet/} : C^{op} \to (\infty,1)Cat are the functors sending a morphism ccc \to c' of CC to the composition functors C /cC /cC_{/c} \to C_{/c'} and C c/C c/C_{c'/} \to C_{c/}.

Take care to note that lax colimits correspond to oplax cones, just as in the 2-categorical case.

Examples

  • If J:[1](,1)CatJ : [1] \to (\infty,1)Cat is the diagram depicting a functor f:ABf : A \to B, then the lax limits are the comma categories laxlim(J)(fB)laxlim(J) \simeq (f \downarrow B) and oplaxlim(J)(Bf)oplaxlim(J) \simeq (B \downarrow f). Dually, the lax colimits are mapping cylinders: laxcolim(J)([1]×A)⨿ {0}×ABlaxcolim(J) \simeq ([1] \times A) \amalg_{\{0\} \times A} B and oplaxcolim(J)([1]×A)⨿ {1}×ABoplaxcolim(J) \simeq ([1] \times A ) \amalg_{\{1\} \times A} B.

  • The covariant (∞,1)-Grothendieck construction sends any F:C(,1)CatF : C \to (\infty,1)Cat to oplaxcolim(F)oplaxcolim(F), and the contravariant version sends F:C op(,1)CatF : C^{op} \to (\infty,1)Cat to laxcolim(F)laxcolim(F).

References

Last revised on March 27, 2021 at 02:27:36. See the history of this page for a list of all contributions to it.