nLab lax (∞,1)-colimit



Higher category theory

higher category theory

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Limits and colimits



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

(,1)Cat(\infty,1)Cat-valued diagrams

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.

These operations also have simple descriptions in terms of fibrations


Let f:C(,1)Catf : C \to (\infty,1)Cat. Then

  • laxlim(f)Fun C(C,el C(f))laxlim(f) \simeq Fun_C(C, el_C(f))
  • oplaxlim(f)Fun C op(C op,el¯ C op(f))oplaxlim(f) \simeq Fun_{C^{op}}(C^{op}, \overline{el}_{C^{op}}(f))
  • laxcolim(f)el¯ C op(f)laxcolim(f) \simeq \overline{el}_{C^\op}(f)
  • oplaxcolim(f)el C(f)oplaxcolim(f) \simeq el_C(f)

where elel and el¯\overline{el} are the covariant and contravariant (∞,1)-Grothendieck construction, and Fun KFun_K is the hom-category functor on (,1)Cat /K(\infty,1)Cat_{/K}.


For the case of lax limits (and dually oplax limits), we can compute

Map(A,laxlim(f)) cCMap(A,Fun(C /c,f(c))) cCMap(A×C /c,f(c)) Nat(A×C /,f)Map C cocart(AC [1],el C(f)) Map C(A×C,el C(f))Map(A,Fun C(C,el C(f))) \begin{aligned} Map(A, laxlim(f)) &\simeq \int_{c \in C} Map(A, Fun(C_{/c}, f(c))) \simeq \int_{c \in C} Map(A \times C_{/c}, f(c)) \\&\simeq Nat(A \times C_{/\bullet}, f) \simeq Map_C^{cocart}(A \otimes C^{[1]}, el_C(f)) \\&\simeq Map_C(A \times C, el_C(f)) \simeq Map(A, Fun_C(C, el_C(f))) \end{aligned}

using the fact A×C [1]C:(a,t)t 1A \times C^{[1]} \to C : (a,t) \mapsto t_1 is the free cocartesian fibration generated by A×CCA \times C \to C.

For the case of oplax colimits (and dually lax colimits), we can observe oplaxcolimoplaxcolim, el Cel_C, and the forgetful functor (,1)Cat /C cocart(,1)Cat(\infty,1)Cat_{/C}^{cocart} \to (\infty,1)Cat preserve colimits and both sides of the isomorphism send [1]×C(c,)[1]×C c/[1] \times C(c, -) \mapsto [1] \times C_{c/}. Thus, they agree on the smallest full subcategory of (,1)Cat C(\infty,1)Cat^C containing these functors and closed under small colimits, which is the entirety of (,1)Cat C(\infty,1)Cat^C.

It’s worth noting morphisms in (,1)Cat /C(\infty,1)Cat_{/C} between cocartesian fibrations can be thought of as lax transformations, so this agrees with the intuition that laxlim(f)laxlim(f) should be the category of lax natural transformations 1f1 \Rightarrow f and similar.


  • Let t:C(,1)Catt : C \to (\infty,1)Cat be the terminal functor. Then laxcolim(t)C oplaxcolim(t) \simeq C^{op} and oplaxcolim(t)Coplaxcolim(t) \simeq C,

  • 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.


Last revised on July 6, 2022 at 22:21:41. See the history of this page for a list of all contributions to it.