nLab lax (∞,1)-colimit

Idea

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

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

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

$$laxcolim(f) \simeq \int^{c \in C} (C_{c/})^{op} \times f(c)$$

$$oplaxcolim(f) \simeq \int^{c \in C} C_{c/} \times f(c)$$

$$laxlim(f) \simeq \int_{c \in C} Fun(C_{/c}, f(c))$$

$$oplaxlim(f) \simeq \int_{c \in C} Fun((C_{/c})^{op}, f(c))$$

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

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

Proposition

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

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

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

Proof

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

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

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

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

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

Examples

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

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

Related concepts

• (infinity,1)-limit

References

• David Gepner, Rune Haugseng, Thomas Nikolaus, Lax colimits and free fibrations in $\infty$-categories (arXiv:1501.02161)

• Jacob Lurie, Higher Topos Theory