# nLab lax (∞,1)-colimit

Contents

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Limits and colimits

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

## Examples

• If $J :  \to (\infty,1)Cat$ 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: $laxcolim(J) \simeq ( \times A) \amalg_{\{0\} \times A} B$ and $oplaxcolim(J) \simeq ( \times A ) \amalg_{\{1\} \times A} B$.

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