nLab lax biend

Idea

A lax biend is, together with the related notion of a pseudobiend, one of the analogues of ends in 2-category theory.

Definition

Similar to how limits are defined as representing objects of the functor $\mathrm{Cones}_{(-)}(D)$ of cones over a diagram $D$, lax biends are representing objects of a pseudofunctor $\mathsf{LaxWedges}_{(-)}(D)$. Below we define all objects involved, arriving at the definition of a lax biend in Section 2.4.

2.1 Preliminaries

In this section we recall some facts and constructions in the setting of bicategories.

2.2 Lax Wedges

Let $\mathcal{C}$ and $\mathcal{I}$ be bicategories, $D\colon\mathcal{I}^\mathrm{op}\times\mathcal{I}\longrightarrow\mathcal{C}$ be a pseudofunctor, and $X$ be an object of $\mathcal{C}$.

A lax wedge is a lax extranatural transformation $\theta\colon\Delta_{X}\overset{\bullet}{\Rightarrow}D$ from the constant pseudofunctor associated to $X$ to $D$.

2.3 Functoriality of Lax Wedges

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2.4 Lax Biends

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References

• Fosco Loregian, Coend Calculus (arXiv).