---
date: 2022-03-25 20:35
modification date: Friday 25th March 2022 20:35:36
title: bicategory
aliases: [bicategory, lax functor]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #higher-algebra #homotopy/stable-homotopy/equivariant 
- Refs:
	- #todo/add-references
- Links:
	- [group actions on categories](group%20actions%20on%20categories.md)
	- [pseudofunctor](Unsorted/pseudofunctor.md)

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# bicategory

![](attachments/Pasted%20image%2020220325203554.png)

# Lax functors

Definition A.123. Let $\mathcal{C}$ and $\mathcal{D}$ be bicategories. $A$ lax functor $F: \mathcal{C} \rightarrow \mathcal{D}$ consists of the following data:

(1) A function $F: \operatorname{ob}(\mathcal{C}) \rightarrow \mathrm{ob}(\mathcal{D})$.
(2) For every $x, y \in \mathrm{ob}(\mathcal{C})$, a functor $F_{x y}: \mathcal{C}(x, y) \rightarrow \mathcal{D}(F x, F y)$.
(3) For 1-cells $f: x \rightarrow y$ and $g: y \rightarrow z$, natural transformations (i.e., 2-cells) $F g \circ F f \rightarrow F(g \circ f) .$
(4) For 0 -cells $x \in \mathrm{ob}(\mathcal{C})$, natural transformations (i.e., 2 -cells) $\mathrm{id}_{F x} \rightarrow F\left(\mathrm{id}_{x}\right)$.
(5) Associativity and unitality diagrams for the 2 -cells described in the preceding items.

![](attachments/Pasted%20image%2020220325203808.png)