Confusion about the definition of bicategory

Question

I know there are some definitions of bicategories in nlab or wikipedia, but it is very difficult for me to understand. I prefer to understand it in a more intuitive way.

In wikipedia, there is an overview of bicategories:

In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism.

I have some confusion about this sentence.

Firstly, let's look at the definition of a regular category.

A regular category $C$ consists of objects and morphisms. And these morphisms must satisfy associativity and identity axioms:

$(h∘g)∘f = h∘(g∘f)$

$id_x∘f = f$

$g∘id_x = g$

In other words, if $C$ is a legal regular category, the axioms above must hold.

Then we look at 2-category.

A 2-category $C$ consists of 0-cells, 1-cells between 0-cells and 2-cell between 1-cells.

The first question is:

In 2-category, does these axioms need to hold for 1-cells?

I think they must hold, but I did not see any literature states it explicitly...

If it holds, then we can conclude that every hom-cateory $C(a,a)$ in $C$ is a strict monoidal category, right? Because strict monoidal laws in $C(a,a)$ follows from 2-category axioms for 1-cells in $C$.

Then we look that bicategory.

As I quoted above, "the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism", so the second question is:

Can I judge that the following laws must hold for 1-cell in $C$?

$(h∘g)∘f ≅ h∘(g∘f)$

$id_x∘f ≅ f$

$g∘id_x ≅ g$

If it holds, then we can conclude that every hom-cateory $C(a,a)$ in $C$ is a monoidal category, right? Because monoidal laws in $C(a,a)$ follows from bicategory axioms for 1-cells in $C$.

But as 2-category, I did not see any literature states it explicitly...

Maybe I make mistakes somewhere....

Thanks.

Answer 1

If you want to see all the definitions spelled out in detail, I would recommend the book 2-Dimensional Categories by Johnson and Yau, which is freely available on arXiv.

In particular (referring to the numbering in v2):

• Definition 2.1.3 lists the data of a bicategory, and the axioms that have to be satisfied. You made a start with listing the necessary isomorphisms, but they have to "work together" in natural ways.
• Definition 2.3.4 lists the data of a 2-category, and the axioms that must be satisfied. Indeed, this is just a bicategory where the things that were isomorphisms before become equalities (see also definition 2.3.2).
• Example 2.1.19 links bicategories with monoidal categories, a monoidal category is essentially a one-object bicategory. Example 2.1.20 exposes $\mathcal{B}(X, X)$ as a monoidal category, where $X$ is an object in a bicategory $\mathcal{C}$ (so this is what you mentioned as well).
• Examples 2.3.11 and 2.3.12 are the strict versions of the two examples mentioned in the previous point.
• You may also be interested in the diagram on page 39, in "motivation 2.3.1".