It is well known that a category is the same as a monad in the 2-category of spans.
So I am wondering if there is a similar statement hold for higher categories: can a bicategory be given as a weak 2-monad in certain tricategory of some span-like things?
The first thing I can think of if a tricatgegory is the following
- objects are sets;
- 1-cells are category-span between sets $A, B$, i.e. a category $\mathcal{C}$ with functors $\mathcal{C}\to A$ and $\mathcal{C}\to B$;
- 2-cells are functors between category-spans between two sets of above type;
- 3-cells are (invertible) natrual transformation between two sets.
Locally, this is a the strict 2-category. The composition of 1-cells are given by pullbacks. The composition is even strict associative up to canonical isomorphism. Disregarding this canonical isomorphism, the resulting tricategory might be a category enriched in a strict 2-category.