I am interested in the category $\mathcal{Pfn}$ of partial functions (using sets as objects), which is well-known to be bicomplete.
And it is also known that one can consider $\mathcal{Pfn}$ as a $2$-category by noticing that every hom-set $Hom(A,B)$ is partially ordered using set-theoretical inclusion. In this sense it holds that $Pfn$ is a locally posetal $2$-category.
I would like to know what it is known about lax colimits in this $2$-category $Pfn$.
Is it true that every lax diagram has a lax colimit in $Pfn$?
If so, any reference (or explanation) for a detailed proof on how to build the lax colimit in $Pfn$?
To finish let me ask something related about the so many notions of colimits available for $2$-categories. I wonder whether there is some place where one can find such definitions in a slightly simplified framework:
- Is there some reference explining the different notions of colimits only for the setting of locally posetal $2$-categories?
For example, in Page 37 of this report by Wolfram Kahl it is introduced the notion of lax colimit only in the context of locally posetal categories; but unfortunately, this report does not say anything about all other colimit alternatives (bicolimits, pseudo colimits, etc.)