Adjoining adjoints in a 2-category

Question

For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$?

Below are some of my thoughts. Although I'll be happy to other ideas too.

There are two ways which I believe give a (not very explicit) construction of a right adjoint adjoining 2-functor. As far as I can see both give the universal such 2-functor, so there should be an isomorphism between them.

1) To construct $C^*$, first take an underlying 2-graph of $C$. Consider it as a 2-computad, and add to it 1-cells which will be the right adjoints, and 2-cells which will be units and counits for the adjunctions. Then take the free 2-category on the resulting 2-computad, and factor it out by an appropriate congruence, which comes from the structure of the original 2-category and the triangle equalities. There is an obvious $C \to C^*$.

2) Let $C_0$ be the underlying 1-category of $C$. Take the universal right adjoint adjoining 2-functor $C_0 \to C'$. The explicit construction of $C'$ is in

R. J. M. Dawson, R. Paré, D. A. Pronk, Adjoining adjoints, Advances inbMathematics 178 (2003), pp. 99-140.

Then let $C^*$ be the pushout of $C \leftarrow C_0 \to C'$ in 2-Cat. We obtain $C \to C^*$.

The question is whether $C \to C^*$ is faithful and locally faithful.

It is shown in the three author paper that $C_0 \to C'$ is faithful and locally faithful. So in (2) we are taking a pushout of faithful and locally faithful 2-functors.

The same question can be considered in which one wants to adjoin adjoints only to a given class of 1-morphisms of $C$.

Answer 1

I think, that under some mild conditions, there is a more natural and explicit construction (I am writing this off the top of my head, you have to carefully check my statements). The idea is that "adjoining a right adjoint to a morphism, makes the morphism a relation".

First, let us consider an easier (1-dimensional) case, when $\mathbb{C}$ is 2-discrete. If $\mathbb{C}$ is regular, then there is a faithful functor: $$\mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ that embeds $\mathbb{C}$ into its 2-posetal category of internal relations $\mathit{Rel}(\mathbb{C})$ and has the property that the image of every morphism has a right adjoint (in fact the converse is also true --- if a morphism in $\mathit{Rel}(\mathbb{C})$ has a right adjoint, then it comes from $\mathbb{C}$).

An internal relation in $\mathbb{C}$ is a span of morphisms $A \leftarrow R \rightarrow B$, where the legs are jointly monic. If we take a pushout of such a span, then we obtain a cospan representation of a relation. A 2-dimensional analogue of an internal relation is a codiscrete cofibred cospan.

So, let us assume, that $\mathbb{C}$ is a cofibrational 2-category. Then one may consider the 2-category $\mathit{Mod}(\mathbb{C})$ of codiscrete cofibred cospans in $\mathbb{C}$ with the (co)fibrational composition (i.e. the dual of discrete fibred spans in $\mathbb{C}^{op}$). There is an embedding: $$\mathbb{C} \rightarrow \mathit{Mod}(\mathbb{C})$$ sending a morphism $f \colon A \rightarrow B$ to the cocomma cospan (i.e. collage) over $A \overset{id}\leftarrow A \overset{f}\rightarrow B$ which has a right adjoint in $\mathit{Mod}(\mathbb{C})$.

(If $\mathbb{C}$ is fibrational, one may also consider the dual construction --- the embedding of $\mathbb{C}$ into the 2-category $\mathit{DFib}(\mathbb{C})$ of discrete fibred spans in $\mathbb{C}$.)