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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 $C$.

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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 $\mathbb{C}$.)

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    $\begingroup$ I think I remember Jeffrey Morton telling me something similar --- that for an n-category C, the category of Spans in C (or perhaps spans of spans, k times) is the universal n-category receiving a functor from C in which every morphism is (k-) dualizable. If I have misremembered this, then it is entirely my fault, and not Jeffrey's. $\endgroup$ Mar 12, 2014 at 1:47
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    $\begingroup$ This construction is only universal under additional constraint (Beck-Chevalley), as explained in the paper cited in the question (btw here is the link) $\endgroup$ Mar 12, 2014 at 5:08
  • $\begingroup$ This is another, in a way more conceptual approach. However it works only when some conditions are imposed on C. In addition, "(co)fibrational composition" usually is not strictly associative. Or is there a strict 2-category Mod(C) when C is a strict 2-category? $\endgroup$ Mar 12, 2014 at 14:57
  • $\begingroup$ A generalization of the universal adjoint adding construction for a bicategory C is announced in the cited paper. It also says that the construction would give a strict 2-category when the original C is strict. But I don't know if anything has been published yet. $\endgroup$ Mar 12, 2014 at 15:05
  • $\begingroup$ @DimitriChikhladze, you may always strictify a weak 2-category to a strict 2-category, however, I think for $\mathit{Mod}(\mathbb{C})$ there should be a (simple) canonical choice of an equivalent strict 2-category. I will come to this comment later, when time allows... $\endgroup$ Mar 12, 2014 at 15:34

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