At the $1$-categorical level we can 'demote' a category to a set by letting it be discrete, and every category has a canonical discrete subcategory that we can view as its 'demotion' to a set given by all the objects and only identity arrows.


Is there a similar notion of 'demotion' for bicategories to $1$-categories?


We can define a 'discrete' bicategory to be a bicategory with only identity $2$-cells, but this immediately forces it to be strict since certain $2$-cells are part of the defining data of a bicategory; if the associators and unitors are identities then horizontal composition commutes on the nose and $1$-cell identities vanish under composition.


This means that $2$-categories (strict bicategories) have canonical demotions to $1$-categories given by the above definition, but bicategories don't.


Specifically, this prevents us from having a canonical discrete sub-$2$-category $\mathcal{C}$ (as defined above) for an arbitrary bicategory $\mathfrak{C}$ since the unitors and associators in $\mathfrak{C}$ may not be identities, so we'll have to choose new composition/identity functors in $\mathcal{C}$. I believe that the new functors will be a postcomposition of the previous composition/identity functors with the isomorphism quotient mapping $q:1-cell\_\mathfrak{C}\to[1-cell\_\mathfrak{C}]$, but this means that we have no canonical embedding back since (for example) $f\circ(g\circ h)$ and $(f\circ g)\circ h$ will be equal in $\mathcal{C}$ but not (in general) equal in $\mathfrak{C}$.



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> Is there some other, more well-behaved definition of a 'discrete' bicategory such that all bicategories have a canonical discrete sub-bicategory we can view as it's demotion to a $1$-category? If not, is this because the 'weakness' involved in the definition of a bicategory is not 'visible' at the $1$-categorical level?
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We might be able to circumvent these issues at the $2$-categorical level by appealing to a coherence theorem and obtaining an equivalent strict $2$-category, then taking the discrete sub-$2$-category of the strictification, but this answer is less interesting to me because full strictification stops working at the $2$-categorical level.



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> Is there some notion of a 'discrete' weak $n$-category in the literature such that all weak $n$-categories have canonical discrete sub-$n$-categories we can view as their 'demotion' to an $n-1$-category? If not, is this somehow because the new 'weakness' that appears at each level is invisible from the previous level?
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It seems like the step from $2$- to $3$-categories actually introduces some fundamental new weakness that stepping from $1$- to $2$-categories doesn't, so an answer for $3$-categories (perhaps involving Gray sub-$3$-categories?) may suffice to answer both questions.



Discrete bicategories/$n$-categories

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> every category has a canonical discrete subcategory that we can view as its 'demotion' to a set given by all the objects and only identity arrows.
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This construction is already fishy; note that it's not invariant under equivalence of categories.


Working invariantly, the notion of a discrete category is also not invariant under equivalence of categories. An invariant notion (something that reproduces exactly the categories equivalent to discrete categories) would be the $0$-truncated groupoids / [setoids](https://ncatlab.org/nlab/show/equivalence+relation) / equivalence relations. The inclusion of setoids into categories has a left adjoint, which one could call $\pi\_0$, sending a small category to the equivalence relation on its objects generated by the relation of being connected by a morphism. It does not, as far as I can tell, have a right adjoint, so there is no canonical setoid mapping into a category.


