The representation theorem in $1$-category theory states that a presheaf $\mathcal{F}$ is representable iff $\int_\mathcal{C}\mathcal{F}$ has a terminal object. When one tries to formulate a bicategorical analogue of this theorem, however, one quickly runs into trouble with the notion of "biterminality":
- Firstly, recall also that an object $\ast$ of a bicategory $\mathcal{C}$ is biterminal in $\mathcal{C}$ if, for each object $U$ of $\mathcal{C}$, the category $\mathsf{Hom}_{\mathcal{C}}(U,\ast)$ is equivalent to the punctual category with a single object and a single identity morphism.
- Now, here's a disturbing fact: $(X,\mathrm{id}_{X})$ is not biterminal in the slice¹ bicategory $\mathcal{C}_{/X}$.
One may see this as follows: take $\mathcal{C}$ to be the bicategory generated by the following data: we have
- Two objects $A$ and $X$,
- Two morphisms $\phi,f\colon A\to X$, and
- Two natural transformations $\alpha,\beta\colon\phi\Rightarrow f$.
We can then check that $\mathsf{Hom}_{\mathcal{C}_{/X}}(A\overset{\scriptstyle\phi}{\longrightarrow}X,X\overset{\scriptstyle\mathrm{id}_{X}}{\longrightarrow}X)$ is a discrete category with two objects. Thus $(X,\mathrm{id}_{X})$ is not biterminal in $\mathcal{C}_{/X}$.
A natural question then arises: is there an appropriate alternative to biterminality solving this issue and giving also a bicategorical analogue to the representation theorem?
For what it's worth, here are two failed attempts:
Attempt I. Taking a hint from the $\infty$-categorical setting (HTT, Definition 1.2.12.3), we may tentatively define an object $X$ of a bicategory $\mathcal{C}$ to be “strongly biterminal” in $\mathcal{C}$ if the canonical projection $\mathcal{C}_{/X}\twoheadrightarrow\mathcal{C}$ is a biequivalence.
This notion turns out be doomed as well, however: the canonical projection $p\colon (\mathcal{C}_{/X})_{(X,\mathrm{id}_{X})}\twoheadrightarrow\mathcal{C}_{/X}$ may fail to be a biequivalence. Indeed, by the Whitehead Theorem for Bicategories (JY, Theorem 7.4.1), $p$ is a biequivalence iff it is:
- $(1)$ Essentially surjective on objects;
- $(2)$ Essentially full on $1$-morphisms;
- $(3)$ Fully faithful on $2$-morphisms;
and one can check (again with a similarly "formal" example as the above one) that, while it does satisfy $(1)$ and $(3)$, it may fail to satisfy $(2)$.
Attempt II. In their absolutely wonderful book, Johnson–Yau define the notion of an inc-lax terminal object in a bicategory $\mathcal{C}$ as an object $X$ of $\mathcal{C}$ such that there exists a lax transformation $$\kappa\colon\mathrm{id}_{\mathcal{C}}\Longrightarrow\Delta_{X}$$ such that, for each $A\in\mathrm{Obj}(\mathcal{C})$, the component $\kappa_{A}\colon A\longrightarrow X$ of $\kappa$ at $A$ is initial in $\mathsf{Hom}_{\mathcal{C}}(A,X)$; see [JY, Definition 7.2.3]). This is a very nice notion, and indeed one can show that $(X,\mathrm{id}_{X})$ is inc-lax terminal in $\mathcal{C}_{/X}$.
However, the "bicategorical representation theorem" fails for inc-lax terminality: given a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}$, its bicategory of elements² can have an inc-lax terminal object $\pi\colon\mathsf{h}_{X}\Longrightarrow\mathcal{F}$ and still fail to be representable. (In fact, one does get a pseudonatural transformation $\alpha\colon\mathcal{F}\Longrightarrow\mathsf{h}_{X}$, but $\alpha$ may fail to be an equivalence: for instance, it would be necessary for the postcompostion functor $\pi_*\colon\mathsf{PseudoNat}(\mathsf{h}_{A},\mathsf{h}_{X})\longrightarrow\mathsf{PseudoNat}(\mathsf{h}_{A},\mathcal{F})$ to be fully faithful, which might not be true. Comparing this to the $1$-categorical case is enlightening: the terminality of a morphism $\pi\colon\mathsf{h}_{X}\Rightarrow\mathcal{F}$ in $\mathsf{El}(\mathcal{F})$ for $\mathcal{F}$ a presheaf corresponds precisely to the statement that the map $\pi_*\colon\mathrm{Nat}(h_A,h_X)\longrightarrow\mathrm{Nat}(h_A,F)$ is a bijection.)
The question (at last).
Is there an appropriate notion of «terminality» for objects of bicategories repairing the above shortcomings? That is, such that the following conditions are true?
- $(X,\mathrm{id}_{X})$ is «terminal» in $\mathcal{C}_{/X}$;
- The above "bicategorical representation theorem" becomes true with this notion of «terminality».
¹$\mathcal{C}_{/X}$ is the bicategory of elements² of $\mathsf{Hom}_{\mathcal{C}}(-,X)$, which agrees with the lax slice $\mathrm{id}_{\mathcal{C}}\downarrow X$ of $\mathcal{C}$ over $X$ defined in [JY, Section 7.1]. For more details, see Example 1.1.5 and Remark 1.1.6 of this PDF.
²This is the bicategorical Grothendieck construction (as defined in arXiv:1212.6283) of $\mathcal{F}$ composed with the inclusion $\mathsf{Cats}\hookrightarrow\mathsf{Bicats}$, which can be shown to be biequivalent to the "bicategory of pseudopresheaves on $\mathcal{C}$ over $\mathcal{F}$" (similarly to how the category of elements of a presheaf $\mathcal{F}$ is equivalent to the full subcategory of $\mathsf{PSh}(\mathcal{C})_{/\mathcal{F}}$ spanned by the representable presheaves on $\mathcal{C}$). Again, see Section 1.2 and Item 2 of Proposition 1.3.1 of the above PDF for a precise statement.
