Abstract. The notion of a strict birepresentation is defined in the context of 2-categories and 2-functors, and in this context birepresentations that underlie strict/pseudo/lax/oplax bilimits are shown to be not strictifiable.

Recall the definition of a birepresentation (Street 1980, (1.11)), restricted to the context of 2-categories and 2-functors:

Definition. Let \(K\) be a 2-category. A birepresentation of a 2-functor \(\newcommand{\Cat}{\textit{Cat}} \newcommand{\col}{\colon} \newcommand{\id}{ \mathrm{id} } \newcommand{\op}{ \mathrm{op} } \newcommand{\xto}[1]{\stackrel{#1}{\to}} F\col K \to \Cat\) is an object \(r \in K\) together with an equivalence

\[\rho\col K(r,-) \xto{\simeq} F\]

in \([K,\Cat]_\mathrm{s,p}\), the 2-category of 2-functors \(K \to \Cat\), pseudonatural transformations and modifications.

One typically thinks of a birepresentation as a weaker version of a 2-representation (i.e. \(\Cat\)-enriched representation; Kelly 2005, §1.10) where an equivalence is required instead of an isomorphism. This, however, is not the only aspect in which the former is weaker than the latter (when compared in the aforementioned context where both notions are defined). A birepresentation is given by a pseudonatural family of natural transformations that has a pseudonatural pseudoinverse, whereas a 2-representation is given by a strictly natural family of natural transformations that has a strictly natural inverse. This means we can also consider the following ‘strict’ variant of a birepresentation:

Definition. A birepresentation \((r,\rho)\) is strict if \(\rho\) is an equivalence in \([K,\Cat]_\mathrm{s,s}\), the 2-category of 2-functors \(K \to \Cat\), strictly natural transformations and modifications.

With this notion at our disposal, we may ask whether birepresentations are strictifiable; for this discussion, let us take this to simply mean that if a \(\Cat\)-valued 2-functor on a 2-category has a birepresentation, then it has a strict birepresentation. We will see that they are not. Note that since the representable 2-functor \(K(r,-)\) is flexible (Bird et al. 1989, Proposition 4.6), hence semiflexible (Blackwell et al. 1989, above Theorem 4.7), any pseudonatural family of natural transformations \(\rho\col K(r,-) \to F\) is equivalent to a strictly natural one. The problem is going to be that we cannot get a strictly natural pseudoinverse.

This strictifiability question arose while I was defining strict bilimit. If birepresentations that underlie strict bilimits were strictifiable, then the definition of a strict bilimit there would have been phrased in terms of a strict birepresentation rather than a birepresentation, since the former is simpler as a gadget as well as with regard to the alignment with the context of strict 2-categories and strict 2-functors. But it turned out they weren’t strictifiable, nor were birepresentations that underlie pseudo, lax or oplax bilimits, as will be shown shortly. This made it clear that strict bilimits had to be defined as (non-strict) birepresentations, for otherwise they wouldn’t be aligned with pseudo, lax and oplax bilimits. The non-strictifiability observation was not detailed in the linked post on strict bilimits as it was not essential for that tale. It has been made the subject of this separate note instead.

It will now be demonstrated that birepresentations underlying strict/pseudo/lax/oplax bilimits are not strictifiable; a fortiori, birepresentations are not strictifiable.

Let \(E = \{\xymatrix@M=0pt@C=2em{ 0 \ar@/^.5pc/[r]^-{\mathrm{a}} \ar@{}[r]|*[]{\style{display: inline-block; transform: rotate(90deg)}{\cong}} % https://tex.stackexchange.com/questions/279231/vertical-cong-or-sym-in-xymatrix https://math.meta.stackexchange.com/questions/27798/is-it-possible-for-me-to-rotate-symbols-and-operations \ar@/_.5pc/[r]_-{\mathrm{b}} & 1 }\}\) denote the 2-category of the walking 2-isomorphism. Let \(\Delta_1\col E \to \Cat\) denote the constant-point 2-functor.

Proposition.

  1. There exists no strict natural transformation \(\Delta_1 \to E(0,-)\).

  2. There exists a pseudonatural equivalence \(\Delta_1 \to E(0,-)\).

Proof. 1. If \(C\col \Delta_1 \to E(0,-)\) were a strict natural transformation, then the squares

\[\vcenter{\xymatrix{ \Delta_1(0) = 1 \ar[r]^-{C_0 = !} \ar[d]_-{G(\mathrm{a}) = !} & E(0,0) = \{\id\} \ar[d]^-{E(0,\mathrm{a})} \\ \Delta_1(1) = 1 \ar[r]_-{C_1} & E(0,1) }} \quad \text{and} \quad \vcenter{\xymatrix{ \Delta_1(0) = 1 \ar[r]^-{C_0 = !} \ar[d]_-{G(\mathrm{b}) = !} & E(0,0) = \{\id\} \ar[d]^-{E(0,\mathrm{b})} \\ \Delta_1(1) = 1 \ar[r]_-{C_1} & E(0,1) }}\]

in \(\Cat\) must commute strictly. Clearly for no possible value of \(C_1 = \Delta_\mathrm{a},\Delta_\mathrm{b}\) can both squares commute, so there exists no strict natural transformation \(\Delta_1 \to E(0,-)\).

2. On the other hand it is quickly checked that any choice of \(C_1 = \Delta_\mathrm{a},\Delta_\mathrm{b}\) extends uniquely to a pseudonatural transformation \(C\col \Delta_1 \to E(0,-)\), due crucially to the 2-isomorphism \(\mathrm{a} \cong \mathrm{b}\) in \(E\). Both pseudonatural transformations have a pseudoinverse in the unique (strictly natural) transformation \(E(0,-) \to \Delta_1\), thanks again to the 2-isomorphism in \(E\). They are therefore pseudonatural equivalences. This proves the proposition.

As an aside, a fortiori, \(\Delta_1\col E \to \Cat\) is an elementary example of a non-semiflexible 2-functor.

Corollary. For each foo ∈ {strict, pseudo, lax, oplax}, the empty diagram in the 2-category \(E\) has a foo bicolimit by a birepresentation, but not by a strict birepresentation.

Proof. Recall that such a foo bicolimit by a birepresentation (resp. strict birepresentation) is a 0-cell \(r \in E\) together with an equivalence

\[E(r,-) \simeq [\emptyset,\Cat]_\mathrm{s,foo}(\emptyset, a \mapsto E(da,-))\]

in \([E,\Cat]_\mathrm{s,p}\) (resp. \([E,\Cat]_\mathrm{s,s}\)). Note that RHS \(\cong \Delta_1\) in \([E,\Cat]_\mathrm{s,s}\). Since \(E(1,0) = \emptyset\) while \(\Delta_1(0) = 1\), the 0-cell \(1 \in E\) cannot be the vertex of a foo bicolimit of the empty diagram. By the proposition, the 0-cell \(0 \in E\) is the vertex of a foo bicolimit of the empty diagram by a birepresentation, but not by a strict birepresentation. This proves the corollary.

That is, the empty diagram in the 2-category \(E^\op\) has a strict/pseudo/lax/oplax bilimit by a birepresentation but not by a strict birepresentation, as desired.

References