Abstract. Strict bilimit is defined in the context of 2-categories and 2-functors, and examples of a strict bilimit not weakly admitted by bilimits and not equivalent to a strict limit are constructed.

In what follows, the term biuniversal limits shall generically refer to limit notions defined in terms of birepresentations (bilimits, lax bilimits, the strict bilimits of this note, etc.), and 2-universal limits generically to limit notions defined in terms of 2-representations (strict limits, pseudolimits, lax limits, etc.).

There is a substantial theory of 2-universal limits in the context of 2-categories and 2-functors1, mostly developed in the late 1980s in Australia. It features a taxonomy of 2-universal limit notions under the all-encompassing umbrella of (weighted)2 strict limits (a.k.a. 2-limits), facilitated by the fundamental3 result that pseudo, lax and oplax limits can be rendered as strict limits. The taxonomy further involves classes of limits such as flexible limits and pie limits.

In contrast, no corresponding theory of biuniversal limits seems to have been documented in the context of 2-categories and 2-functors. In fact, there appears to be no mention of ‘strict bilimit’ at all in the literature.4 This comes as a surprise, because not only is defining strict bilimit straightforward, but it is also effortlessly seen that it subsumes pseudo, lax and oplax bilimit (see below). As a possible explanation of the absence, perhaps all examples of 2-dimensional limits that have been of interest were either bilimits or strict limits, so that there was little practical need for the notion of strict bilimit. This thought led me to the following question.

(Q) Are there examples of strict bilimits at all that cannot be “covered” by other limit notions in use (this comes down to: by bilimits and strict limits)?

Some time ago, I asked Emily de Oliveira Santos5 about strict bilimits and particularly about examples of strict bilimits that “are not” strict limits. From the discussions with Emily and the answers to the ensuing questions she kindly asked on MathOverflow, I was able to learn that strict bilimits “are not” strict limits in both of the two possible interpretations of the phrase:

  1. (Emily:) There are examples of a bilimiting strict cone that isn’t a 2-limiting strict cone.

  2. (Martin Brandenburg:) There are examples of a diagram that has a strict bilimit but not a strict limit.

The second result, which implies the first, confirms that strict bilimits are not “covered” by strict limits. So the example-seeking question I asked to Emily is completely answered.

The refined question (Q) of this note, on the other hand, is unfortunately not settled as a bonus by Brandenburg’s examples, all of which happen to be discrete diagrams (i.e. the indexing 2-categories are sets). This is because a strict bilimit of a discrete diagram is, note, also simultaneously a bilimit, a lax bilimit as well as an oplax bilimit. In order to settle (Q), we need an example of a strict bilimit that is not only not “covered” by strict bilimits, but also not “covered” by bilimits (as to lax and oplax bilimits, bilimits subsume them; see below). Such an example will be constructed in the sequel. In precise terms, we will see that: \(\newcommand{\CAT}{\textit{Cat}} \newcommand{\col}{\colon} \newcommand{\emto}{\hookrightarrow} \newcommand{\s}{ \mathrm{s} } \newcommand{\p}{ \mathrm{p} } \newcommand{\xto}[1]{\stackrel{#1}{\to}} \newcommand{\id}{ \mathrm{id} } \newcommand{\op}{ \mathrm{op} } \newcommand{\co}{ \mathrm{co} } \newcommand{\foo}{ \mathrm{foo} } \newcommand{\baa}{ \mathrm{bar} } \newcommand{\hyAlg}{\textrm{-Alg}} \newcommand{\lup}{\mathrm{l}} \newcommand{\oup}{\mathrm{o}} \newcommand{\twoCat}{ {2\textit{Cat}} } \newcommand{\F}{\mathcal{F}} \newcommand{\W}{\mathcal{W}} \newcommand{\MonCAT}{\textit{MonCat}}\)

Proposition. There are 2-categories \(A\) and \(K\), and 2-functors \(W\col A \to \CAT\) and \(d\col A \to K\), such that

  1. \(d\) has a \(W\)-weighted strict bilimit,
  2. \(d\) has no \(W\)-weighted strict limit, and
  3. the weight \(W\) is not weakly admitted (see below) by bilimits.

The contents of this note are as follows. In Section 1, strict bilimit is defined. In Section 2, strict bilimits are seen to subsume pseudo, lax and oplax bilimits. In Section 3, bilimits are seen to subsume lax and oplax bilimits. In Section 4, the notion of a class of strict bilimits weakly admitting another is defined, as an as-weak-as-possible form of the concept that the former “covers” the latter. In Section 5, an example of a strict bilimit not weakly admitted by bilimits is exhibited. In Section 6, the above proposition is proved.

1. Definition of strict bilimit

The two general notions of biuniversal limits in use, bilimits (perhaps better termed pseudobilimits in the context of our present topic) and lax bilimits, were introduced in Street (1980) in the contexts of bicategories and pseudofunctors6 and of bicategories and lax functors6 respectively. Let us review the definition of pseudobilimit, and then define strict bilimit by imitation.

If \(K\) is a bicategory and \(\foo,\baa \in \{\mathrm{s(trict), p(seudo), l(ax), o(plax)}\}\), then \([K,\CAT]_\mathrm{\foo,\baa}\) shall denote the 2-category of foo functors \(K \to \CAT\), bar natural transformations and modifications.7 For example, \([K,\CAT]_\mathrm{p,p}\) denotes the 2-category of pseudofunctors \(K \to \CAT\), pseudonatural transformations and modifications.

Definition (Street 1980, (1.11)). Let \(K\) be a bicategory. A birepresentation of a pseudofunctor \(F\col K \to \CAT\) is an object \(r \in K\) together with an equivalence

\[\begin{equation}\label{eq:birepresentation} \rho\col K(r,-) \xto{\simeq} F \end{equation}\]

in \([K,\CAT]_\mathrm{p,p}\).

Definition (Street 1980, (1.12)). Let \(A\) and \(K\) be bicategories, and \(W\col A \to \CAT\) and \(d\col A \to K\) pseudofunctors. A \(W\)-weighted (pseudo)bilimit of \(d\) is a birepresentation of the pseudofunctor

\[\begin{equation}\label{eq:pseudocones} K^\mathrm{op} \to \CAT\col k \mapsto [A,\CAT]_\mathrm{p,p}(W,K(k,d-)). \end{equation}\]

For better comparison of this with the definition of strict bilimit which will now follow, note that when \(A,K\) are 2-categories and \(W,d\) are 2-functors, then \(\eqref{eq:pseudocones}\) is identical to the 2-functor

\[K^\mathrm{op} \to \CAT\col k \mapsto [A,\CAT]_\mathrm{s,p}(W,K(k,d-)).\]

Definition. Let \(A\) and \(K\) be 2-categories, and \(W\col A \to \CAT\) and \(d\col A \to K\) 2-functors. A \(W\)-weighted strict bilimit of \(d\) is a birepresentation of the 2-functor

\[K^\mathrm{op} \to \CAT\col k \mapsto [A,\CAT]_\mathrm{s,s}(W,K(k,d-)).\]

Remark. Note that in the context of 2-categories and 2-functors where a strict bilimit is defined, the component \(\eqref{eq:birepresentation}\) of a birepresentation is an equivalence not just in \([K,\CAT]_\mathrm{p,p}\) but in fact in \([K,\CAT]_\mathrm{s,p}\). One may wonder whether in the definition of a strict bilimit, we shouldn’t use a stricter version of a birepresentation where the equivalence component is taken rather from \([K,\CAT]_\mathrm{s,s}\), since everything else in the context is strict. Note however that the other biuniversal limit notions – pseudo, lax and oplax bilimits – are given in terms of ordinary birepresentations, of course even when the context is restricted to 2-categories and 2-functors. Strict bilimits, for the sake of the name, have to align with these siblings. Hence they are also defined in terms of ordinary birepresentations. The possibility that birepresentations underlying strict, pseudo, lax, or oplax limits may be “strictifiable” is eliminated in this later post.

2. Strict bilimits subsume pseudo, lax and oplax bilimits

Strict bilimits subsume pseudo, lax and oplax bilimits, for the exact same reason strict limits subsume8 pseudo, lax and oplax limits, as follows.

Theorem (special case of Blackwell et al. 1989, Theorem 3.16 for pseudo and lax; Bird et al. 1989, p. 7 for oplax). If \(A\) is a small 2-category, then the three inclusion 2-functors

\[[A,\CAT]_\mathrm{s,s} \emto [A,\CAT]_\mathrm{s,p},[A,\CAT]_\mathrm{s,l},[A,\CAT]_\mathrm{s,o}\]

have left adjoints \(Q_\mathrm{p},Q_\mathrm{l},Q_\mathrm{o}\) respectively.

When \(K\) is a 2-category, let \([K^\op,\CAT]_\mathrm{s,p\&eqv}\) denote the wide and locally full sub-2-category of \([K^\op,\CAT]_\mathrm{s,p}\) on equivalences.

Corollary. Let \(A\) be a small 2-category, \(K\) a locally small 2-category, and \(W\col A \to \CAT\) and \(d\col A \to K\) 2-functors. Let \(\foo \in \{\textup{p(seudo)}, \textup{l(ax)}, \textup{o(plax)}\}\). For each 0-cell \(r \in K\), there is an isomorphism of categories9

\[\begin{gathered}{} [K^\op,\CAT]_\mathrm{s,p\&eqv}(K(-,r),[A,\CAT]_\mathrm{s,s}\Bigl(Q_\foo(W),\lambda a.K(-,da)\Bigr)) \\ \cong \\ [K^\op,\CAT]_\mathrm{s,p\&eqv}(K(-,r),[A,\CAT]_\mathrm{s,foo}\Bigl(W,\lambda a.K(-,da)\Bigr)). \end{gathered}\]

Proof. The left adjoint \(Q_\foo\) gives an isomorphism

\[[A,\CAT]_\mathrm{s,s}(Q_\foo(W),\lambda a.K(-,da)) \cong [A,\CAT]_\mathrm{s,foo}(W,\lambda a.K(-,da))\]

in \([K^\op,\CAT]_\mathrm{s,s}\), hence in \([K^\op,\CAT]_\mathrm{s,p\&eqv}\). To this isomorphism, the 2-functor

\[[K^\op,\CAT]_\mathrm{s,p\&eqv}(\lambda x.K(x,r),-)\col [K^\op,\CAT]_\mathrm{s,p\&eqv} \to \CAT\]

applies, giving the desired isomorphism10 in \(\CAT\). This proves the corollary.

That is, in simplified words, a \(W\)-weighted foo bilimit of \(d\) with vertex \(r\) is precisely a \(Q_\foo(W)\)-weighted strict bilimit of \(d\) with vertex \(r\). This way, strict bilimits subsume pseudo, lax and oplax bilimits.

3. Pseudobilimits subsume lax and oplax bilimits

Pseudolimits, unlike strict limits, do not subsume lax limits (Blackwell et al. 1989, Remark 5.5). But pseudobilimits do subsume lax bilimits, as already indicated in Street (1980, under (1.17)), and dually oplax bilimits. We will see this as a consequence of the following theorem from 2-monad theory (Blackwell et al. 1989).

Let \(t\) be a 2-monad with a rank on a complete and cocomplete 2-category \(K\). Let \(t\textrm{-Alg}_\mathrm{s}\), \(t\textrm{-Alg}_\mathrm{p}\), \(t\textrm{-Alg}_\mathrm{l}\), \(t\textrm{-Alg}_\mathrm{o}\) denote the 2-categories of \(t\)-algebras whose 1-cells are strict, pseudo, lax, oplax morphisms respectively. Let \(Q_\mathrm{l}\) be a left adjoint to the inclusion 2-functors \(t\textrm{-Alg}_\mathrm{s} \emto t\textrm{-Alg}_\mathrm{l}\), which exists by Theorem 3.13 of Blackwell et al. (1989) and whose special case we have already used above.

Theorem (Blackwell et al. 1989, Corollary 5.4). The composite 2-functor \(t\textrm{-Alg}_\mathrm{l} \xto{Q_\mathrm{l}} t\textrm{-Alg}_\mathrm{s} \emto t\textrm{-Alg}_\mathrm{p}\) is left biadjoint to the inclusion \(t\textrm{-Alg}_\mathrm{p} \emto t\textrm{-Alg}_\mathrm{l}\).

The oplax version of this theorem is derived as follows. Note that the 3-functor \((-)^\co\col \twoCat \to \twoCat^*\)11 sends the 2-monad \(t\col K \to K\) in \(\twoCat\) to the 2-monad \(t^\co\col K^\co \to K^\co\) in \(\twoCat^*\), or equivalently in \(\twoCat\).

Proposition. We have isomorphisms of 2-categories

\[\begin{equation}\label{eq:alg-o-l-co} t\textrm{-Alg}_\mathrm{o} \cong t^\mathrm{co}\textrm{-Alg}_\mathrm{l}^\mathrm{co},\enspace t\textrm{-Alg}_\mathrm{p} \cong t^\mathrm{co}\textrm{-Alg}_\mathrm{p}^\mathrm{co},\enspace t\textrm{-Alg}_\mathrm{s} \cong t^\mathrm{co}\textrm{-Alg}_\mathrm{s}^\mathrm{co} \end{equation}\]

all given by identity on underlying data. ∎

Corollary.

  1. There is a 2-functor \(Q_\mathrm{o}\col t\textrm{-Alg}_\mathrm{o} \to t\textrm{-Alg}_\mathrm{s}\) that is left adjoint to the inclusion \(t\textrm{-Alg}_\mathrm{s} \emto t\textrm{-Alg}_\mathrm{o}\).

  2. The composite 2-functor \(t\textrm{-Alg}_\mathrm{o} \xto{Q_\mathrm{o}} t\textrm{-Alg}_\mathrm{s} \emto t\textrm{-Alg}_\mathrm{p}\) is left biadjoint to the inclusion \(t\textrm{-Alg}_\mathrm{p} \emto t\textrm{-Alg}_\mathrm{o}\).

Proof. 1. Since the 2-functor \(Q_\lup\col t^\co\hyAlg_\lup \to t^\co\hyAlg_\s\) is left adjoint to the inclusion \(t^\co\hyAlg_\s \emto t^\co\hyAlg_\lup\), the 2-functor \(Q_\lup^\co\col t^\co\hyAlg_\lup^\co \to t^\co\hyAlg_\s^\co\) is left adjoint to the inclusion \(t^\co\hyAlg_\s^\co \emto t^\co\hyAlg_\lup^\co\) in the 3-category \(\twoCat^*\), and equivalently in \(\twoCat\). It follows that the 2-functor \(t\textrm{-Alg}_\mathrm{o} \to t\textrm{-Alg}_\mathrm{s}\) corresponding under \(\eqref{eq:alg-o-l-co}\) to the 2-functor \(Q_\lup^\co\col t^\co\hyAlg_\lup^\co \to t^\co\hyAlg_\s^\co\) is left adjoint to the inclusion \(t\textrm{-Alg}_\mathrm{s} \emto t\textrm{-Alg}_\mathrm{o}\), as desired.

2. Let \(Q_\oup\col t^\co\hyAlg_\lup^\co \to t^\co\hyAlg_\s^\co\) denote the 2-functor corresponding via \(\eqref{eq:alg-o-l-co}\) to the 2-functor \(Q_\oup\col t\hyAlg_\oup \to t\hyAlg_\s\). Because the latter \(Q_\oup\) is left adjoint to the inclusion 2-functor \(t\hyAlg_\s \emto t\hyAlg_\oup\), the former \(Q_\oup\) is left adjoint to the inclusion 2-functor \(t^\co\hyAlg_\s^\co \emto t^\co\hyAlg_\lup^\co\). It follows that the 2-functor \(Q_\oup^\co\col t^\co\hyAlg_\lup \to t^\co\hyAlg_\s\) is left adjoint to the inclusion \(t^\co\hyAlg_\s \emto t^\co\hyAlg_\lup\). By the theorem, the composite 2-functor \(t^\co\textrm{-Alg}_\mathrm{l} \xto{Q_\oup^\co} t^\co\textrm{-Alg}_\mathrm{s} \emto t^\co\textrm{-Alg}_\mathrm{p}\) is left biadjoint to the inclusion \(t^\co\textrm{-Alg}_\mathrm{p} \emto t^\co\textrm{-Alg}_\mathrm{l}\). It follows that the 2-functor

\[(t^\co\textrm{-Alg}_\mathrm{l}^\co \xto{Q_\oup} t^\co\textrm{-Alg}_\mathrm{s}^\co \emto t^\co\textrm{-Alg}_\mathrm{p}^\co) = (t^\co\textrm{-Alg}_\mathrm{l} \xto{Q_\oup^\co} t^\co\textrm{-Alg}_\mathrm{s} \emto t^\co\textrm{-Alg}_\mathrm{p})^\co\]

is left biadjoint to the 2-functor

\[\begin{align*} (t^\co\textrm{-Alg}_\mathrm{p} \emto t^\co\textrm{-Alg}_\mathrm{l})^\co &= (t^\co\textrm{-Alg}_\mathrm{p}^\co \emto t^\co\textrm{-Alg}_\mathrm{l}^\co) \end{align*}\]

in the 3-category \(\twoCat^*\), and equivalently in \(\twoCat\). Via the isomorphisms \(\eqref{eq:alg-o-l-co}\), this is precisely what 2. asserts. This proves the corollary.

Let \(I_\mathrm{p}\) denote the inclusion 2-functor \([A,\CAT]_\mathrm{s,s} \emto [A,\CAT]_\mathrm{s,p}\).

Corollary. Let \(A\) be a small 2-category.

  1. The composite 2-functor \(I_\mathrm{p}Q_\mathrm{l}\col [A,\CAT]_\mathcal{s,l} \to [A,\CAT]_\mathrm{s,p}\) is left biadjoint to the inclusion \([A,\CAT]_\mathrm{s,p} \emto [A,\CAT]_\mathrm{s,l}\).

  2. The composite 2-functor \(I_\mathrm{p}Q_\mathrm{o}\col [A,\CAT]_\mathcal{s,o} \to [A,\CAT]_\mathrm{s,p}\) is left biadjoint to the inclusion \([A,\CAT]_\mathrm{s,p} \emto [A,\CAT]_\mathrm{s,o}\).

Proof. We know that there are a complete and cocomplete 2-category \(K\) and a finitary 2-monad \(t\) on \(K\) such that \([A,\CAT]_\mathrm{s,s} = t\textrm{-Alg}_\s\), \([A,\CAT]_\mathrm{s,p} = t\textrm{-Alg}_\p\), \([A,\CAT]_\mathrm{s,l} = t\textrm{-Alg}_\lup\) (Blackwell et al. 1989, Remark 3.15) and similarly \([A,\CAT]_\mathrm{s,o} = t\textrm{-Alg}_\oup\). Therefore 1. is a special case of the last theorem, and 2. is a special case of the 2. of the previous corollary.

A basic form of the promised conclusion is as follows. When \(K\) is a 2-category, let \([K^\op,\CAT]_\mathrm{s,p\&eqv}\) denote the wide and locally full sub-2-category of \([K^\op,\CAT]_\mathrm{s,p}\) on equivalences.

Corollary. Let \(A\) be a small 2-category, \(K\) a locally small 2-category, and \(W\col A \to \CAT\) and \(d\col A \to K\) 2-functors. Let \(\foo \in \{\textup{l(ax)}, \textup{o(plax)}\}\). For each 0-cell \(r \in K\), there is an equivalence

\[\begin{gathered}{} [K^\op,\CAT]_\mathrm{s,p\&eqv}(K(-,r),[A,\CAT]_\mathrm{s,p}\Bigl(I_\p Q_\mathrm{foo}(W),\lambda a.K(-,da)\Bigr)) \\ \simeq \\ [K^\op,\CAT]_\mathrm{s,p\&eqv}(K(-,r),[A,\CAT]_\mathrm{s,foo}\Bigl(W,\lambda a.K(-,da)\Bigr)) \end{gathered}\]

in \(\CAT\).

Proof. The biadjunction of the last theorem gives an equivalence

\[[A,\CAT]_\mathrm{s,p}(I_\p Q_\mathrm{foo}(W),\lambda a.K(-,da)) \simeq [A,\CAT]_\mathrm{s,foo}(W,\lambda a.K(-,da))\]

in \([K^\op,\CAT]_\mathrm{s,p}\). To this equivalence, the 2-functor

\[[K^\op,\CAT]_\mathrm{s,p}(\lambda x.K(x,r),-)\col [K^\op,\CAT]_\mathrm{s,p} \to \CAT\]

applies, giving the desired equivalence10 in \(\CAT\).

That is, in simplified words, a \(W\)-weighted foo (= lax, oplax) bilimit of \(d\) is precisely a \(I_\p Q_\mathrm{foo}(W)\)-weighted pseudobilimit of \(d\) with the same vertex 0-cell, up to isomodifications. This way, pseudobilimits subsume lax and oplax bilimits.

4. A class of strict (bi)limits ‘weakly admitting’ another

In what follows, the notion that a class of strict (bi)limits weakly/strongly admits another will be defined, and subsequently compared with other conceivable manifestations of the concept that one class “covers” another. While our concern for this notion stems from comparing classes of strict bilimits (specifically pseudobilimits vs all strict bilimits, see below), the notion will also be defined for classes of strict limits. This is to allow for more definite comparisons with related concepts, as we have more resources on strict limits.

The expression that a category (of a kind) admits limits (of a kind) appears commonly in the literature as an alternative to saying that the category has those limits. The terminology of the following definition has been adopted as a variation of this use of ‘admit’.

Definition. Let \(\mathcal{F}\) and \(\mathcal{W}\) be classes of 2-functorial weights, considered as classes of strict limits in 2-categories. We say \(\mathcal{F}\) weakly admits \(\mathcal{W}\) if every 2-category that admits strict limits of type \(\mathcal{F}\) admits strict limits of type \(\mathcal{W}\).12 We say \(\mathcal{F}\) (strongly) admits \(\mathcal{W}\) if, in addition to weakly admitting \(\mathcal{W}\), every 2-functor that preserves strict limits of type \(\mathcal{F}\) preserves strict limits of type \(\mathcal{W}\).

Consider now \(\mathcal{F}\) and \(\mathcal{W}\) as classes of strict bilimits in 2-categories. We say \(\mathcal{F}\) weakly admits \(\mathcal{W}\) if every 2-category that admits strict bilimits of type \(\mathcal{F}\) admits strict bilimits of type \(\mathcal{W}\). We say \(\mathcal{F}\) (strongly) admits \(\mathcal{W}\) if, in addition to weakly admitting \(\mathcal{W}\), every 2-functor that preserves strict bilimits of type \(\mathcal{F}\) preserves strict bilimits of type \(\mathcal{W}\).

While this terminology is in principle ambiguous as to whether it is with respect to strict limits or with respect to strict bilimits, the context makes this often clear. For example, in the question ‘do pseudolimits admit strict equalisers?’, obviously the notion for strict limits is intended. When a clarification is needed, we can say \(\mathcal{F}\) (weakly) admits \(\mathcal{W}\) as classes of strict limits or as classes of strict bilimits.

Note that a class of strict limits \(\F\) (weakly) admits another class of strict limits \(\W\) if and only if the (weak) saturation13 of \(\F\) contains \(\W\). The ‘(weakly) admit’ terminology is intended as a simpler way of saying the latter. The same applies to strict bilimits.14

Remark. A closely related concept is that of a class of strict (bi)limits being constructible from another. It is used e.g. in Bird et al. (1989). But I do not know whether the literature offers a formal definition of this concept. As Bird et al. essentially remark (under their Proposition 1.1), if strict \(\W\)-limits are constructible from strict \(\F\)-limits, then strict \(\F\)-limits are to15 strongly admit strict \(\W\)-limits. In the same spirit, if strict \(\W\)-bilimits are constructible from strict \(\F\)-bilimits, then strict \(\F\)-bilimits are to strongly admit strict \(\W\)-bilimits. It is not clear to me whether the converses should be true (assuming we have a formal definition of ‘constructible’).16

Remark. Another related notion is that strict (bi)limits of one class are (a sepcial case of) those of another (or alternatively phrased, the latter subsume the former). If strict \(\F\)-(bi)limits subsume strict \(\W\)-(bi)limits, then trivially strict \(\F\)-(bi)limits admit strict \(\W\)-(bi)limits. The converse is not true, at least for strict limits; for example, while inverters are clearly not a special case of inserters and equifiers17, they are admitted by inserters and equifiers (e.g. Bird et al. 1989, Proposition 1.1). Of course, if strict \(\F\)-(bi)limits are a saturated18 class, then the converse does hold.

Note that weak admission is the weakest among the concepts (of one class “covering” another) we have considered. It will be therefore used when classes of strict bilimits are separated below, as it correspondingly gives the strongest form of separation.

5. Bilimits don’t weakly admit strict bilimits

Let \(\MonCAT_\p\) denote the 2-category of monoidal categories and strong monoidal functors.

Proposition. \(\MonCAT_\p\) does not have strict biequalisers.

Proof. Consider the diagram \(\xymatrix@M=0pt@C=2em{ \{0\} \ar@<.5ex>[r]^-0 \ar@<-.5ex>[r]_-1 & \{0,1\} }\) in \(\MonCAT_\p\), where \(\{0\}\) and \(\{0,1\}\) are regarded as indiscrete monoidal categories (with any choice of a monoidal structure on \(\{0,1\}\)). Clearly no monoidal category can be the vertex of a cone on this diagram, because every monoidal category is inhabited. In particular, this diagram has no strict bilimit. This proves the proposition.

Since we know \(\MonCAT_\p\) is a pseudobilimit-complete 2-category (it is in fact pseudolimit-complete; see Blackwell et al. 1989, Theorem 2.6), it is an example of a pseudobilimit-complete 2-category that does not have strict biequalisers. (In particular, it is an example of a pseudobilimit-complete 2-category that is not strict-limit complete.) Therefore:

Corollary. Pseudobilimits don’t weakly admit strict biequalisers. In particular, they don’t weakly admit strict bilimits. ∎

6. Non-pseudo strict bilimits not equivalent to strict limits

We have observed that strict bilimits are properly more general (= weaker) than strict limits, as well as properly more general (= admissive) than pseudobilimits. While strict limits and pseudobilimits thus individually do not “cover” strict bilimits, there still remains the logical possibility that they together might “cover” strict bilimits: that every instance19 of a strict bilimit not weakly admitted by pseudobilimits might be equivalent to a strict limit. This possibility will now be eliminated (i.e. prove the Proposition in the introductory words of this note).

A counterexample serving this purpose can in fact be given by a simple 2-category that has exactly one 0-cell, one non-identity 1-cell and two non-identity 2-cell, as we will see below. Instead of directly describing this particular 2-category, a more general construction will be presented that turns any inhabited 2-category with strict equalisers into a 2-category that now lacks strict equalisers but has strict biequalisers, therefore serving our purpose (because strict biequalisers are not weakly admitted by pseudobilimits, as we have seen above).20

Construction. Let \(K\) be a 2-category. We will define a 2-category \(K'\).

The 0-cells of \(K'\) are the 0-cells of \(K\). For each 1-cell \(a\col x \to y\) in \(K\), its two copies \(a^0,a^1\col x \to y\) are 1-cells in \(K'\), and all 1-cells in \(K'\) are of this form. The 2-cells \(f^p \to g^q\) \((p,q \in \{0,1\})\) in \(K'\) are the 2-cells \(f \to g\) in \(K\).

The identity 1-cell on a 0-cell \(x \in K'\) is the 1-cell \(\id_x^0\). If \(f^p\col x \to y\) and \(g^q\col y \to z\) are 1-cells, then their composite is \(g^qf^p := (gf)^{\max\{p,q\}}\col x \to z\). The identity as well as vertical and horizontal composite 2-cells in \(K'\) are given by the respective operations in \(K\). This defines \(K'\).21

Proposition.

  1. \(K'\) is a 2-category.

  2. The forgetful 2-functor \(u\col K' \to K\) is a biequivalence of 2-categories.

  3. Let \(W\col A \to \CAT\) be a 2-functor. If \(K\) has strict \(W\)-(co)limits, then \(K'\) has strict \(W\)-bi(co)limits.

  4. Diagrams of the form \(\xymatrix@M=0pt@C=2em{ x \ar@<.5ex>[r]^-{f^0} \ar@<-.5ex>[r]_-{g^1} & y }\) admits no strict equaliser in \(K'\).

Proof. 1. The composition of 1-cells is associative, for \(\max\{-_1,-_2\}\) is associative. Identity 1-cells are unital, for \(0\) is unital with respect to \(\max\{-_1,-_2\}\). The vertical and horizontal compositions of 2-cells are associative, and identity 2-cells are unital, because the same is the case for the underlying 2-cells in \(K\). For the likewise reason, the horizontal composition of 2-cells preserves identity 2-cells as well as vertical composition. Therefore \(K'\) is a 2-category.

2. The 2-functor \(u\col K' \to K\) is bijective on 0-cells, 1-homwise surjective (splitly) and 2-homwise bijective, hence22 a biequivalence23.

3. We will prove this for limits. It then holds for colimits by duality, for clearly \((K^\op)' = K'^\op\).

Let \(d'\col A \to K'\) be a 2-functor. In light of 2., we have an equivalence of categories, i.e. an equivalence in the 2-category \(\CAT\),

\[K'(x',y') \simeq K(ux',uy')\]

that is strictly natural in \(x',y' \in K'_0\).24 It follows that we have an equivalence

\[K'(x',d'-) \simeq K(ux',ud'-)\]

in the 2-category \([A,\CAT]_\mathrm{s,s}\) that is strictly natural in \(x' \in K'_0\). This induces an equivalence of categories

\[[A,\CAT]_\mathrm{s,s}(W,K'(x',d'-)) \simeq [A,\CAT]_\mathrm{s,s}(W,K(ux',ud'-))\]

that is strictly natural in \(x' \in K'_0\).25

Now, let \(l \in K_0\) and an equivalence of categories

\[K(x,l) \simeq [A,\CAT]_\mathrm{s,s}(W,K(x,ud'-))\]

strictly natural in \(x \in K_0\) be a strict \(W\)-limit of \(ud'\col A \to K\). Let \(l'\) be the unique 0-cell in \(K'\) such that \(l = ul'\). Then we have the chain of equivalences of categories

\[\begin{align*} K'(x',l') \simeq K(ux',ul') = K(ux',l) &\simeq [A,\CAT]_\mathrm{s,s}(W,K(ux',ud'-)) \\ &\simeq [A,\CAT]_\mathrm{s,s}(W,K'(x',d'-)) \end{align*}\]

strictly natural in \(x' \in K'_0\), providing the 0-cell \(l' \in K'\) with the structure of a strict \(W\)-bilimit of \(d'\). This proves 3.

4. If \(c \xto{h^p} x\) is a strict cone on the diagram, then necessarily \(p = 1\). Now whenever \(c \xto{h^1} x\) and \(l \xto{e^1} x\) are two strict cones on the diagram and \(i^q\col c \to l\) is a 1-cell such that the triangle in

\[\xymatrix{ c \ar[d]_-{i^q} \ar[rd]^-{h^1} \\ l \ar[r]_-{e^1} & x \ar@<.5ex>[r]^-{f^0} \ar@<-.5ex>[r]_-{g^1} & y }\]

commutes, then the triangle in

\[\xymatrix{ c \ar[d]_-{i^{1-q}} \ar[rd]^-{h^1} \\ l \ar[r]_-{e^1} & x \ar@<.5ex>[r]^-{f^0} \ar@<-.5ex>[r]_-{g^1} & y }\]

must also commute. Therefore no strict cone on the diagram can satisfy the uniqueness condition of 2-universality. This proves 4.

Corollary.

  1. If \(K\) is inhabited, then \(K'\) does not have strict equalisers.

  2. If \(K\) is inhabited and has strict equalisers, then \(K'\) has strict biequalisers but lacks strict equalisers.

  3. If \(K\) is strict-limit complete, then \(K'\) is strict-bilimit complete but lacks strict equalisers (so is not strict-limit complete).

Proof.

  1. As soon as a 0-cell \(x \in K'\) exists, the diagram

    \[\xymatrix{ x \ar@<.5ex>[r]^-{\id_x^0} \ar@<-.5ex>[r]_-{\id_x^1} & x }\]

    can be formed, which admits no strict equaliser by the proposition’s 4.

  2. Immediate by 1. and the proposition’s 3.

  3. Since \(K\) is strict-limit complete, it has a limit of the empty diagram, so is inhabited. Hence also immediate by 1. and the proposition’s 3. This proves the corollary.

In light of the 2. of the corollary, all we need in order to obtain the counterexample sought above is an inhabited 2-category with strict equalisers. Perhaps the simplest such 2-category is \(1\), which evidently has all strict limits, in particular strict equalisers. In fact, \(1'\) is the promised counterexample by a 2-category with exactly one 0-cell, one non-identity 1-cell and two non-identity 2-cell. 2-categories that induce “non-mini” counterexamples include \(\CAT\), which is known to be also strict-limit complete. Thus both \(1'\) and \(\CAT'\) are in fact examples of a 2-category that is strict-bilimit complete but lacks strict equalisers, and are therefore more than sufficient to be our counterexamples.

These examples honestly settle the Proposition from the beginning of this note, so we are happy, at least on the theoretical side. On the other hand, they are no examples meant to be “practical”: these 2-categories are obtained by deliberately complicating (specifically, duplicating 1-cells of) the perfectly good original presentations. This raises the following question.

Question. Is there a “naturally occurring” example of a strict bilimit that is not weakly admissible by pseudobilimits and not equivalent to a strict limit?

References

  1. A 2-category refers to a strict 2-category, and a 2-functor refers to a strict 2-functor (as is standard). ↩

  2. All 2-dimensional limit notions in this note shall be tacitly weighted, unless otherwise specified. ↩

  3. Fundamental, in that without it there wouldn’t be the taxonomy as is under the single notion of strict limits. ↩

  4. Not only the general notion of strict bilimit, but also its particular instances (that aren’t bilimits) such as strict biequaliser or strict bipullback seem impossible to come across. ↩

  5. While I was searching about strict bilimits in puzzlement, I came across a beautiful table of 2-dimensional limit notions on MathOverflow by Emily. The table systematically listed almost all combinations from {strict,pseudo,lax,oplax} \(\times\) {limits,bilimits} – all except, sure enough, strict bilimits. This prompted me to make the enquiry. (By the way, Emily now added strict bilimits to the table.) ↩

  6. Street (1980) uses the older terms ‘homomorphism’ and ‘morphism’ for pseudofunctor and lax functor respectively. ↩ ↩2

  7. Though some combinations of \(K\), foo and bar are not practical and will certainly not be used in this note. ↩

  8. Bird et al. (1989, p. 7) ↩

  9. In the following, \(\lambda a.K(-,da)\) is just another notation for \(a \mapsto K(-,da)\). It is used here because the latter notation can cause confusion when a comma ‘\(,\)’ immediately preceeds it. ↩

  10. Because 2-functors preserve equivalences. ↩ ↩2

  11. This 3-functor appears e.g. in Garner (2006, p. 142). \(\twoCat^*\) denotes the 3-cell dual of the 3-category \(\twoCat\). ↩

  12. Trivia: this reminds me of the semantic consequence relation between theories in logic. ↩

  13. The weak saturation (by strict limits) of \(\F\) is the class of those 2-functorial weights \(F\) such that every 2-category that admits strict limits of type \(\F\) admits strict limits of type \(F\). The (strong) saturation (by strict limits) of \(\F\) is the class of those 2-functorial weights \(F\) such that every 2-category that admits strict limits of type \(\F\) admits strict limits of type \(\F\) and every 2-functor that preserves strict limits of type \(\F\) admits strict limits of type \(\F\). These saturations were defined in Albert and Kelly (1988, p. 3) in the notations \(\F^\dagger\) and \(\F^*\) respectively, where the term ‘closure’ was used rather than the more recent term ‘saturation’. ↩

  14. Where the weak and strong saturations by strict bilimits are of course defined analogously to the respective saturations13 by strict limits: The weak saturation (by strict bilimits) of \(\F\) is the class of those 2-functorial weights \(F\) such that every 2-category that admits strict bilimits of type \(\F\) admits strict bilimits of type \(\F\). The (strong) saturation (by strict bilimits) of \(\F\) is the class of those 2-functorial weights \(\F\) such that every 2-category that admits strict bilimits of type \(\F\) admits strict bilimits of type \(F\) and every 2-functor that preserves strict bilimits of type \(\F\) admits strict bilimits of type \(\F\). ↩

  15. I say this in this way in order to indicate that this assertion is informal (at least for me who does not have a formal definition of ‘constructible’). ↩

  16. By extension of the previous trivia12, this seems to resemble the completeness question in logic. ↩

  17. See e.g. Bird et al. (1989, p. 4) for descriptions of the weights for inverters, inserters and equifiers. ↩

  18. A class of (bi)limits is saturated if it coincides with its saturation. (Bird et al. 1989, Proposition 3.2, where it is called ‘closed’ instead) ↩

  19. Here, by an ‘instance’ of a strict bilimit I mean an actual bilimiting strict cone on a diagram in a 2-category. ↩

  20. This may be a slight overkill, but I only realised the existence of the mini example after I had described (in fact, through) the general construction. An upside is that the generalisation, as usual, adds insights as to why the example works. ↩

  21. An abstract description of this construction is not sought today. ↩

  22. See e.g. Johnson and Yau (2021), Theorem 7.4.1. ↩

  23. It is not a 2-equivalence, since it is not 1-homwise bijective; see Johnson and Yau (2021), Theorem 7.5.8 for the necessity of the 1-homwise bijectiveness. ↩

  24. \(K'_0\) denotes the collection of 0-cells in \(K'\). ↩

  25. Because \([A,\CAT]_\mathrm{s,s}(W,-)\) is a 2-functor (from \([A,\CAT]_\mathrm{s,s}\) to \(\CAT\)). ↩