Abstract
In ordinary category theory, limits are known to be equivalent to terminal objects in the slice category of cones. In this paper, we prove that the 2categorical analogues of this theorem relating 2limits and 2terminal objects in the various choices of slice 2categories of 2cones are false. Furthermore we show that, even when weakening the 2cones to pseudo or laxnatural transformations, or considering bitype limits and biterminal objects, there is still no such correspondence.
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1 Introduction
In this paper we address the question of whether the natural 2categorical analogue to the 1categorical result giving a correspondence between limits and terminal objects in a slice category of cones, holds.
1.1 Motivation, the 1Dimensional Case and the Case of Cat
A limit of a functor \( {F}:{I}\rightarrow {{\mathcal {C}}} \) comprises the data of an object \(L\in {\mathcal {C}}\) together with a natural transformation \( {\lambda }:{\Delta L}\Rightarrow {F} \), called the limit cone, which satisfies the following universal property: for each \(X\in {\mathcal {C}}\), the map \( {\lambda _{*}\circ \Delta }:{{\mathcal {C}}(X,L)}\rightarrow {\text {\textsf {Cat}}(I,{\mathcal {C}})(\Delta X,F)} \) given by postcomposition with \(\lambda \) is an isomorphism of sets. We may also form the slice category \({\Delta }\downarrow {F}\) of cones over F, and it is a folklore result that a limit of F is equivalently a terminal object in \({\Delta }\downarrow {F}\).
As an example, and in progressing up in dimension, let us now consider products in Cat – the category of small categories and functors. The universal property of the product \(({\mathcal {C}}\times {\mathcal {D}},\pi _{\mathcal {C}},\pi _{\mathcal {D}})\) of two categories \({\mathcal {C}}\) and \({\mathcal {D}}\) gives, for each pair of functors \( {F}:{{\mathcal {X}}}\rightarrow {{\mathcal {C}}} \) and \( {G}:{{\mathcal {X}}}\rightarrow {{\mathcal {D}}} \), a functor \( {\left<F,G\right>}:{{\mathcal {X}}}\rightarrow {{\mathcal {C}}\times {\mathcal {D}}} \) unique among those satisfying \(\pi _{{\mathcal {C}}}\left<F,G\right>=F\) and \(\pi _{{\mathcal {D}}}\left<F,G\right>=G\).
However, the category \(\text {\textsf {Cat}}\) has further structure. Indeed, it is a 2category with 2morphisms the natural transformations between the functors. This 2dimensional structure is compatible with the product of categories. More precisely, there is a bijection of natural transformations as depicted below, which is implemented by whiskering with the projection functors.
Observe that the natural transformations \(\alpha \) and \(\beta \) correspond to functors \(\alpha :{\mathcal {X}}\times \mathbb {2}\rightarrow {\mathcal {C}}\) and \(\beta :{\mathcal {X}}\times \mathbb {2}\rightarrow {\mathcal {D}}\), where \(\mathbb {2}\) is the category \(\{0\rightarrow 1\}\). In this light, the bijection (1.2) of natural transformations can be retrieved by applying the universal property of (1.1) to the functors \(\alpha :{\mathcal {X}}\times \mathbb {2}\rightarrow {\mathcal {C}}\) and \(\beta :{\mathcal {X}}\times \mathbb {2}\rightarrow {\mathcal {D}}\).
Taken together, the bijections of (1.1) and (1.2) assemble into an isomorphism of categories
This then is the defining feature of a 2dimensional limit: there are two aspects of the universal property, one for morphisms and one for 2morphisms.
In the case of the product above, the indexing category is just a 1category. Since \(\text {\textsf {Cat}}\) is a 2category, one could instead consider indexing diagrams by a 2category I. In order to define a general 2dimensional limit in \(\text {\textsf {Cat}}\), we need a category of higher morphisms between two 2functors. This is the category of 2natural transformations and 3morphisms, called modifications, between them. With these notions, a 2limit of a 2functor \(F:I\rightarrow \text {\textsf {Cat}}\) can be defined as a pair \(({\mathcal {L}},\lambda )\) of a category \({\mathcal {L}}\) and a 2natural transformation \( {\lambda }:{\Delta {\mathcal {L}}}\Rightarrow {F} \) which are such that postcomposition with \(\lambda \) gives an isomorphism of categories
1.2 2Dimensional Conjectures
A 2limit of a general 2functor \( {F}:{I}\rightarrow {{\mathcal {A}}} \) is defined in the same fashion as indicated in (1.3) above; see Definition 2.4. This notion was first introduced, independently, by Auderset [1] and BorceuxKelly [2], and was further developed by Street [10], Kelly [7, 8] and Lack in [9]. Motivated by the 1categorical case, it is natural to ask whether 2limits can be characterised as 2dimensional terminal objects in the slice 2category of 2cones. The appropriate notion of terminality here is that of a 2terminal object – an object such that every homcategory to this object is isomorphic to the terminal category \(\mathbb {1}\).
Having seen all other concepts involved, let us introduce now the slice 2category \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over a 2functor \(F:I\rightarrow {\mathcal {A}}\). This 2category has as objects pairs \((X,\mu )\) of an object \(X\in {\mathcal {A}}\) together with a 2cone \(\mu :\Delta X\Rightarrow F\), and as morphisms those morphisms \( {f}:{X}\rightarrow {Y} \) of \({\mathcal {A}}\) making the 2cones commute.
The 2morphisms are given by 2morphisms in \({\mathcal {A}}\) which satisfy a certain whiskering identity.
This slice 2category seems appropriate to our conjecture since the 1dimensional aspect of the universal property of a 2terminal object in there is exactly the same as the 1dimensional aspect of the universal property of a 2limit. In the special case of \(\text {\textsf {Cat}}\), by generalising the argument we have seen for products, the 1dimensional aspect of the universal property of a 2limit in Cat suffices to reconstruct its 2dimensional aspect. This holds more broadly in every 2category admitting tensors by \(\mathbb {2}\), as demonstrated in Proposition 2.11. Given this, we conjecture:
Conjecture 1
Let I and \({\mathcal {A}}\) be 2categories, and let \( {F}:{I}\rightarrow {{\mathcal {A}}} \) be a 2functor. Let \(L\in {\mathcal {A}}\) be an object and \( {\lambda }:{\Delta L}\Rightarrow {F} \) be a 2natural transformation. The following two statements are equivalent:

(i)
The pair \((L,\lambda )\) is a 2limit of the functor F.

(ii)
The pair \((L,\lambda )\) is a 2terminal object in the slice 2category \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over F.
Although it gives the authors no great pleasure to mislead the reader so, such a conjecture is false. While a 2limit is always a 2terminal object in the slice 2category of 2cones (see Proposition 2.9), the converse is not necessarily true (see Counterexample 2.10). The reason for this failure is that the 2dimensional aspect of the universal property of a 2terminal object in the slice 2category is weaker than the 2dimensional aspect of the universal property of a 2limit. This manifests in, among other things, the inability of the slice 2category to detect enough modifications between two 2cones with the same summit.
This is not, however, the last word for this conjecture. The theory of 2categories affords us more room than that of categories to coherently weaken various notions. Instead of considering a 2category of 2cones where the morphisms render triangles like that of (1.4) commutative, we are free to ask that the data of a morphism comprises also the data of a general, or perhaps invertible, modification filling that triangle. This leads to the notions of laxslice and pseudoslice 2categories of 2cones, respectively. Unlike the original, strict, slice 2category given above, the lax and pseudoslice 2categories detect all, or more, modifications between 2cones. With this in mind, we might conjecture:
Conjecture 2
Let I and \({\mathcal {A}}\) be 2categories, and let \( {F}:{I}\rightarrow {{\mathcal {A}}} \) be a 2functor. Let \(L\in {\mathcal {A}}\) be an object and \( {\lambda }:{\Delta L}\Rightarrow {F} \) be a 2natural transformation. The following two statements are equivalent:

(i)
The pair \((L,\lambda )\) is a 2limit of the 2functor F.

(ii)
The pair \((L,\lambda )\) is a 2terminal object in the laxslice (or pseudoslice) 2category of 2cones over F.
Unfortunately, this too is incorrect. The failure here is twofold: the pseudoslice may still fail to detect enough modifications, as before, while simultaneously allowing too many new morphisms to appear. Similar issues plague the laxslice, and as we will see in Sect. 3, 2terminal objects in either are generally unrelated to 2limits.
At this point, it is natural to ask whether the failure of Conjectures 1, 2 has something to do with the rigidity of the notion of 2limits. We might, for instance, ask that our 2cones have naturality triangles filled by a general 2morphism, or perhaps any invertible 2morphism, not just the identity. This leads us to consider laxlimits and pseudolimits. One might imagine that these limits have special relationships with the lax and pseudoslices of matching cones, respectively. Specifically, one might hope that the peculiarities of these weaker notions of 2dimensional limit conspire somehow to support the following conjectures.
Conjecture 3
Let I and \({\mathcal {A}}\) be 2categories, and let \( {F}:{I}\rightarrow {{\mathcal {A}}} \) be a 2functor. Let \(L\in {\mathcal {A}}\) be an object and \( {\lambda }:{\Delta L}\Rightarrow {F} \) be a pseudonatural (resp. laxnatural) transformation. The following two statements are equivalent:

(i)
The pair \((L,\lambda )\) is a pseudolimit (resp. laxlimit) of the functor F.

(ii)
The pair \((L,\lambda )\) is a 2terminal object in the strict/pseudo/laxslice 2category of pseudocones (resp. laxcones) over F.
As in the case of 2limits, pseudo and laxlimits are in particular 2terminal objects in the strictslice of appropriate cones (see Proposition 4.3). However, all other implications are generally false, as established in Sect. 4.
We might then ask whether the failure of Conjectures 1, 2, 3 has had, all along, something to do with the rigidity of universal properties expressed by isomorphisms of categories. One might, on this view, hope to generate analogous and valid conjectures by weakening these isomorphisms to equivalences of categories – conjectures concerning bitype limits and biterminal objects. However, as discussed in Sect. 5, even these do not hold.
Finally, one might wonder about the case of weighted 2limits, which is a wellestablished notion for limits in enriched category theory in the literature. The theory of weighted limits was developed by Auderset [1], Street [10], and Kelly [8] in the case of 2categories, and by BorceuxKelly [2] as well as Kelly [7, Chapter 3] in the case of general enriched categories. However, as conical 2limits, pseudolimits, and laxlimits are special cases of such weighted 2limits as noted in [8, §3 and §5], we can see that the analogues of Conjectures 1, 2, 3 for weighted 2limits must in general fail too.
1.3 Outline
The structure of the paper is as follows. In Sect. 2, we introduce the notions of 2limits and strictslices of 2cones. We prove that a 2limit is always 2terminal in the strictslice of 2cones, but we provide a counterexample demonstrating that the converse fails in general. However, for the converse to hold, it is sufficient for the ambient 2category to admit tensors by \(\mathbb {2}\)—as is the case of the 2category Cat. In Sect. 3, we turn our attention to the larger 2categories of pseudo and laxslices of 2cones. We provide counterexamples demonstrating that 2limits are in fact unrelated to 2terminal objects in these—neither notion generally implies the other. In Sect. 4, we introduce pseudo and laxlimits, and investigate their relationships with 2terminal objects in the different slices. Finally, in Sect. 5, we address the case of bitype limits. We show that these are in particular always biterminal in the pseudoslice of appropriate cones, and then adapt the results we have for the 2type cases to the bitype cases.
All of the counterexamples presented in this paper, with the exception of Example 4.5, are indexed by finite (1)categories: \(\mathbb {2}\) and the pullback shape specifically. These counterexamples were generally constructed to exhibit certain data, for example a modification that is not detected by the strictslice or a 2morphism which gives an errant morphism in the laxslice. The resulting diagram shapes were inessential to this process, and there are certainly many more counterexamples yet.
In the Tables 1 and 2, we summarise our counterexamples and reductions for Conjectures 1, 2, 3—see Sect. 5 for the matching tables for the bitype conjectures. Only results marked with a \(\checkmark \) are true, everything else establishes a counterexample. Note that the objects in the slices considered vary by the column: the type of objects should match the type of the limit cone.
In consulting these tables, some readers might be confused that the adjectives “pseudo” and “lax” do not appear in the same order in the rows as they appear in the columns. We should be careful to consider that these adjectives play very different roles when attached to the various slice 2categories as they do when attached to a notion of limit. Adding these adjectives to the labels of the rows changes the morphisms of the slices, while adding these adjectives to the labels of the columns changes the type of the cones, i.e. it changes the objects of the slices. The unexpected ordering of the columns of the tables has been chosen to be this way, since all counterexamples for pseudoslices are reductions of counterexamples for laxslices.
1.4 Positive Results for Characterisations of 2Dimensional Limits
We may always view a 2category as a horizontal double category with only trivial vertical morphisms, and in the double categorical setting we are now afforded a stronger notion of terminality. In this broader context, GrandisParé show in [5, 6] that (weighted) 2limits of a 2functor F are equivalently double terminal objects in the double category of (weighted) cones over the horizontal double functor induced by F; see also [4, §5.6]. Similar work in this direction is done by Verity in his thesis [11]. With this proliferation of positive results, it is surprising that the failures of Conjectures 1, 2, 3 are not documented in the literature. GrandisParé are certainly aware of such a failure as they write the following in their recent paper [6]:
On the other hand, there seems to be no natural way of expressing the 2dimensional universal property of weighted (strict or pseudo) limits by terminality in a 2category.
Unfortunately however, GrandisParé do not record their formulation of the “natural way” nor whatever obstacles they encountered. We feel that Conjectures 1, 2, 3 express a natural expectation of the relationship between 2limits and 2terminal objects in a 2category, and we hope that our counterexamples illustrate clearly the failure of all such conjectures.
Closer examination of these counterexamples reveals the need to capture additional information not present in the slice 2category of cones. The double categorical approach of Grandis, Paré, and Verity certainly suffices for this task, but in our paper [3] we give a purely 2categorical characterisation of 2limits by constructing two different slice 2categories of cones which have the joint property that objects which are simultaneously 2terminal in both correspond precisely to 2limits. One of these slice 2categories is predictably the slice 2category of cones, but in fact the other slice 2category alone succeeds in precisely characterising 2limits through biinitial objects of a specific form. This second slice 2category, however, is a shifted version of the usual slice 2category of cones: its objects are modifications between cones. An advantage of this approach will be highlighted in forthcoming work by the second author, where a notion of \((\infty ,2)\)limits can then be defined in a fully \((\infty ,2)\)categorical language without requiring the development of the accompany theory of double \((\infty ,1)\)categories.
The counterexamples in this paper are indicative of a larger failure in the extension of 1categorical theorems to the setting of 2category theory. More generally, the existence and characterisations of bilimits may be viewed as an instance of the corresponding problems for birepresentations of general pseudopresheaves, and it is here that the analogy breaks down: while a representation for a presheaf corresponds to an initial object in the category of elements, the data of a birepresentation for a pseudopresheaf is not wholly captured by a biinitial object in the 2category of elements.
At the level of 2dimensional representations, in [3] we weaken the strict setting to that of pseudofunctors and pseudonatural transformations and generalise the results of Grandis, Paré, and Verity to the case of birepresentations. In particular we give a double categorical characterisation of birepresentations of pseudopresheaves in terms of double biinitial objects in the double category of elements. Furthermore, we succeed in providing a purely 2categorical characterisation of birepresentations in terms of objects which are simultaneously biinitial in the familiar 2category of elements and in a new 2category of morphisms. In fact, we are able to demonstrate that birepresentations can actually be characterised as biinitial objects of a specific form in the 2category of morphisms alone. These results are the content of [3, Theorem 6.8]. The counterexamples of this paper establish the necessity of the presence of both 2categories in the theorems there, as bilimits are birepresentations. As a corollary of these theorems we obtain a purely 2categorical characterisation of weighted bilimits in [3, Theorem 7.19].
Finally, the positive results of Propositions 2.11, 5.5 are special cases of more general results for birepresentations: [3, Theorem 6.14] shows that in the presence of tensors by \(\mathbb {2}\), if the pseudopresheaf preserves such tensors, then birepresentations are precisely biinitial objects in the 2category of elements.
2 2Limits do not Correspond to 2Terminal Objects in the StrictSlice
In this section, we start by comparing 2limits with 2terminal objects in the strictslice 2category of 2cones. After introducing all the terms involved, we show that a 2limit is in particular a 2terminal object in the strictslice, but we provide a counterexample for the other implication. However, when the ambient 2category admits tensors by \(\mathbb {2}\), such as is the case of \(\text {\textsf {Cat}}\), these two notions do coincide.
A 2category has not only the structure of a category, with objects and morphisms, but additionally has 2morphisms between parallel morphisms. These 2morphisms may be composed both vertically, along a common morphism boundary, and horizontally, along a common object boundary. To differentiate on 2morphisms, we write \(*\) for horizontal composition and use juxtaposition to denote vertical composition. A 2functor between 2categories comprises maps of objects, morphisms, and 2morphisms strictly compatible with the 2categorical structures. There are also notions of 2 and 3morphisms between 2categories, which we introduce now.
Definition 2.1
Let \(F,G:I\rightarrow {\mathcal {A}}\) be 2functors. A 2natural transformation \(\mu :F\Rightarrow G\) comprises the data of a morphism \(\mu _i:Fi\rightarrow Gi\) of \({\mathcal {A}}\) for each \(i\in I\), which must satisfy

(1)
for all morphisms \( {f}:{i}\rightarrow {j} \) of I, we have \((Gf)\mu _i=\mu _j(Ff)\), and

(2)
for all 2morphisms \( {\alpha }:{f}\Rightarrow {g} \) of I, we have \(G\alpha *\mu _i=\mu _j*F\alpha \).
Definition 2.2
Let \( {F,G}:{I}\rightarrow {{\mathcal {A}}} \) be 2functors and let \(\mu ,\nu :F\Rightarrow G\) be 2natural transformations. A modification comprises the data of a 2morphism \(\varphi _i:\mu _i\Rightarrow \nu _i\) for each \(i\in I\), which satisfy \(Gf*\varphi _i=\varphi _j*Ff\), for all morphisms \(f:i\rightarrow j\) of I.
With these definitions, 2functors, 2natural transformations, and modifications assemble into a 2category.
Notation 2.3
Let I and \({\mathcal {A}}\) be 2categories. We denote by \([I,{\mathcal {A}}]\) the 2category of 2functors \(I\rightarrow {\mathcal {A}}\), 2natural transformations between them, and modifications.
We are now ready to define 2dimensional limits.
Definition 2.4
Let I and \({\mathcal {A}}\) be 2categories, and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. A 2limit of F comprises the data of an object \(L\in {\mathcal {A}}\) together with a 2natural transformation \(\lambda :\Delta L\Rightarrow F\), which are such that, for each object \(X\in {\mathcal {A}}\), the functor
given by postcomposition with \(\lambda \) is an isomorphism of categories.
In what follows, we call a 2natural transformation \(\Delta X\Rightarrow F\) from a constant functor a 2cone over F.
Remark 2.5
There are two aspects of the universal property of a 2limit, which arise from the isomorphism of categories \( {\lambda _*\circ \Delta }:{{\mathcal {A}}(X,L)}\rightarrow {[I,{\mathcal {A}}](\Delta X, F)} \) at the level of objects and at the level of morphisms. We reformulate this more explicitly as follows. For every \(X\in {\mathcal {A}}\),

(1)
for every 2cone \(\mu :\Delta X\Rightarrow F\), there is a unique morphism \(f_{\mu }:X\rightarrow L\) in \({\mathcal {A}}\) such that \(\lambda \Delta f_{\mu }=\mu \),

(2)
for every modification between 2cones \(\mu ,\nu :\Delta X\Rightarrow F\), there is a unique 2morphism \(\alpha :f_{\mu }\Rightarrow f_{\nu }\) in \({\mathcal {A}}\) such that \(\lambda *\Delta \alpha =\varphi \).
We now define strictslice 2categories of 2cones and 2terminal objects.
Definition 2.6
Let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. The strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over F is defined to be the following pullback in the (1)category of 2categories and 2functors.
This 2category \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) is given by the following data:

(i)
an object in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) is a pair \((X,\mu )\) of an object \(X\in {\mathcal {A}}\) together with a 2natural transformation \(\mu :\Delta X\Rightarrow F\),

(ii)
a morphism \(f:(X,\mu )\rightarrow (Y,\nu )\) consists of a morphism \(f:X\rightarrow Y\) in \({\mathcal {A}}\) such that \(\nu \Delta f=\mu \),

(iii)
a 2morphism \(\alpha :f\Rightarrow g\) between morphisms \(f,g:(X,\mu )\rightarrow (Y,\nu )\) is a 2morphism \(\alpha :f\Rightarrow g\) in \({\mathcal {A}}\) such that \(\lambda *\Delta \alpha ={{\,\mathrm{id}\,}}_{\mu }\).
Definition 2.7
Let \({\mathcal {A}}\) be a 2category. An object \(L\in {\mathcal {A}}\) is 2terminal if for all \(X\in {\mathcal {A}}\) there is an isomorphism of categories \({\mathcal {A}}(X,L)\cong \mathbb {1}\).
As for 2limits, there are also two aspects of the universal property of a 2terminal object. Since we are interested here by 2terminal objects in a strictslice 2category of 2cones, we give a more explicit description of their universal property. We will then compare this description with the universal property of 2limits (c.f. Remark 2.5).
Remark 2.8
Given a 2functor \(F:I\rightarrow {\mathcal {A}}\), we describe the two aspects of the universal property of a 2terminal object in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\). Such a 2terminal object comprises the data of an object \(L\in {\mathcal {A}}\) together with a 2cone \(\lambda :\Delta L\Rightarrow F\) which satisfy the following:

(1)
for every \(X\in {\mathcal {A}}\) and every 2cone \(\mu :\Delta X\Rightarrow F\), there is a unique morphism \(f_{\mu }:X\rightarrow L\) in \({\mathcal {A}}\) such that \(\lambda \Delta f_{\mu }=\mu \),

(2)
for every \(X\in {\mathcal {A}}\) and every 2cone \(\mu :\Delta X\Rightarrow F\), the unique 2morphism \(f_{\mu }\Rightarrow f_{\mu }\) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) is the identity \({{\,\mathrm{id}\,}}_{f_{\mu }}\).
In particular, we can see that the 2dimensional aspect above seems somehow degenerate in comparison to (2) of Remark 2.5. However, the 1dimensional aspect is the same as the one expressed in Remark 2.5 (1). This gives the following result.
Proposition 2.9
Let I and \({\mathcal {A}}\) be 2categories, and let \( {F}:{I}\rightarrow {{\mathcal {A}}} \) be a 2functor. If \((L,\lambda :\Delta L\Rightarrow F)\) is a 2limit of F, then \((L,\lambda )\) is 2terminal in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over F.
Proof
By Remark 2.5 (1) and Remark 2.8 (1), we observe that the 1dimensional aspects of the universal property of a 2limit and of a 2terminal object in the strictslice coincide. Both say that, for every \(X\in {\mathcal {A}}\) and every 2cone \(\mu :\Delta X\Rightarrow F\), there exists a unique morphism \(f_{\mu }:X\rightarrow L\) in \({\mathcal {A}}\) such that \(\lambda \Delta f_{\mu }=\mu \).
It remains to show (2) of Remark 2.8, that is, that the unique 2morphism \(f_{\mu }\Rightarrow f_{\mu }\) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) is the identity \({{\,\mathrm{id}\,}}_{f_{\mu }}\). Any 2morphism \( {\alpha }:{f_{\mu }}\Rightarrow {f_{\mu }} \) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) must satisfy \(\lambda *\Delta \alpha ={{\,\mathrm{id}\,}}_{\lambda \Delta f}\) by Definition 2.6 (iii). In particular, we also have \(\lambda *\Delta {{\,\mathrm{id}\,}}_f={{\,\mathrm{id}\,}}_{\lambda \Delta f}=\lambda *\Delta \alpha \). By the uniqueness in Remark 2.5 (2), it follows that \(\alpha ={{\,\mathrm{id}\,}}_{f}\). \(\square \)
However, it is not true that every 2terminal object in the strictslice of 2cones is a 2limit. One reason for this is that the strictslice only sees the identity modifications between 2cones (compare (2.6) with (2.5)). With this in mind, to illustrate this failure we give an example of a modification between two 2cones which does not arise from a 2morphism.
Counterexample 2.10
Let I be the pullback shape diagram \(\{ \bullet \longrightarrow \bullet \longleftarrow \bullet \}\). Let \({\mathcal {A}}\) be the 2category generated by the data
subject to the relations \(b\lambda _0=c\lambda _1\) and \(b*\gamma _0=c*\gamma _1\). Take \(F:I\rightarrow {\mathcal {A}}\) to be the diagram
Claim
The object \((L, {\lambda }:{\Delta L}\Rightarrow {F} )\) is 2terminal in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over F, but the functor
given by postcomposition with \(\lambda \) is not surjective on morphisms, thus \((L,\lambda )\) is not a 2limit of F.
Proof
The objects of the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) are given by the 2cones over F:
Each of these objects admits precisely one morphism to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) given by
There are no nontrivial 2morphisms to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\), since there are no nontrivial 2morphisms between X and L in \({\mathcal {A}}\). This proves that \((L,\lambda )\) is 2terminal in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\).
However, the 2morphisms \(\gamma _0\) and \(\gamma _1\) give the data of a modification , i.e. a morphism in \([I,{\mathcal {A}}](\Delta X, F)\). But there is no 2morphism between f and g in \({\mathcal {A}}\) that maps to \(\gamma \) via \( {\lambda _*\circ \Delta }:{{\mathcal {A}}(X,L)}\rightarrow {[I,{\mathcal {A}}](\Delta X, F)} \). Hence \((L,\lambda )\) is not the 2limit of F. \(\square \)
A 2terminal object in the strictslice is, however, a 2limit when the 2category \({\mathcal {A}}\) admits tensors by the category \(\mathbb {2}=\{0\rightarrow 1\}\). Indeed, it follows from this condition that the 1dimensional aspect of the universal property of a 2limit implies the 2dimensional one (see [8, §3]). A 2category \({\mathcal {A}}\) is said to admit tensors by a category \({\mathcal {C}}\) when, for each object \(X\in {\mathcal {A}}\), there exists an object \(X\otimes {\mathcal {C}}\in {\mathcal {A}}\) together with isomorphisms of categories
2natural in \(X,Y\in {\mathcal {A}}\). In particular, this implies that there is a bijection between morphisms \(X\otimes {\mathcal {C}}\rightarrow Y\) in \({\mathcal {A}}\) and functors \({\mathcal {C}}\rightarrow {\mathcal {A}}(X,Y)\).
Proposition 2.11
Suppose \({\mathcal {A}}\) is a 2category that admits tensors by \(\mathbb {2}\), and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. Then an object \((L,\lambda :\Delta L\Rightarrow F)\) is a 2terminal object in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) if and only if it is a 2limit of F.
Proof
We already saw one of the implications in Proposition 2.9. Let us prove the other.
Suppose that \((L,\lambda )\) is a 2terminal object in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\). We show that \((L,\lambda )\) satisfies the two conditions (1) and (2) of Remark 2.5, expressing the two aspects of the universal property of a 2limit. It is clear that (1) holds since it is the same condition as the one expressing the 1dimensional aspect of the universal property of a 2terminal object, as in Remark 2.8 (1). It remains to show (2).
Before proceeding, let us examine the effect of admitting tensors by \(\mathbb {2}\). Let \(X\in {\mathcal {A}}\). The universal property of tensoring by \(\mathbb {2}\) in \({\mathcal {A}}\) gives a canonical bijection between morphisms \(X\otimes \mathbb {2}\rightarrow L\) in \({\mathcal {A}}\) and functors \(\mathbb {2}\rightarrow {\mathcal {A}}(X,L)\), but these functors coincide with 2morphisms from X to L. The 2category \([I,{\mathcal {A}}]\) is also tensored by \(\mathbb {2}\), since \({\mathcal {A}}\) is, and the tensor is given objectwise [7, §3.3]. In particular, \(\Delta (X\otimes \mathbb {2})=\Delta X\otimes \mathbb {2}\) as constant functors, by the objectwise definition of tensoring by \(\mathbb {2}\). As \([I,{\mathcal {A}}]\) admits tensors by \(\mathbb {2}\), we have a canonical bijection between 2cones \(\Delta X\otimes \mathbb {2}\Rightarrow F\) and functors \(\mathbb {2}\rightarrow [I,{\mathcal {A}}](\Delta X,F)\), which in turn coincide with modifications between \(\Delta X\) and F.
By Remark 2.8 (1), for every 2cone \(\varphi :\Delta X\otimes \mathbb {2}\Rightarrow F\), there exists a unique morphism \(\alpha :X\otimes \mathbb {2}\rightarrow L\) in \({\mathcal {A}}\) such that \(\lambda \Delta \alpha =\varphi \). Using the above, we can reformulate this statement as follows: for every modification \(\varphi \) between 2cones \(\Delta X\Rightarrow F\), there is a unique 2morphism \(\alpha \) between morphisms \(X\rightarrow L\) such that \(\lambda \Delta \alpha =\varphi \). But this is exactly (2) of Remark 2.5. \(\square \)
The category \(\text {\textsf {Cat}}\) of categories and functors is cartesian closed. Therefore, it is enriched over itself and so is, in particular, tensored over \(\text {\textsf {Cat}}\). In other words, the 2category \(\text {\textsf {Cat}}\) of categories, functors, and natural transformations admits tensors by all categories, and these tensors are given by cartesian products. In particular, Proposition 2.11 yields the following result.
Corollary 2.12
Let \(F:I\rightarrow \text {\textsf {Cat}}\) be a 2functor into \(\text {\textsf {Cat}}\). A pair \(({\mathcal {L}},\lambda :\Delta {\mathcal {L}}\Rightarrow F)\) is a 2terminal in the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) of 2cones over F if and only if it is a 2limit of F.
3 2Terminal Objects in Lax and PseudoSlices are not Related to 2Limits
We have seen in Sect. 2 that 2terminal objects in the strictslice of 2cones over a 2functor do not, in general, succeed in capturing both aspects of the universal properties of 2limits. In particular, the problem is that the strictslice of 2cones does not see the modifications between two 2cones with same summit. In attempt to rectify this, we might consider richer slice 2categories containing more data in their morphisms: the laxslice and the pseudoslice of 2cones. However, 2terminal objects in these new slice 2categories seem to be unrelated to 2limits. As we present below, there are 2limits that are not 2terminal objects in the laxslice (resp. pseudoslice), and conversely so.
We start by introducing lax and pseudonatural transformations between 2functors, and modifications between them.
Definition 3.1
Let I and \({\mathcal {A}}\) be 2categories, and let \(F,G:I\rightarrow {\mathcal {A}}\) be 2functors between them. A laxnatural transformation \(\mu :F\Rightarrow G\) comprises the data of

(i)
a morphism \( {\mu _i}:{Fi}\rightarrow {Gi} \), for each \(i\in I\),

(ii)
a 2morphism \( {\mu _f}:{(Gf)\mu _{i}}\Rightarrow {\mu _{j}(Ff)} \), for each morphism \(f:i\rightarrow j\) in I,
which satisfy the following conditions:

(1)
for all \(i\in I\), \(\mu _{{{\,\mathrm{id}\,}}_i}={{\,\mathrm{id}\,}}_{\mu _i}\),

(2)
for all composable morphisms f, g in I, \(\mu _{gf}=(\mu _g*Ff)(Gg*\mu _f)\),

(3)
for all 2morphisms \(\alpha :f\Rightarrow g\) in I, we have that \(\mu _g(G\alpha *\mu _i)=(\mu _j*F\alpha )\mu _f\).
A pseudonatural transformation is a laxnatural transformation \( {\mu }:{F}\Rightarrow {G} \) whose every 2morphism component \(\mu _f\) is invertible.
Definition 3.2
Let \( {F,G}:{I}\rightarrow {{\mathcal {A}}} \) be 2functors, and let \( {\mu ,\nu }:{F}\Rightarrow {G} \) be laxnatural transformations between them. A modification comprises the data of a 2morphism \( {\varphi _i}:{\mu _i}\Rightarrow {\nu _{i}} \) for each \(i\in I\), which satisfy \(\nu _f(Gf*\varphi _i)=(\varphi _j*Ff)\mu _f\), for all morphisms \( {f}:{i}\rightarrow {j} \) in I.
Similarly, we have a notion of modification between pseudonatural transformations.
Remark 3.3
Note that a 2natural transformation \( {\mu }:{F}\Rightarrow {G} \) as defined in Definition 2.1 is precisely a laxnatural transformation whose every 2morphism component \(\mu _f\) is an identity. Moreover, modifications in the sense just defined between two laxnatural transformations which happen to be 2natural coincide with the modifications of Definition 2.2.
As in the case of 2natural transformations, lax and pseudonatural transformations and modifications assemble into 2categories whose objects are 2functors.
Notation 3.4
Let I and \({\mathcal {A}}\) be 2categories. We can define two 2categories whose objects are the 2functors \(I\rightarrow {\mathcal {A}}\):

(i)
the 2category \({{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}]\), whose 1 and 2morphisms are laxnatural transformations and modifications,

(ii)
the 2category \({{\,\mathrm{Ps}\,}}[I,{\mathcal {A}}]\), whose 1 and 2morphisms are pseudonatural transformations and modifications.
The laxslice and pseudoslice of 2cones over a 2functor \( {F}:{I}\rightarrow {{\mathcal {A}}} \) can be defined as pullbacks, as in Definition 2.6, where we replace the upperleft corner with the 2categories \({{\,\mathrm{Lax}\,}}[\mathbb {2},[I,{\mathcal {A}}]]\) and \({{\,\mathrm{Ps}\,}}[\mathbb {2},[I,{\mathcal {A}}]]\), respectively. These constructions do not change the objects of the slice, but add more morphisms between them.
Definition 3.5
Let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. The laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) of 2cones over F is defined to be the following pullback in the (1)category of 2categories and 2functors.
This 2category \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) is given by the following data:

(i)
an object in \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) is a pair \((X,\mu )\) of an object \(X\in {\mathcal {A}}\) together with a 2natural transformation \(\mu :\Delta X\Rightarrow F\),

(ii)
a morphism \((f,\varphi ):(X,\mu )\rightarrow (Y,\nu )\) consists of a morphism \(f:X\rightarrow Y\) in \({\mathcal {A}}\) together with a modification ,

(iii)
a 2morphism \(\alpha :(f,\varphi )\Rightarrow (g,\psi )\) between morphisms \((f,\varphi ),(g,\psi ):(X,\mu )\rightarrow (Y,\nu )\) is a 2morphism \(\alpha :f\Rightarrow g\) in \({\mathcal {A}}\) such that \(\psi (\nu *\Delta \alpha )=\varphi \).
Similarly, we can define the pseudoslice \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\) of 2cones over F by replacing the upperleft corner \({{\,\mathrm{Lax}\,}}[\mathbb {2},[I,{\mathcal {A}}]]\) in the pullback above with \({{\,\mathrm{Ps}\,}}[\mathbb {2},[I,{\mathcal {A}}]]\). The pseudoslice corresponds to the sub2category of the laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) containing all objects and only the morphisms \((f,\varphi )\) for which the modification \(\varphi \) is invertible, and which is locallyfull on 2morphisms.
Remark 3.6
Note that the strictslice \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\) as defined in Definition 2.6 corresponds to the locallyfull sub2category of the lax or pseudoslice containing all objects and only the morphisms \((f,\varphi )\) for which the modification \(\varphi \) is an identity.
We now give two counterexamples which show that

not every 2limit is 2terminal in the laxslice of 2cones (Counterex. 3.7),

not every 2terminal object in the laxslice of 2cones is a 2limit (Counterex. 3.9).
These statements imply that, unlike in the case of strictslices, 2terminal objects in the laxslice are not at all related to 2limits. We derive counterexamples to show that the same is true for pseudoslices, namely that 2terminal objects in the pseudoslice of 2cones are not related to 2limits.
We first give an example of a 2limit that is not 2terminal in the laxslice of 2cones. To illustrate this failure we seek a case where the laxslice sees too many morphisms between the 2cones. In the counterexample below, we show that a 2morphism that is part of the 2dimensional aspect of the universal property of a 2limit might create undesirable morphisms in the laxslice of 2cones.
Counterexample 3.7
Let I be the pullback shape diagram \(\{ \bullet \longrightarrow \bullet \longleftarrow \bullet \}\). Let \({\mathcal {A}}\) be the 2category generated by the data
subject to the relation \(b\lambda _0=c\lambda _1\). Take \(F:I\rightarrow {\mathcal {A}}\) to be the diagram
Claim
The object \((L, {\lambda }:{\Delta L}\Rightarrow {F} )\) is the 2limit of F, but it is not 2terminal in the laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) of 2cones over F.
Proof
Let us begin by enumerating all the 2cones over F:
We can see that \((L,\lambda )\) is a 2limit of F, since we have
and \({\mathcal {A}}(L,L)=\{{{\,\mathrm{id}\,}}_L\}\) and \([I,{\mathcal {A}}](\Delta L,F)=\{\lambda \}\).
However, there are two distinct morphisms from \((X,\lambda *g)\) to \((L,\lambda )\) in the laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\), which are given by
Therefore \((L,\lambda )\) is not 2terminal in \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\). \(\square \)
Reduction 3.8
By requiring \(\alpha \) to be invertible in Counterexample 3.7, we can similarly show that \((L,\lambda )\) is the 2limit of F, but is not 2terminal in the pseudoslice \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\) of 2cones over F.
Next we give an example of a 2terminal object in the laxslice of 2cones that is not a 2limit. This counterexample is designed to capture a particular arrangement of two 2cones over a 2functor together with a single nontrivial modification between them. This modification gives rise to a morphism in the laxslice between these two 2cones, exhibiting the target 2cone as 2terminal in the laxslice. However, the source 2cone is not in the image of the postcomposition functor by the target 2cone, which shows that the latter is not a 2limit.
Counterexample 3.9
Let I be the pullback shape diagram \(\{ \bullet \longrightarrow \bullet \longleftarrow \bullet \}\). Let \({\mathcal {A}}\) be the 2category generated by the data
subject to the relations \(b\lambda _0=c\lambda _1\) and \(b\alpha _0=c\alpha _1\). Take \(F:I\rightarrow {\mathcal {A}}\) to be the diagram
Claim
The object \((L, {\lambda }:{\Delta L}\Rightarrow {F} )\) is 2terminal in the laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) of 2cones over F, but the functor
given by postcomposition with \(\lambda \) is not surjective on objects, thus \((L,\lambda )\) is not a 2limit of F.
Proof
The objects of the laxslice \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) are given by the 2cones over F:
Each of these objects admits precisely one morphism to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\) given by
There are no nontrivial 2morphisms to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny }}_{s}{F}\), since there are no nontrivial 2morphisms between X and L in \({\mathcal {A}}\). This proves that \((L,\lambda )\) is 2terminal in \({\Delta }\downarrow ^{\text {\tiny }}_{l}{F}\).
However, the 2cone \( {\alpha }:{\Delta X}\Rightarrow {F} \) is an object of \([I,{\mathcal {A}}](\Delta X,F)\), but it is not in the image of \(\lambda _*\circ \Delta \). \(\square \)
Reduction 3.10
By requiring \(\gamma _0\) and \(\gamma _1\) to be invertible in Counterexample 3.9, we can similarly show that \((L,\lambda )\) is 2terminal in the pseudoslice \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\) of 2cones over F, but it is not a 2limit of F.
4 The Cases of Pseudo and LaxLimits
Recall that part of the definition of a 2limit involved 2cones which were 2natural transformations. In Sect. 3, we presented weaker notions of 2dimensional natural transformations, namely pseudo and laxnatural transformations. These give other ways of taking a 2dimensional limit of a 2functor, by changing the shape of the 2dimensional cones. The corresponding notions are called pseudolimits and laxlimits.
In this section, we show that results similar to those we have seen in Sects. 2, 3 hold for pseudo and laxlimits. In the laxlimit case, we show that:

every laxlimit is 2terminal in the strictslice of laxcones (Remark 4.3),

not every 2terminal object in the strictslice of laxcones is a laxlimit (Counterex. 4.5),

not every laxlimit is 2terminal in the laxslice of laxcones (Counterex. 4.10),

not every 2terminal object in the laxslice of laxcones is a laxlimit (Counterex. 4.14).
From the last two, we also derive the result that laxlimits are not related to 2terminal objects in the pseudoslice of laxcones. With all of the results and counterexamples we have established thus far, we are able to derive proofs and counterexamples covering the conjectures related to pseudolimits.
We first introduce the notions of pseudo and laxlimits.
Definition 4.1
Let I and \({\mathcal {A}}\) be 2categories, and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor.

(i)
A pseudolimit of F comprises the data of an object \(L\in {\mathcal {A}}\) together with a pseudonatural transformation \(\lambda :\Delta L\Rightarrow F\) such that, for each object \(X\in {\mathcal {A}}\), the functor
$$\begin{aligned} {\lambda _*\circ \Delta }:{{\mathcal {A}}(X,L)}\rightarrow {{{\,\mathrm{Ps}\,}}[I,{\mathcal {A}}](\Delta X, F)} \end{aligned}$$given by postcomposition with \(\lambda \) is an isomorphism of categories.

(ii)
A laxlimit of F comprises the data of an object \(L\in {\mathcal {A}}\) together with a laxnatural transformation \(\lambda :\Delta L\Rightarrow F\) such that, for each object \(X\in {\mathcal {A}}\), the functor
$$\begin{aligned} {\lambda _*\circ \Delta }:{{\mathcal {A}}(X,L)}\rightarrow {{{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}](\Delta X, F)} \end{aligned}$$given by postcomposition with \(\lambda \) is an isomorphism of categories.
In order to consider slices in which these pseudo and laxlimit cones live, we need to change the shape of the cone objects of the slices considered in Definitions 2.6, 3.5.
Remark 4.2
We can also define the strict, pseudo, and laxslices of pseudocones (resp. laxcones) over F, by considering objects of the form \((X,\mu )\) where \( {\mu }:{\Delta X}\Rightarrow {F} \) is a pseudonatural (resp. laxnatural) transformation. These constructions can be achieved by replacing \([I,{\mathcal {A}}]\) in the pullbacks of Definitions 2.6, 3.5 with \({{\,\mathrm{Ps}\,}}[I,{\mathcal {A}}]\) (resp. \({{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}]\)). For example, the pseudoslice of laxcones is the following pullback.
There is an analogue of Proposition 2.9 in the case of pseudolimits (resp. laxlimits), whose proof may be derived by replacing 2natural transformations with pseudonatural ones (resp. laxnatural ones).
Proposition 4.3
A pseudolimit (resp. laxlimit) of a 2functor is 2terminal in the strictslice of pseudocones (resp. laxcones).
However, not every 2terminal object in the strictslice of pseudo or laxcones is a pseudo or laxlimit. In particular, Counterexample 2.10 exhibits such an object in the pseudolimit case:
Reduction 4.4
Since there are no invertible 2morphisms in Counterexample 2.10, this is also an example of a 2terminal object in the strictslice of pseudocones that is not a pseudolimit.
Let us recall that Counterexample 2.10 has 2morphisms \(\gamma _0\) and \(\gamma _1\) that introduce two additional laxcones with summit X over F. These new laxcones do not admit a morphism to \((L,\lambda )\) in the strictslice of laxcones. This arrangement demonstrates that \((L,\lambda )\) is not 2terminal in the strictslice of laxcones. Therefore, we can not use this counterexample for the laxlimit case.
The issue at heart here is that modifications between laxcones over a 2functor may be turned into laxcones over this same 2functor. Thus, to find an example of a 2terminal object in the strictslice of laxcones that is not a laxlimit, we must find a case where such a transformation is not possible. In order to create such an example, we need the diagram shape to have objects that are both the source and the target of a nontrivial morphism.
Counterexample 4.5
Let I be the 2category freely generated by the data
i.e. the nontrivial morphisms in I are given by all possible composites of x and y, e.g. xyxy. Let \({\mathcal {A}}\) be the 2category freely generated by the data
subject to the relations \((\lambda _x*g)(a*\gamma _0)=\gamma _1(\lambda _x*f)\), and \((\lambda _y*g)(b*\gamma _1)=\gamma _0(\lambda _y*f)\). Again, we have all possible composites of the morphisms a and b in \({\mathcal {A}}\), and all possible pastings of the 2morphisms \(\lambda _x\) and \(\lambda _y\). Take \(F:I\rightarrow {\mathcal {A}}\) to be the diagram defined on the generators x and y of I by
Remark 4.6
Note that the morphisms \(\lambda _0:L\rightarrow A\) and \(\lambda _1:L\rightarrow B\) together with the 2morphisms \(\lambda _x:a\lambda _0\Rightarrow \lambda _1\) and \(\lambda _y:b\lambda _1\Rightarrow \lambda _0\) suffice to give the data of a laxnatural transformation \(\lambda :\Delta L\Rightarrow F\). Indeed, by Definition 3.1 (2), the 2morphism component of \(\lambda \) at some composite of x and y is determined by the corresponding pasting of the 2morphism components \(\lambda _x\) and \(\lambda _y\).
Claim
The object \((L,\lambda :\Delta L\Rightarrow F)\) is 2terminal in the strictslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{s}{F}\) of laxcones over F, but the functor
given by postcomposition with \(\lambda \) is not surjective on morphisms, thus \((L,\lambda )\) is not a laxlimit of F.
Proof
The objects of the strictslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{s}{F}\) are given by the laxcones over F:
Note that the 2morphisms \(\gamma _0\) and \(\gamma _1\) do not induce laxcones over F with summit X, since there are no 2morphisms from \(\lambda _1 g\) to \(\lambda _0 f\), and from \(\lambda _0 g\) to \(\lambda _1 f\) in \({\mathcal {A}}\), respectively. There are also no laxcones over F with summit A or B since there are no nontrivial 2morphisms between any two composites of a and b in \({\mathcal {A}}\). Each of the objects above admits precisely one morphism to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny lx}}_{s}{F}\) given by
There are no nontrivial 2morphisms to \((L,\lambda )\) in \({\Delta }\downarrow ^{\text {\tiny lx}}_{s}{F}\), since there are no nontrivial 2morphisms between X and L in \({\mathcal {A}}\). This proves that \((L,\lambda )\) is 2terminal in \({\Delta }\downarrow ^{\text {\tiny lx}}_{s}{F}\).
However, the 2morphisms \(\gamma _0\) and \(\gamma _1\) give the data of a modification , i.e. a morphism in \({{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}](\Delta X, F)\). But there is no 2morphism between f and g in \({\mathcal {A}}\) that maps to \(\gamma \) via \( {\lambda _*\circ \Delta }:{{\mathcal {A}}(X,L)}\rightarrow {{{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}](\Delta X, F)} \). Hence \((L,\lambda )\) is not the laxlimit of F. \(\square \)
Remark 4.7
Note that, in the above counterexample, it is essential to have all free composites of x and y in I, and also of a and b in \({\mathcal {A}}\). Indeed, if we impose any conditions on the composites of x and y, e.g. x and y are mutual inverses, then the 2functor F must preserve these, and these new conditions on a and b in \({\mathcal {A}}\) add undesirable laxcones with summits A and B.
However, when the 2category \({\mathcal {A}}\) admits tensors by \(\mathbb {2}\), there is an analogue of Proposition 2.11 in the pseudolimit (resp. laxlimit) case, whose proof may be derived by replacing 2natural transformations by pseudonatural ones (resp. laxnatural ones).
Proposition 4.8
Suppose \({\mathcal {A}}\) is a 2category that admits tensors by \(\mathbb {2}\), and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. Then an object is 2terminal in the strictslice of pseudocones (resp. laxcones) over F if and only if it is a pseudolimit (resp. laxlimit) of F.
We dedicate the rest of the section to exploring counterexamples which together refute all the remaining conjectures relating pseudo and laxlimits to the lax and pseudoslices of appropriate cones. We begin by recalling Counterexample 3.7, which also shows that not every pseudolimit is a 2terminal object in the laxslice of pseudocones:
Reduction 4.9
Since there are no invertible 2morphisms in Counterexample 3.7, this is also an example of a pseudolimit that is not 2terminal in the laxslice of pseudocones.
Let us recall that Counterexample 3.7 has a 2morphism \(\alpha \). This 2morphism was an obstruction to \((L,\lambda )\) being 2terminal in the laxslice, but not to being a 2limit. In the move to laxcones, however, this 2morphism introduces an additional laxcone over F with summit X that is not in the image of \(\lambda _*\circ \Delta \). Therefore, \((L,\lambda )\) cannot be a laxlimit of F, and we need a new counterexample for the laxlimit case.
Our new counterexample should have a nontrivial 2morphism to serve as an obstruction to 2terminality in the laxslice. But, to ensure that this 2morphism is in the image of postcomposition by the tobe laxlimit cone, we must also introduce new relations.
Counterexample 4.10
Let \(I=\mathbb {2}\), and let \({\mathcal {A}}\) be the 2category generated by the data
subject to the relations \(f\alpha _{0}=f\alpha _{1}\) and \(f*\alpha ={{\,\mathrm{id}\,}}_{f\alpha _{0}}\). Consider the 2functor \( {f}:{\mathbb {2}}\rightarrow {{\mathcal {A}}} \) given by the morphism \( {f}:{A}\rightarrow {B} \).
Claim
The object \((A, {{{\,\mathrm{id}\,}}_f}:{\Delta A}\Rightarrow {f} )\) is the laxlimit of f in \({\mathcal {A}}\), but it is not 2terminal in the laxslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\) of laxcones over f.
Proof
Let us begin by enumerating all the laxcones over f:
Note that the last two abovelisted objects differ in their cone components to A: in the first case the leg is \(\alpha _0\), and in the second case it is \(\alpha _1\).
We can see that \((A,{{\,\mathrm{id}\,}}_f)\) is a laxlimit of f, since we have
and \({\mathcal {A}}(A,A)=\{{{\,\mathrm{id}\,}}_A\}\) and \([\mathbb {2},{\mathcal {A}}](\Delta A,f)=\{{{\,\mathrm{id}\,}}_f\}\).
However, there are two distinct morphisms from \((X,{{\,\mathrm{id}\,}}_{f\alpha _{1}})\) to \((A,{{\,\mathrm{id}\,}}_{f})\) in the laxslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\) of laxcones, given by
where we have used the fact \(f*\alpha ={{\,\mathrm{id}\,}}_{f\alpha _{1}}\) in displaying the latter morphism. Therefore, \((A,{{\,\mathrm{id}\,}}_f)\) is not 2terminal in the laxslice of laxcones over f. \(\square \)
Remark 4.11
Note that the object \((A,{{\,\mathrm{id}\,}}_f)\) is also a pseudolimit of f. This gives a second example of a pseudolimit that is not 2terminal in the laxslice of pseudocones.
Reduction 4.12
By requiring \(\alpha \) to be invertible in Counterexample 4.10, we can similarly show that \((A,{{\,\mathrm{id}\,}}_{f})\) is a laxlimit (resp. pseudolimit) of f, which is not 2terminal in the pseudoslice of laxcones (resp. pseudocones).
Moreover, one can derive from Counterexample 3.9 that not every 2terminal object in the laxslice of pseudocones is a pseudolimit:
Reduction 4.13
Since there are no invertible 2morphisms in Counterexample 3.9, this is also an example of a 2terminal object in the laxslice of pseudocones that is not a pseudolimit.
Counterexample 3.9 in fact also applies in the laxcone case. Although the computations are more involved, one can check this is also an example of a 2terminal object in the laxslice of laxcones that it is not a laxlimit. However, there is a more striking example for this case. When considering diagrams of shape \(\mathbb {2}\), it turns out that even a single nontrivial laxcone over a morphism exhibits a 2terminal object in the laxslice of laxcones that is not a laxlimit of the morphism.
Counterexample 4.14
Let \(I=\mathbb {2}\), and let \({\mathcal {A}}\) be the 2category generated by the data
Consider the 2functor \( {f}:{\mathbb {2}}\rightarrow {{\mathcal {A}}} \) given by the morphism \( {f}:{A}\rightarrow {B} \).
Claim
The object \((A, {{{\,\mathrm{id}\,}}_f}:{\Delta A}\Rightarrow {f} )\) is 2terminal in the laxslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\) of laxcones over f, but the functor
is not surjective on objects, thus \((A,{{\,\mathrm{id}\,}}_f)\) is not a laxlimit of f.
Proof
The objects of the laxslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\) are the following laxcones over f:
Each of these objects admits precisely one morphism to \((A,{{\,\mathrm{id}\,}}_f)\) in \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\), given by
As there are no nontrivial 2morphisms between X and A in \({\mathcal {A}}\), there are no nontrivial 2morphisms to \((A,{{\,\mathrm{id}\,}}_f)\) in \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\). This shows that \((A,{{\,\mathrm{id}\,}}_f)\) is 2terminal in the laxslice \({\Delta }\downarrow ^{\text {\tiny lx}}_{l}{f}\) of laxcones.
Next, observe that the laxcone \( {\alpha }:{f\alpha _{0}}\Rightarrow {\alpha _{1}} \) is an object of \({{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}](\Delta X,f)\), but it is not in the image of \(({{\,\mathrm{id}\,}}_f)_*\circ \Delta \). Hence \((A,{{\,\mathrm{id}\,}}_f)\) is not a laxlimit of f. \(\square \)
Reduction 4.15
By requiring \(\alpha \) to be invertible in Counterexample 4.14, we can similarly show that \((A,{{\,\mathrm{id}\,}}_f)\) is 2terminal in the pseudoslice of laxcones (resp. pseudocones) over f, but that it is not a laxlimit (resp. pseudolimit) of f.
5 BiType Limits for the Completionist
At this point we have seen that 2terminal objects in all slices and 2dimensional limits do not generally align. In this last section, which we present for completeness, we address one final weakening of the central definitions we have thus far considered. In defining 2dimensional limits with various strengths of cones, we have always asked for an isomorphism of categories to govern the universal property. However, we might seek to relax this requirement by asking instead that the relevant functor induces only an equivalence of categories. This leads to the following definitions.
Definition 5.1
Let I and \({\mathcal {A}}\) be 2categories, and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. A bilimit of F comprises the data of an object \(L\in {\mathcal {A}}\) together with a 2natural transformation \(\lambda :\Delta L\Rightarrow F\) which are such that, for each object \(X\in {\mathcal {A}}\), the functor
given by postcomposition with \(\lambda \) is an equivalence of categories.
Similarly, we can define pseudobilimit (resp. laxbilimit) by replacing \([I,{\mathcal {A}}]\) in the above with \({{\,\mathrm{Ps}\,}}[I,{\mathcal {A}}]\) (resp. \({{\,\mathrm{Lax}\,}}[I,{\mathcal {A}}])\).^{Footnote 1}
Remark 5.2
The two aspects of a universal property of a bilimit may be reformulated more explicitly by expanding the content of the equivalence of categories above. For every \(X\in {\mathcal {A}}\),

(1)
for every 2cone \(\mu :\Delta X\Rightarrow F\), there is a morphism \(f:X\rightarrow L\) in \({\mathcal {A}}\) and an invertible modification ,

(2)
for all morphisms \(f,g:X\rightarrow L\), and for every modification , there is a unique 2morphism \(\alpha :f\Rightarrow g\) in \({\mathcal {A}}\) such that \(\lambda *\Delta \alpha =\Theta \).
Definition 5.3
Let \({\mathcal {A}}\) be a 2category. An object \(L\in {\mathcal {A}}\) is biterminal if for all \(X\in {\mathcal {A}}\) there is an equivalence of categories \({\mathcal {A}}(X,L)\mathrel {\simeq }\mathbb {1}\).
In formulating analogous conjectures for the bilimit and biterminal cases, we should pay careful attention to Remark 5.2 (1). Observe that, given a 2cone \( {\mu }:{\Delta X}\Rightarrow {F} \), the 1dimensional aspect of the universal property of a bilimit \((L,\lambda )\) gives only a morphism \((X,\mu )\rightarrow (L,\lambda )\) of the pseudoslice of 2cones \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\). It is thus inappropriate to look at biterminal objects in the strictslice of 2cones when attempting to recover a general bilimit.
Much as was the case for 2limits, bilimits are in general biterminal objects in the pseudoslice of 2cones. In fact, all of the positive results of Sect. 2 follows in this context. We defer all proofs to the paper [3] which deals with such relationships in greater generality. The first result can be deduced from [3, Corollary 7.22] and the second is [3, Corollary 7.25].
Proposition 5.4
Let I and \({\mathcal {A}}\) be 2categories, and let \( {F}:{I}\rightarrow {{\mathcal {A}}} \) be a 2functor. If \((L,\lambda :\Delta L\Rightarrow F)\) is a bilimit of F, then \((L,\lambda )\) is biterminal in the pseudoslice \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\) of 2cones over F.
Proposition 5.5
Suppose \({\mathcal {A}}\) is a 2category that admits tensors by \(\mathbb {2}\), and let \(F:I\rightarrow {\mathcal {A}}\) be a 2functor. Then an object is biterminal in the pseudoslice \({\Delta }\downarrow ^{\text {\tiny }}_{p}{F}\) of 2cones over F if and only if it is a bilimit of F.
Remark 5.6
Proposition 5.4, 5.5 also hold true when the 2cones are replaced by pseudo and laxcones.
With every sunrise there is a sunset, and just as the positive results extended themselves to this weaker context, so too do the negative. Since isomorphims of categories are, in particular, equivalences of categories, a 2type limit or a 2terminal object is, in particular, a bitype limit or a biterminal object. Moreover, an examination of all of the counterexamples and reductions referenced in the tables below shows that there are no nontrivial invertible 2morphisms in the 2categories involved. This allows us to deduce the following facts. First, the notions of 2type limits (resp. 2terminal objects) and bitype limits (resp. biterminal objects) in each of these counterexamples coincide. Second, for each of Counterexamples 2.10, 4.5, the strictslice and pseudoslice coincide. Therefore, all counterexamples and reductions for 2type limits and 2terminality seen in previous sections are also counterexamples and reductions for bitype limits and biterminality, as summarised in the Tables 3 and 4.
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Notes
Note that what we call bilimit here does not match the typical notion of bilimit present in the literature, which is usually considered in a weaker context than that of (strict) 2natural transformations. Pseudobilimits therefore coincide with the usual notion of bilimits.
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Acknowledgements
Both authors are indebted to Emily Riehl for her close readings of and thoughtful inputs on several early drafts of this paper. In addition, both authors are grateful to Jérôme Scherer for his careful input on an early draft. Finally, both authors also wish to extend their gratitude to Alexander Campbell and Emily Riehl for their enthusiasm for what became Counterexample 4.14, which provided the impetus to write this paper.
Funding
Open Access funding enabled and organized by Projekt DEAL. This work was realised while both authors were at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester. The firstnamed author benefited from support by the National Science Foundation under Grant No. DMS1440140, while at residence in MSRI. The secondnamed author was supported by the Swiss National Science Foundation under the Project P1ELP2_188039. The firstnamed author was additionally supported by the National Science Foundation Grant DMS1652600, as well as the JHU Catalyst Grant.
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clingman, t., Moser, L. 2Limits and 2Terminal Objects are too Different. Appl Categor Struct 30, 1283–1304 (2022). https://doi.org/10.1007/s1048502209691z
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DOI: https://doi.org/10.1007/s1048502209691z