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April 18, 2014

Elementary Observations on 2-Categorical Limits

Posted by Emily Riehl

Guest post by Christina Vasilakopoulou

In the eighth installment of the Kan Extension Seminar, we discuss the paper “Elementary Observations on 2-Categorical Limits” by G.M. Kelly, published in 1989. Even though Kelly’s classic book Basic Concepts of Enriched Category Theory, which contains the abstract theory related to indexed (or weighted) limits for arbitrary 𝒱\mathcal{V}-categories, was available since 1982, the existence of the present article is well-justifiable.

On the one hand, it constitutes an independent account of the fundamental case 𝒱\mathcal{V}=Cat\mathbf{Cat}, thus it motivates and exemplifies the more general framework through a more gentle, yet meaningful exposition of 2-categorical limits. The explicit construction of specific notable finite limits such as inserters, equifiers etc. promotes the comprehension of the definitions, via a hands-on description. Moreover, these finite limits and particular results concerning 2-categories rather than general enriched categories, such as the construction of the cotensor as a PIE limit, are central for the theory of 2-categories. Lastly, by introducing indexed lax and pseudo limits along with Street’s bilimits, and providing appropriate lax/ pseudo/ bicategorical completeness results, the paper serves also as an indespensable reference for the later “2-Dimensional Monad Theory” by Blackwell, Kelly and Power.

I would like to take this opportunity to thank Emily as well as all the other participants of the Kan Extension Seminar. This has been a unique experience of constant motivation and inspiration for me!

Basic Machinery

Presently, our base of enrichment is the cartesian monoidal closed category Cat\mathbf{Cat} of (small) categories, with the usual adjunction ×𝒜[𝒜,]-\times\mathcal{A}\dashv[\mathcal{A},-]. The very definition of an indexed limit requires a good command of the basic Cat\mathbf{Cat}-categorical notions, as seen for example in “Review of the Elements of 2-categories” by Kelly and Street. In particular, a 2-natural transformation α:GH\alpha:G\Rightarrow H between 2-functors consists of components which not only satisfy the usual naturality condition, but also the 2-naturality one expressing compatibility with 2-cells. Moreover, a modification between 2-natural transformations m:αβm:\alpha\Rrightarrow\beta has components families of 2-cells m A:α Aβ A:GAHAm_A:\alpha_A\Rightarrow\beta_A:GA\to HA compatible with the mapped 1-cells of the domain 2-category, i.e. m BGf=Hfm Am_B\cdot Gf=Hf\cdot m_A (where \cdot is whiskering).

A 2-functor F:𝒦CatF:\mathcal{K}\to\mathbf{Cat} is called representable, when there exists a 2-natural isomorphism α:𝒦(K,)F. \alpha:\mathcal{K}(K,-)\xrightarrow{\quad\sim\quad}F. The components of this isomorphism are α A:𝒦(K,A)FA\alpha_A:\mathcal{K}(K,A)\cong FA in Cat\mathbf{Cat}, and the unit of the representation is the corresponding `element’ 1FK\mathbf{1}\to FK via Yoneda.

For a general complete symmetric monoidal closed category 𝒱\mathcal{V}, the usual functor category [𝒜,][\mathcal{A},\mathcal{B}] for two 𝒱\mathcal{V}-categories is endowed with the structure of a 𝒱\mathcal{V}-category itself, with hom-objects ends [𝒜,](T,S)= A𝒜(TA,SA) [\mathcal{A},\mathcal{B}](T,S)=\int_{A\in\mathcal{A}} \mathcal{B}(TA,SA) (which exist at least when 𝒜\mathcal{A} is small). In our context of 𝒱\mathcal{V}=Cat\mathbf{Cat} it is not necessary to employ ends and coends at all, and the hom-category [𝒦,](G,H)[\mathcal{K},\mathcal{L}](G,H) of the functor 2-category is evidently the category of 2-natural transformations and modifications. However, we note that computations via (co)ends simplify and are essential for constructions and (co)completeness results for enrichment in general monoidal categories.

The definition of weighted limits for 2-categories

To briefly motivate the definition of a weighted limit, recall that an ordinary limit of a (Set\mathbf{Set}-) functor G:𝒫𝒞G:\mathcal{P}\to\mathcal{C} is characterized by an isomorphism 𝒞(C,limG)[𝒫,𝒞](ΔC,G) \mathcal{C}(C,limG)\cong[\mathcal{P},\mathcal{C}](\Delta C, G) natural in CC, where ΔC:𝒫𝒞\Delta C:\mathcal{P}\to\mathcal{C} is the constant functor on the object CC. In other words, the limit is the representing object of the presheaf [𝒫,𝒞](Δ,G):𝒞 opSet. [\mathcal{P},\mathcal{C}](\Delta -,G):\mathcal{C}^\op\to\mathbf{Set}. Since components of a natural transformation ΔCG\Delta C\Rightarrow G (i.e. cones) can be viewed as components of a natural Δ1𝒞(C,G):𝒞Set\Delta\mathbf{1}\Rightarrow\mathcal{C}(C,G-):\mathcal{C}\to\mathbf{Set}, the above defining isomorphism can be written as 𝒞(C,limG)[𝒫,Set](Δ1,𝒞(C,G)). \mathcal{C}(C,\mathrm{lim}G)\cong[\mathcal{P},\mathbf{Set}](\Delta\mathbf{1},\mathcal{C}(C,G-)). In this form, ordinary limits can be easily seen as particular examples of conical indexed limits for 𝒱\mathcal{V}=Set\mathbf{Set}, and we are able to generalize the concept of a limit by replacing the functor Δ1\Delta\mathbf{1} by an arbitrary functor (weight) 𝒞Set\mathcal{C}\to\mathbf{Set}.

We may thus think of a 2-functor F:𝒫CatF:\mathcal{P}\to\mathbf{Cat} as a (small) indexing type or weight, and a 2-functor G:𝒫𝒦G:\mathcal{P}\to\mathcal{K} as a diagram in 𝒦\mathcal{K} of shape 𝒫\mathcal{P}: Cat weight F 𝒫 diagramG 𝒦. \begin{matrix} & \mathbf{Cat}\quad \\ {}^{weight} \nearrow_{F} & \\ \mathcal{P} & \overset{G}\underset{diagram}{\rightarrow} & \mathcal{K}. \\ \end{matrix} The 2-functor GG gives rise to a 2-functor p[Fp,𝒦(,Gp)]=[𝒫,Cat](F,𝒦(,G)):𝒦 opCat \int_p [Fp,\mathcal{K}(-,Gp)]=[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(-,G)):\; \mathcal{K}^\op\longrightarrow\mathbf{Cat} which maps a 0-cell AA to the category [𝒫,Cat](F,𝒦(A,G))[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(A,G)). A representation of this contravariant 2-functor is an object {F,K}𝒦\{F,K\}\in\mathcal{K} along with 2-natural isomorphism 𝒦(,{F,G})[𝒫,Cat](F,𝒦(,G))\mathcal{K}(-,\{F,G\})\xrightarrow{\;\sim\;}[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(-,G)) with components isomorphisms between categories 𝒦(A,{F,G})[𝒫,Cat](F,𝒦(A,G)). \mathcal{K}(A,\{F,G\})\cong[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(A,G-)). The unit of this representation is 1[𝒫,Cat](F,𝒦({F,G},G))\mathbf{1}\to[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(\{F,G\},G)) which corresponds uniquely to a 2-natural transformation ξ:F𝒦({F,G},G)\xi:F\Rightarrow\mathcal{K}(\{F,G\},G).

Via this 2-natural isomorphism, the object {F,G}\{F,G\} in 𝒦\mathcal{K} satisfies a universal property which can be expressed in two levels:

  • The 1-dimensional aspect of the universal property states that every natural transformation ρ\rho factorizes as Fρ 𝒦(A,G) ξ 𝒦(h,1) 𝒦({F,G},G) \begin{matrix} F \xrightarrow{\rho} & \mathcal{K}(A,G) \\ {}_\xi \searrow & \uparr_{\mathcal{K}(h,1)} \\ & \mathcal{K}(\{F,G\},G) \\ \end{matrix} for a unique 1-cell h:A{F,G}h:A\to\{F,G\} in 𝒦\mathcal{K}, where the vertical arrow is just pre-composition with hh.

  • The 2-dimensional aspect of the universal property states that every modification θ:ρρ\theta:\rho\Rrightarrow\rho' factorizes as 𝒦(α,1)ξ\mathcal{K}(\alpha,1)\cdot \xi for a unique 2-cell α:hh\alpha:h\Rightarrow h' in 𝒦\mathcal{K}.

The fact that the 2-dimensional aspect (which asserts an isomorphism of categories) does not in general follow from the 1-dimensional aspect (which asserts a bijection between the hom-sets of the underlying categories) is a recurrent issue of the paper. In fact, things would be different if the underlying category functor 𝒱(I,)=() 0:𝒱-CatCat \mathcal{V}(I,-)=(\;)_0:\mathcal{V}\text{-}\mathbf{Cat}\to\mathbf{Cat} were conservative, in which case the 2-dimensional universal property would always imply the 1-dimensional one. Certainly though, this is not the case for 𝒱\mathcal{V}=Cat\mathbf{Cat}: the respective functor discards all the 2-cells and is not even faithful. However, if we know that a weighted limit exists, then the first level of the universal property suffices to detect it up to isomorphism.

Completeness of 2-categories

A 2-category 𝒦\mathcal{K} is complete when all limits {F,G}\{F,G\} exist. The defining 2-natural isomorphism extends the mapping (F,G){F,G}(F,G)\mapsto\{F,G\} into a functor of two variables (the weighted limit functor) {,}:[𝒫,Cat] op×[𝒫,𝒦]𝒦 \{-,-\}:[\mathcal{P},\mathbf{Cat}]^{op}\times[\mathcal{P},\mathcal{K}]\longrightarrow \mathcal{K} as the left parametrized adjoint (actually its opposite) of the functor 𝒦(,?):𝒦 op×[𝒫,𝒦][𝒫,Cat] \mathcal{K}(-,?):\mathcal{K}^{op}\times[\mathcal{P},\mathcal{K}]\to[\mathcal{P},\mathbf{Cat}] mapping an object AA and a functor GG to 𝒦(A,G)\mathcal{K}(A,G-). A colimit in 𝒦\mathcal{K} is a limit in 𝒦 op\mathcal{K}^op, and the weighted colimit functor is *:[𝒫 op,Cat]×[𝒫,𝒦]𝒦. -\ast-:[\mathcal{P}^op,\mathbf{Cat}]\times[\mathcal{P},\mathcal{K}]\longrightarrow\mathcal{K}. Apart from the evident duality, we observe that often colimits are harder to compute than limits. This may partially be due to the fact that {F,G}\{F,G\} is determined by the representable 𝒦(,{F,G})\mathcal{K}(-,\{F,G\}) which gives generalized elements of {F,G}\{F,G\}, whereas the description of 𝒦(F*G,)\mathcal{K}(F\ast G,-) gives us arrows out of F*GF\ast G. For example, limits in Cat\mathbf{Cat} are easy to compute via [𝒜,{F,G}][𝒫,Cat](F,[𝒜,G])[𝒜,[𝒫,Cat](F,G)] [\mathcal{A},\{F,G\}]\cong[\mathcal{P},\mathbf{Cat}](F,[\mathcal{A},G-])\cong[\mathcal{A},[\mathcal{P},\mathbf{Cat}](F,G)] and in particular, taking 𝒜=1\mathcal{A}=\mathbf{1} gives us the objects of the category {F,G}\{F,G\} and 𝒜=2\mathcal{A}=\mathbf{2} gives us the morphisms. On the contrary, colimits in Cat\mathbf{Cat} are not straightforward (except than their property F*GG*FF\ast G\cong G\ast F).

Notice that like ordinary limits are defined, via representability, in terms of limits in Set\mathbf{Set}, we can define weighted limits in terms of limits of representables in Cat\mathbf{Cat}: 𝒦(A,{F,G}){F,𝒦(A,G)},𝒦(F*,G,A){F,𝒦(G,A)}. \mathcal{K}(A,\{F,G\})\cong\{F,\mathcal{K}(A,G-)\},\quad\mathcal{K}(F\ast ,G,A)\cong\{F,\mathcal{K}(G-,A)\}. On the other hand, if the weights are representables, via Yoneda lemma we get {𝒫(P,),G}GP,𝒫(,P)*GGP. \{\mathcal{P}(P,-),G\}\cong GP, \qquad \mathcal{P}(-,P)\ast G\cong GP.

The main result for general 𝒱\mathcal{V}-completeness in Kelly’s book says that a 𝒱\mathcal{V}-enriched category is complete if and only if it admits all conical limits (equivalently, products and equalizers) and cotensor products. Explicitly, conical limits are those with weight the constant 𝒱\mathcal{V}-functor ΔI\Delta I, whereas cotensors are those where the domain enriched category 𝒫\mathcal{P} is the unit category 1 \mathbf{1}, hence the weight and the diagram are determined by objects in 𝒱\mathcal{V} and 𝒦\mathcal{K} respectively. Once again, for 𝒱\mathcal{V}=Cat\mathbf{Cat} an elementary description of both limits is possible.

Notice that when a 2-category admits tensor products of the form 2*A\mathbf{2}\ast A, then the 2-dimensional universal property follows from the 1-dimensional for every limit, because of conservativity of the functor Cat 0(2,)\mathbf{Cat}_0(\mathbf{2},-) and the definition of tensors. Moreover, the former also implies that the category 2\mathbf{2} is a strong generator in Cat\mathbf{Cat}, hence the existence of only the cotensor {2,B}\{\mathbf{2},B\} along with conical limits in a 2-category 𝒦\mathcal{K} is enough to deduce 2-completeness.

Cat\mathbf{Cat} itself has cotensor and tensor products, given by {𝒜,}=[𝒜,]\{\mathcal{A},\mathcal{B}\}=[\mathcal{A},\mathcal{B}] and 𝒜*=𝒜×\mathcal{A}\ast\mathcal{B}=\mathcal{A}\times\mathcal{B}. It is ultimately also cocomplete, all colimits being constructed from tensors and ordinary colimits in Cat 0\mathbf{Cat}_0 (which give the conical limits in Cat\mathbf{Cat} by the existence of the cotensor [2,B][\mathbf{2},B]).

If we were to make use of ends and coends, the explicit construction of an arbitrary 2-(co)limit in 𝒦\mathcal{K} as the (co)equalizer of a pair of arrows between (co)products of (co)tensors coincides with {F,G}= K{FK,GK},F*G= KFK*GK. \{F,G\}=\int_K \{FK,GK\}, \qquad F\ast G=\int^K FK\ast GK. Such an approach simplifies the proofs of many useful properties of limits and colimits, such as {F,{G,H}}{F*G,H},(G*F)*HF*(G*H) \{F,\{G,H\}\}\cong\{F\ast G,H\},\;\;(G\ast F)\ast H\cong F\ast(G\ast H) for appropriate 2-functors.

Famous finite 2-limits

The paper provides the description of some important classes of limits in 2-categories, essentially by exhibiting the unit of the defining representation for each particular case. A table which summarizes the main examples included is the following:

graphs

Let’s briefly go through the explicit construction of an inserter in a 2-category 𝒦\mathcal{K}. The weight and diagram shape are as in the first line of the above table, and denote by BgfCB\overset{f}\underset{g}{\rightrightarrows}C the image of the diagram in 𝒦\mathcal{K}. The standard technique is to identify the form of objects and morphisms of the functor 2-category [𝒫,Cat](F,𝒦(A,G))[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(A,G-)), and then state both aspects of the universal property.

An object is a 2-natural transformation α:F𝒦(A,G)\alpha:F\Rightarrow\mathcal{K}(A,G-) with components α :1𝒦(A,B)\alpha_\bullet:1\to\mathcal{K}(A,B) and α :2𝒦(A,C)\alpha_\star:\mathbf{2}\to\mathcal{K}(A,C) satisfying the usual naturality condition (2-naturality follows trivially, since 𝒫\mathcal{P} only has the identity 2-cell). This amounts to the following data:

  • an 1-cell Aα BA\xrightarrow{\alpha_\bullet}B, i.e. the object in 𝒦(A,B)\mathcal{K}(A,B) determined by the functor α \alpha_\bullet;

  • a 2-cell α 0α α 1{\alpha_\star0}\overset{\alpha_\star}{\Rightarrow}{\alpha_\star1}, i.e. the morphism in 𝒦(A,C)\mathcal{K}(A,C) determined by the functor α \alpha_\star;

  • properties, which make the 1-cells α 0,α 1\alpha_\star0,\alpha_\star1 factorize as α 0=Aα BfC\alpha_\star0=A\xrightarrow{\alpha_\bullet}B\xrightarrow{f}C and α 1=Aα BgC\alpha_\star1=A\xrightarrow{\alpha_\bullet}B\xrightarrow{g}C.

We can encode the above data by a diagram B α f A α C. α g B \begin{matrix} & B & \\ {}^{\alpha_\bullet} \nearrow && {\searrow}^f \\ A\; & \Downarrow{\alpha_\star}& \quad C. \\ {}_{\alpha_\bullet} \searrow && \nearrow_g \\ & B & \\ \end{matrix} Now a morphism is a modification m:αβm:\alpha\Rrightarrow\beta between two objects as above. This has components

  • m :α β m_\bullet:\alpha_\bullet\Rightarrow\beta_\bullet in 𝒦(A,B)\mathcal{K}(A,B);

  • m :α β m_\star:\alpha_\star\Rightarrow\beta_\star given by 2-cells m 0:α 0β 0m_\star^0:\alpha_\star0\Rightarrow{\beta_\star0} and m 1:α 1β 1m_\star^1:\alpha_\star1\Rightarrow\beta_\star1 in 𝒦(A,C)\mathcal{K}(A,C) satisfying naturality m 1α =β m 0m^1_\star\circ\alpha_\star=\beta_\star\circ m^0_\star.

The modification condition m 0=fm m^0_\star=f\cdot m_\bullet and m 1=gm m^1_\star=g\cdot m_\bullet gives the components of m m_\star as whiskered composites of m m_\bullet. We can thus express such a modification as a 2-cell m m_\bullet satisfying gm α =fm β gm_\bullet\circ\alpha_\star=fm_\bullet\circ\beta_\star (graphically expressed by pasting m m_\bullet accordingly to the sides of α ,β \alpha_\star,\beta_\star).

This encoding simplifies the statement of the universal property for {F,G}\{F,G\}, as the object of in 𝒦\mathcal{K} through which any natural transformation and modification uniquely factorize in an appropriate way (in fact, through the unit ξ\xi). A very similar process can be followed for the identification of the other classes of limits. As an illustration, let’s consider some of these limits in the 2-category Cat\mathbf{Cat}.

  • The inserter of two functors F,G:𝒞F,G:\mathcal{B}\to\mathcal{C} is a category 𝒜\mathcal{A} with objects pairs (B,h)(B,h) where BB\in\mathcal{B} and h:FBGBh:FB\to GB in 𝒞\mathcal{C}. A morphism (B,h)(B,h)(B,h)\to(B',h') is an arrow f:BBf:B\to B' in \mathcal{B} such that the following diagram commutes: FB Ff FB h h FB Gh GB. \begin{matrix} FB & \overset{Ff}{\longrightarrow} & FB' \\ {}_h\downarrow && \downarrow_{h'} \\ FB & \underset{Gh}{\longrightarrow} & GB'. \\ \end{matrix} The functor α =P:𝒜\alpha_\bullet=P:\mathcal{A}\to\mathcal{B} is just the forgetful functor, and the natural transformation is given by (α ) (B,h)=h(\alpha_\star)_{(B,h)}=h.

  • The comma-object of two functors F,GF,G is precisely the comma category. If the functors have also the same domain, their inserter is a subcategory of the comma category.

  • The equifier of two natural transformations ϕ 1,ϕ 2:FG:𝒞\phi^1,\phi^2:F\Rightarrow G:\mathcal{B}\to\mathcal{C} is the full subcategory 𝒜\mathcal{A} of \mathcal{B} over all objects BB such that ϕ B 1=ϕ B 2\phi^1_B=\phi^2_B in 𝒞\mathcal{C}.

There is a variety of constructions of new classes of limits from given ones, coming down to the existence of endo-identifiers, inverters, iso-inserters, comma-objects, iso-comma-objects, lax/ oplax/pseudo limits of arrows and the cotensors {2,K}\{\mathbf{2},K\}, {I,K}\{\mathbf{I},K\} out of inserters, equifiers and binary products in the 2-category 𝒦\mathcal{K}. Along with the substantial construction of arbitrary cotensors out of these three classes, P(roducts)I(nserters)E(quifiers) limits are established as essential tools, also relatively to categories of algebras for 2-monads. Notice that equalizers are `too tight’ to fit in certain 2-categories of importance such as Lex\mathbf{Lex}.

Weaker notions of limits in 2-categories

The concept of a weighted 2-limit strongly depends on the specific structure of the 2-category [𝒫,Cat][\mathcal{P},\mathbf{Cat}] of 2-functors, 2-natural transformations and modifications, for the 2-categories 𝒫\mathcal{P} and Cat\mathbf{Cat}. If we alter this structure by considering lax natural transformations or pseudonatural transformations, we obtain the definition of the lax limit {F,G} l\{F,G\}_l and pseudo limit {F,G} p\{F,G\}_p as the representing objects for the 2-functors Lax[𝒫,Cat](F,𝒦(,G)):𝒦 opCat Psd[𝒫,Cat](F,𝒦(,G)):𝒦 opCat. \begin{matrix} Lax[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(-,G)):\mathcal{K}^\op\to\mathbf{Cat} \\ Psd[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(-,G)):\mathcal{K}^\op\to\mathbf{Cat}. \end{matrix} Notice that the functor categories Lax[𝒫,]Lax[\mathcal{P},\mathcal{L}] and Psd[𝒫,]Psd[\mathcal{P},\mathcal{L}] are 2-categories whenever \mathcal{L} is a 2-category, hence the defining isomorphisms are again between categories as before.

An important remark is that any lax or pseudo limit in 𝒦\mathcal{K} can be in fact expressed as a `strict’ weighted 2-limit. This is done by replacing the original weight with its image under the left adjoint of the incusion functors [𝒫,Cat]Lax[𝒫,Cat][\mathcal{P},\mathbf{Cat}]\hookrightarrow Lax[\mathcal{P},\mathbf{Cat}], [𝒫,Cat]Psd[𝒫,Cat][\mathcal{P},\mathbf{Cat}]\hookrightarrow Psd[\mathcal{P},\mathbf{Cat}]. The opposite does not hold: for example, inserters and equifiers are neither lax not pseudo limits.

We can relax the notion of limits in 2-categories even further, and define the bilimit {F,G} b\{F,G\}_b of 2-functors FF and GG as the representing object up to equivalence: 𝒦(A,{F,G} b)Psd[𝒫,Cat](F,𝒦(A,G)). \mathcal{K}(A,\{F,G\}_b)\simeq Psd[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(A,G)). This is of course a particular case of general bilimits in bicategories, for which 𝒫\mathcal{P} and 𝒦\mathcal{K} are requested to be bicategories and FF and GG homomorphism of bicategories. The above equivalence of categories expresses a birepresentation of the homomorphism Hom[𝒫,Cat](F,𝒦(,G)):𝒦 opCatHom[\mathcal{P},\mathbf{Cat}](F,\mathcal{K}(-,G)):\mathcal{K}^op\to\mathbf{Cat}.

Evidently, bilimits (firstly introduced by Ross Street) may exist even when pseudo limits do not, since they require an equivalence rather than isomorphism of hom-categories. The following two results sum up the conditions ensuring whether a 2-category has all lax, pseudo and bilimits.

  • A 2-category with products, inserters and equifiers has all lax and pseudo limits (whereas it may not have all strict 2-limits).

  • A 2-category with biproducts, biequalizers and bicotensors is bicategorically complete. Equivalently, it admits all bilimits if and only if for all 2-functors F:𝒫CatF:\mathcal{P}\to\mathbf{Cat}, G:𝒫𝒦G:\mathcal{P}\to\mathcal{K} from a small ordinary category 𝒫\mathcal{P}, the above mentioned birepresentation exists.

Street’s construction of an arbitrary bilimit requires a descent object of a 3-truncated bicosimplicial object in 𝒦\mathcal{K}. An appropriate modification of the arguments exhibits lax and pseudo limits as PIE limits.

These weaker forms of limits in 2-categories are fundamental for the theory of 2-categories and bicategories. Many important constructions such as the Eilenberg-Moore object as well as the Grothendieck construction on a fibration, arise as lax/oplax limits. They are also crucial in 2-monad theory, for example when studying categories of (strict) algebras with non-strict (pseudo or even lax/oplax) morphisms, which are more common in nature.

Posted at April 18, 2014 8:34 PM UTC

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15 Comments & 1 Trackback

Re: Elementary Observations on 2-Categorical Limits

A number of the theorems in the paper are about how to construct certain limits out of other limits. Some of the constructions are relatively simple:

  • Equation (4.4) identifies the lax limit of an arrow as a certain comma-object, and a cotensor product with 2 as a certain lax limit of an arrow.

  • Prop 4.1 constructs (iso)comma objects out of binary products and (iso)inserters.

  • Prop 4.2 constructs endo-identifiers out of equifiers, inverters out of inserters and endo-identifiers, and iso-inserters out of inserters and inverters.

  • Prop 6.1 constructs bilimits out of regular limits as a special case.

But some of them are more complicated:

  • Equation (3.2) constructs arbitrary limits out of products, equalizers, and cotensors.

  • Prop 4.3 constructs cotensor products out of products, inserters, and equifiers.

  • Prop 5.1 constructs lax (pseudo) limits out of products, (iso)inserters, (iso)equifiers, and cotensor products.

Generally Kelly describes the construction in question and leaves it to the reader to verify that the correct universal property is obtained. I suppose there are usually two ways to do this: by diagram chase, or by using the representable nature of limits to reduce to the case of 2-limits in Cat\mathbf{Cat}, where more explicit descriptions of the constructions are available.

Even using the representability trick, there’s still work to do in verifying that each part of the construction is appropriately functorial. Is there a more intuitive way to think about these constructions?

Posted by: Tim Campion on April 19, 2014 5:35 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

Not sure if this helps, but in each of these cases you can construct a map of finite 2-categories KLK\rightarrow L where KK is the “input” shape you want to take a limit of and LL is the “output” shape of the limit. e.g. for a comma object KK is the bottom right corner imbedded in LL which is the square with a 2-morphism. Or for a conical limit LL is the point.

Then for F:KCF:K\rightarrow C you can form the comma square in 22-Cat\mathbf{Cat}: X C L * F C K \begin{matrix} X & \rightarrow & C^L \\ \downarrow & \swArrow & \downarrow \\ * & \xrightarrow{F} & C^K \end{matrix}

and a final object in XX is the limit.

Posted by: Adam Gal on April 20, 2014 11:53 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I’m interested in this too.

The representable definition of limits, for FF and GG as above, says something like

𝒦(k,{F,G}){F,𝒦(k,G)}\mathcal{K}(k, \{F,G\}) \cong \{F, \mathcal{K}(k,G)\}

2-naturally in k𝒦k \in \mathcal{K}. The point is that the right-hand side is a 2-limit of a Cat\mathbf{Cat}-valued diagram.

Suppose we find some 2-limit formula in Cat\mathbf{Cat} for this particular type of 2-limit. Ignoring the diagram for the moment, this formula might have the form

{F,}{E,{I,}}\{F,-\} \cong \{E,\{I,-\}\}

where EE and II are weights, and II is probably a profunctor.

Of course, we can also talk about II-indexed and EE-indexed limits in 𝒦\mathcal{K}, where these are defined representably, and this representable definition is how we translate the above isomorphism in Cat\mathbf{Cat} to one in 𝒦\mathcal{K}.

Is this convincing?

You might think it’s not really worth arguing that it suffices to prove these formulas in Cat\mathbf{Cat} as a special case: after all, the formula for ordinary 1-categorical limits are typically proven in a general sufficiently complete category simply by comparing the universal properties. But here, I think the specific constructions of inserters and equifiers and so on in Cat\mathbf{Cat} (or if you prefer, of coinserters and coequifiers in the category [𝒫,Cat][\mathcal{P},\mathbf{Cat}] of weights) is helpful: for instance, we’re used to thinking of a category CC as a quotient of the free category on its underlying reflexive directed graph. This is exactly how you express the weight for a cotensor as coequifiers of coinserters of coproducts.

Posted by: Emily Riehl on April 21, 2014 8:30 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I’m starting to think that this representability thing is most helpful as a “meta” principle: it gives you a sense for what should be true by representable re-phrasing.

The precise statement of the principle is that representables preserve and jointly reflect limits: K(k,limF)limK(k,F)K(k, \lim F) \cong \lim K(k,F).

As for its actual use, let me illustrate my concerns with a toy argument that you outlined in class, Emily, which I’ve tinkered with to reflect my understanding. This is an argument that a 1-category with products and equalizers is (1-categorically) complete, using the same fact in Set\mathbf{Set}.

Consider F:DKF \colon D \to K. To find a limit for FF, it suffices to represent the functor limK(k,F):KSet\lim K(k,F): K \to \mathbf{Set}, because representables jointly reflect limits. For any kKk \in K, we have

limK(k,F)eq( dK(k,Fd) ddK(k,Fd)) K(k,eq( dFd ddFd))\begin{matrix} \lim K(k,F) \cong \mathrm{eq}\left( \prod_{d} K(k,Fd) \rightrightarrows \prod_{d \to d'} K(k, Fd')\right)\\ \cong K\left(k, \mathrm{eq}( \prod_d Fd \rightrightarrows \prod_{d \to d'} Fd') \right) \end{matrix}

The first isomorphism is a construction of the limit in Set\mathbf{Set}. The second is because representables K(k,)K(k,-) preserve limits.

I guess I’m uncomfortable about the fact that we don’t really analyze those two arrows we’re equalizing. Surely if we go through the argument carefully, there must be some point where we really have to analyze the construction, and show that the equalizer in KK is really sent to the one in Set\mathbf{Set} by K(k,)K(k,-)!

Let’s see. The last step is really two steps:

  1. Use dK(k,Fd)K(k, dFd)\prod_d K(k, Fd) \cong K(k, \prod_d Fd) and similarly with the other product.
  2. Use eq(K(k,))K(k,eq())\mathrm{eq}(K(k,\dots)) \cong K(k, \mathrm{eq}(\dots)).

In order to apply the second step, we need to know that the the two arrows we’re equalizing are of the form K(k,)K(k, \dots). I suppose this is automatic by the Yoneda lemma. We don’t need to analyze the arrows to conclude this.

But what about when we check that all of this is natural in kk? Will we have to unpack those arrows then? This question becomes more pressing for me when looking at the categorified version of this argument, where Kelly shows that products, equalizers, and cotensors suffice for 2-completeness.

I should probably come clean and admit that ever since I first saw the construction of limits from products from equalizers in introductory category theory, I had thought of it is a “non-functorial” sort of construction – I suppose because of the way the indexing category DD is “deconstructed” to form the products and equalizer diagram. But the construction is functorial enough to show that a functor between complete categories preserving products and equalizers preserves all small limits. And it’s functorial enough for the purposes of this argument, too. But it doesn’t feel right to me, somehow that it should be. So maybe I’m just suffering from an irrational misconception and nothing will convince me!

Posted by: Tim Campion on April 22, 2014 4:41 AM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

Now I understand your point: because we know how to define the equalizer and product formula for a general limit, we can see that the arrows in the equalizer diagram above are of the form K(k,)K(k,-). But it’s not immediately obvious from our description of this formula for a generic Set-valued diagram that this will be the case for the particular diagram K(k,F)K(k,F-) — until we check and see that it is true.

As you suggest, there should be some way to use the Yoneda lemma to see that this isn’t something we need to worry about. Can anyone weigh in and clarify?

I should probably come clean and admit that ever since I first saw the construction of limits from products from equalizers in introductory category theory, I had thought of it is a “non-functorial” sort of construction — I suppose because of the way the indexing category DD is “deconstructed” to form the products and equalizer diagram. But the construction is functorial enough to show that a functor between complete categories preserving products and equalizers preserves all small limits. And it’s functorial enough for the purposes of this argument, too. But it doesn’t feel right to me, somehow that it should be. So maybe I’m just suffering from an irrational misconception and nothing will convince me!

Now, regarding, functoriality of the limit formula, I think the first thing to get used to is the functoriality of the nerve construction N:CatSet Δ opN \colon \mathbf{Cat} \to \mathbf{Set}^{\Delta^{op}}. This is what’s behind the functoriality of the simplicial bar and cosimplicial cobar constructions:

B (*,D,):K DK Δ opB_\bullet(*,D,-) \colon K^D \to K^{\Delta^{op}}

C (*,D,):K DK ΔC^\bullet(*,D,-) \colon K^D \to K^{\Delta}

The product and equalizer limit formula is equally

lim ΔC (*,D,F).\lim_{\Delta} C^\bullet(*,D,F).

Posted by: Emily Riehl on April 22, 2014 6:20 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I’d like to help, but I still don’t understand what the problem is.

Posted by: Mike Shulman on April 22, 2014 7:14 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

This last paragraph of Emily’s is very aptly put, and is a good way to feel one’s way into homotopy limits and colimits.

Posted by: Todd Trimble on April 22, 2014 7:14 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I wait for this since Simon Willerton’s post. Weighted limits are a natural generator of categorical integrals!

To my eye, weighted limits are best appreciated once you see that there are canonical isomorphisms (allow me to switch to a notation I’m more comfortable with!) lim WF cC{Wc,Fc} \underset{\leftarrow}{\lim}\!{}^W F \cong \int_{c\in \mathbf C} \{ W c, F c\} lim WF cCWc*Fc\underset{\to}\lim\!{}^W F \cong \int^{c\in \mathbf C} W c * F c for any diagram FF and weight WW (of course you have to suppose (co)completeness). Using these a great number of evocative properties of weighted limits and colimits are easily proved:

  1. (pointwise) Kan extensions can be described as those weighted (co)limits where the weight is representable: just look at the end/coend form and recognize the power/copower action.

  2. Set\Set-weighted limits are nothing interesting, as they can be reduced to ordinary limits, taken over the category of elements of the weight. There is in fact a canonical isomorphism lim WFlim el WF\lim\!{}^W F\cong \lim\!{}_{\text{el }W} F. This is spelled out precisely in a short book by Dubuc, which I really enjoyed reading. I had a pleasant conversation about this with A. Joyal which made me wonder why in an enriched setting we lack a “category of elements” construction; can this be viewed as implying that we are forced to allow the existence of weighted (co)limits? Can one differentiate the “contexts” where we can get a Grothendieck construction from others where we can’t?

  3. (Co)ends are, on their own right, suitable weighted (co)limits: given H:C op×CSetsH\colon \mathbf C^\text{op}\times \mathbf C\to \mathbf{Sets}, then cH(c,c)lim homH\int_c H(c,c)\cong \underset{\leftarrow}{\lim}^\text{hom} H. I learned from the “private” discussion after the expose’ that this definition extends ends and coends to an arbitrary enriched category; good to know!

  4. One has isomorphisms lim lim ZWFlim Z(lim WF)lim W(lim ZF)\underset{\leftarrow}{\lim}\!{}^{\underset{\to}\lim\!{}^Z W}F\cong \underset{\leftarrow}{\lim}\!{}^Z\Big(\underset{\leftarrow}{\lim}\!{}^W F\Big)\cong \underset{\leftarrow}{\lim}\!{}^W\Big(\underset{\leftarrow}{\lim}\!{}^Z F\Big) lim lim ZWFlim Z(lim WF)lim W(lim ZF)\underset{\to}\lim\!{}^{\underset{\to}\lim\!{}^Z W}F\cong \underset{\to}\lim\!{}^Z\Big(\underset{\to}\lim\!{}^W F\Big)\cong \underset{\to}\lim\!{}^W\Big(\underset{\to}\lim\!{}^Z F\Big) telling that “(co)limits commute with (co)limits”. This leads me to ask: there is a well-established -albeit extremely involved- theory of “which colimits commute with which limits”; can this be extended to ask “which weighted colimits commute with which weighted limits”?

  5. (This is an astoundingly simple exercise using ends): if DC\mathbf D\subset \mathbf C is a full subcategory, then the weighted limit of Fi:DCFMF\circ i \colon \mathbf D\hookrightarrow\mathbf C\xrightarrow{F} \mathbf{M} along a weight W:D𝒱W\colon \mathbf{D}\to \mathcal{V} is isomorphic to the weighted limit of FF along Lan iW:C𝒱\text{Lan}_i W\colon \mathbf{C}\to{\mathcal{V}}:

(1)lim W(F| D)lim Lan iWF\underset{\leftarrow}{\lim}{}\!^W(F|_{\mathbf D})\cong \underset{\leftarrow}{\lim}{}\!^{\text{Lan}_i W}F
Posted by: Fosco Loregian on April 21, 2014 12:18 AM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

  1. (pointwise) Kan extensions can be described as those weighted (co)limits where the weight is representable: just look at the end/coend form and recognize the power/copower action.

Wait – when the weight is representable, isn’t the (co)limit just given by evaluation at that object?

Posted by: Tim Campion on April 22, 2014 3:58 AM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

With regard to your question about weighted limits and colimits commuting: You might want to check out

G. M. Kelly and V. Schmitt, “Notes on Enriched Categories with Colimits of some Class,”

particularly sections 5 and 6. The authors discuss when weighted limits and colimits commute in 𝒱\mathcal{V}, and especially the class of small projective weights. These are those weights ϕ\phi such that ϕ\phi-weighted limits commute with all colimits in 𝒱\mathcal{V}. It turns out that these are the same as the absolute weights, i.e. those weights whose limits are preserved by any functor whatsoever. I don’t know if anyone’s studied this question in general 𝒱\mathcal{V}-categories rather than just 𝒱\mathcal{V} itself though.

Posted by: Tom Avery on April 23, 2014 3:05 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I’ve just learned, from a number of participants in the seminar, about the following characterization of weights for PIE-limits: a 2-functor W:ACatW \colon \mathbf{A} \to \mathbf{Cat} is a PIE weight if and only if the underlying functor W 0:A 0CatSetW_0 \colon \mathbf{A}_0\to \mathbf{Cat} \to \mathbf{Set} — technically the composite of W 0W_0 (which throws away the 2-cells) with the set-of-objects functor, but I’ll just abbreviate to W 0W_0 — is a coproduct of representable functors.

It’s easy to check that the weights for products, inserters, and equifiers satisfies this condition. As observed by Tim and Joe, it makes it easy to see that the weight for EM-objects is also PIE. (The previous proof I knew was “computadic”.)

If I know that WW is a PIE weight, in particular, if I’m given a projective cell complex structure on WW as an object of the functor category [A,Cat][A, \mathbf{Cat}], then it’s clear that W 0W_0 will be a coproduct of representables: the generating projective cofibrations are products of representables with *\emptyset \to \ast and with two identity-on-objects functors.

But how do I prove the converse, that any WW with W 0W_0 of this form is a PIE weight?

Posted by: Emily Riehl on April 21, 2014 2:04 AM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

If I recall correctly, you can use that fact to build WW out of representables using coproducts, coinserters, and coequifiers in the 2-category of weights. From W 0W_0 being a coproduct of representables you can start with the same coproduct of the same representables, then use coinserters and coequifiers to make the morphisms come out right.

Posted by: Mike Shulman on April 21, 2014 5:41 AM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

Okay, let’s see if I really understand how this works.

(1) Let’s take W:1CatW \colon \mathbf{1} \to \mathbf{Cat} to be a weight for the cotensor with a category CC, i.e., WW is the functor C\bullet \mapsto C. Then W 0 obC1(,)W_0 \cong \coprod_{ob C} \mathbf{1}(\bullet,-) so we deduce that WW is a PIE weight.

Let’s now express WW-limits as a composite of products, inserters, and equifiers — equally express WW as coequifiers of coinserts of coproducts. From the isomorphism W 0 obC1(,)W_0 \cong \coprod_{ob C} \mathbf{1}(\bullet,-) and the Yoneda lemma, we get a map

obC1(,)W\coprod_{ob C} \mathbf{1}(\bullet,-) \to W

in the category of weights (no longer an isomorphism). For each morphism of C=WC = W\bullet, we can form a coinserter, which I’ll do all at once using the diagram:

morC1(,) obC1(,)\coprod_{mor C} \mathbf{1}(\bullet,-) \rightrightarrows \coprod_{ob C} \mathbf{1}(\bullet,-)

One of the arrows projects to the domain of the indexing morphism of CC and the other projects to the codomain. Note WW forms a cone under this diagram so we get a map from the coinserter II to WW.

As a functor I:1CatI \colon \mathbf{1} \to \mathbf{Cat}, II picks out the category with the same objects as CC and whose morphisms are freely generated by morphisms in CC. So we next form the coequifier

comp~pair~morC1(,) I \begin{array}{ccc} \quad & \rightarrow & \\ \coprod_{\text{comp~pair~mor} C} \mathbf{1}(\bullet,-) & \Downarrow \Downarrow & I \\ & \rightarrow & \end{array}

where one of the 2-cells picks out the free composite of a composable pair of morphisms and the other picks out the actual composite. This is the thing that’s isomorphic to WW.

(2) Consider W:𝒫CatW \colon \mathcal{P} \to \mathbf{Cat}, the weight for the comma object. Then W 0𝒫(,)𝒫(*,)W_0 \cong \mathcal{P}(\bullet,-) \coprod \mathcal{P}(\ast,-). We again have a map

𝒫(,)𝒫(*,)W\mathcal{P}(\bullet,-) \coprod \mathcal{P}(\ast,-) \to W

in the category of weights. The coinserter of

𝒫(,)𝒫(,)𝒫(*,)\mathcal{P}(\star,-) \rightrightarrows \mathcal{P}(\bullet,-) \coprod \mathcal{P}(\ast,-)

is isomorphic to WW.

(3) Consider W:𝒫CatW \colon \mathcal{P} \to \mathbf{Cat}, the weight for the equifier. Then W 0𝒫(,)W_0 \cong \mathcal{P}(\bullet,-). We have a map

𝒫(,)W\mathcal{P}(\bullet,-) \to W

that’s not an isomorphism and also not injective: 𝒫(,)\mathcal{P}(\bullet, \star) is the parallel pair while WW\star is the walking arrow. But this is corrected by forming the coequifier:

𝒫(,) 𝒫(,) \begin{array}{ccc} \quad & \rightarrow & \\ \mathcal{P}(\star,-) & \Downarrow \Downarrow & \mathcal{P}(\bullet, -) \\ & \rightarrow \end{array}

which is isomorphic to WW.

These examples aside, I’m not sure I fully understand the general principle at work here. From the characterization of W 0W_0 we are lead to consider a coproduct of representable 2-functors 𝒫𝒞𝒶𝓉\mathcal{P} \to \mathcal{Cat} whose objects agree with the objects of WW. This and the Yoneda lemma defines a 2-natural transformation from the coproduct of representables to WW that is bijective on objects in each component. We can “correct the morphisms” in each component using coinserters and coequifiers, as illustrated above, but it seems we’re not really paying attention to the 1-cells and 2-cells in the image of WW; we’re just fixing the objects.

But now that I think about it, I suppose we’ve just constructed some 2-functor (via a coequifier of a coinserter of a coproduct of representables) and a 2-natural transformation to WW that is an isomorphism on each component. Because a 2-natural transformation is an isomorphism iff its components are, it follows that this 2-functor must be isomorphic to WW. So I guess that’s it. Huh.

Posted by: Emily Riehl on April 21, 2014 8:15 PM | Permalink | Reply to this

Re: Elementary Observations on 2-Categorical Limits

I realize belatedly that in the first example I forgot to correct the identities. It’s probably good enough to restrict the codomain of the coinserter to the coproduct over non-identity morphisms. But you get the idea.

Posted by: Emily Riehl on April 21, 2014 8:32 PM | Permalink | Reply to this
Read the post On Two-Dimensional Monad Theory
Weblog: The n-Category Café
Excerpt: Describes the approach to two-dimensional universal algebra taken in the paper of Blackwell, Kelly, and Power on two-dimensional monad theory.
Tracked: April 28, 2014 10:32 PM

Re: Elementary Observations on 2-Categorical Limits

The aforementioned appreciations are of great consideration , and additional standpoint may be studied in Adv. Math. 2014 ; 252 : 708-747 , concerning 2-dimensional universal algebra and comparatives with 2-categorical limits , enhancing the role of F-categories interplay between weak and strong morphisms.

Posted by: SABINO GUILLERMO ECHEBARRIA MENDIETA on April 30, 2014 11:39 AM | Permalink | Reply to this

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