nLab
2-limit

Context

\infty-Limits


2-Category theory

Contents

Idea

A 2-limit is the type of limit that is appropriate in a (weak) 2-category. (Since general 2-categories are often called bicategories, 2-limits are often called bilimits.)

There are three notable changes when passing from ordinary 1-limits to 2-limits:

  1. In order to satisfy the principle of equivalence, the “cones” in a 2-limit are required to commute only up to 2-isomorphism.

  2. The universal property of the limit is expressed by an equivalence of categories rather than a bijection of sets. This means that

    1. every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and

    2. every transformation between cones also factors through a 2-cell in the limit. We will give some examples below.

  3. Since 2-categories are enriched over Cat (this is precise in the strict case, and weakly true otherwise), CatCat-weighted limits become important. This means that both the diagrams we take limits of and the shape of “cones” that limits represent can involve 22-cells as well as 11-cells.

Definition

Let KK and DD be 2-categories, and J:DCatJ\colon D\to Cat and F:DKF\colon D\to K be 2-functors. A JJ-weighted (2-)limit of FF is an object LKL\in K equipped with a pseudonatural equivalence

K(X,L)[D,Cat](J,K(X,F)). K(X,L) \simeq [D,Cat](J,K(X,F-)).

where [D,Cat][D,Cat] denotes the 2-category of 2-functors DCatD\to Cat, pseudonatural transformations between them, and modifications between those.

A 2-limit in the opposite 2-category K opK^{op} is called a 2-colimit in KK. Everything below applies dually to 2-colimits, the higher analogues of colimits. (But somebody might want to make a separate page that gives appropriate examples of these.)

Strictness and terminology

If KK and DD are strict 2-categories, JJ and FF are strict 2-functors, and if we we replace this pseudonatural equivalence by a (strictly 2-natural) isomorphism and the 2-category [D,Cat][D,Cat] by the 2-category [D,Cat] strict[D,Cat]_{strict} of strict 2-functors and strict 2-natural transformations, then we obtain the definition of a strict 2-limit. This is precisely a Cat-weighted limit in the sense of ordinary enriched category theory. See strict 2-limit for details.

On the other hand, if KK, DD, JJ, and FF are strict as above, and we replace the equivalence by an isomorphism but keep the weak meaning of [D,Cat][D,Cat], then we obtain the notion of a strict pseudolimit. Strict pseudolimits are, in particular, 2-limits, whereas strict 2-limits are not always (although some, such as PIE-limits and flexible limits, are). In a strict 2-category, these types of strict limits are often technically useful in constructing the “up-to-isomorphism” 2-limits we consider here.

When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a (non-strict) 2-limit. Therefore, we usually call these simply “limits.” To emphasize the distinction with the strict 2-limits in a strict 2-category, the “up-to-isomorphism” 2-limits were historically often called bilimits (by analogy with bicategory for “weak 2-category”). However, this terminology is somewhat unfortunate, not only because it doesn’t generalize well to nn, but because it leads to words like “biproduct,” which also has the completely unrelated meaning of an object that is both a product and a coproduct (which is common in additive categories).

Unfortunately, we probably shouldn’t use “weak limit” to emphasize the “up-to-isomorphism” nature of these limits, because that also has the completely unrelated meaning of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.

Examples

2-limits over diagrams of special shape

Any ordinary type of limit can be “2-ified” by boosting its ordinary universal property up to a 2-categorical one. In the following examples we work in a 2-category KK.

  • A terminal object in KK is an object 1 such that K(X,1)K(X,1) is equivalent to the terminal category for any object XX. This means that for any XX there is a morphism X1X\to 1 and for any two morphisms f,g:X1f,g:X\to 1 there is a unique isomorphism fgf\cong g.

  • A product of two objects A,BA,B in KK is an object A×BA\times B together with a natural equivalence of categories K(X,A×B)K(X,A)×K(X,B)K(X,A\times B) \simeq K(X,A)\times K(X,B). This means that we have projections p:A×BAp:A\times B\to A and q:A×BBq:A\times B\to B such that (1) for any f:XAf:X\to A and g:XBg:X\to B, there exists an h:XA×Bh:X\to A\times B and isomorphisms phfp h\cong f and qhgq h\cong g, and (2) for any h,k:XA×Bh,k:X\to A\times B and 2-cells α:phpk\alpha:p h \to p k and β:qhqk\beta: q h \to q k, there exists a unique γ:hk\gamma:h \to k such that pγ=αp \gamma = \alpha and qγ=βq \gamma = \beta.

  • A pullback of a co-span AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B consists of an object A× CBA\times_C B and projections p:A× CBAp:A\times_C B\to A and q:A× CBBq:A\times_C B\to B together with an isomorphism ϕ:fpgq\phi:f p \cong g q, such that (1) for any m:XAm:X\to A and n:XBn:X\to B with an isomorphism ψ:fmgn\psi:f m \cong g n, there exists an h:XA× CBh:X\to A\times_C B and isomorphisms α:phm\alpha:p h \cong m and β:qhn\beta:q h \cong n such that gβ.ϕh.fα 1=ψg\beta . \phi h . f \alpha^{-1} = \psi, and (2) given any two morphisms h,k:XA× CBh,k:X\to A\times_C B and 2-cells μ:phpk\mu:p h \to p k and ν:qhqk\nu:q h \to q k such that fμ=gνf \mu = g \nu (modulo the given isomorphism fpgqf p \cong g q), i.e., ϕk.fμ=gν.ϕh\phi k . f\mu = g\nu . \phi h, there exists a unique 2-cell γ:hk\gamma:h\to k such that pγ=μp \gamma = \mu and qγ=νq \gamma = \nu. This is sometimes called the pseudo-pullback but that term more properly refers to a particular strict 2-limit.

  • An equalizer of f,g:ABf,g:A\to B consists of an object EE and a morphism e:EAe:E\to A together with an isomorphism fegef e \cong g e, which is universal in a sense the reader should now be able to write down. This is sometimes called the pseudo-equalizer but that term more properly refers to a particular strict 2-limit. Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by descent objects.

There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just “boosted-up” versions of 1-limits.

  • The comma object of a cospan AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a universal object (f/g)(f/g) and projections p:(f/g)Ap:(f/g)\to A and q:(f/g)Bq:(f/g)\to B together with a transformation (not an isomorphism) fpgqf p \to g q. In Cat, comma objects are comma categories. Comma objects are sometimes called lax pullbacks but that term more properly refers to the lax version of a pullback; see “lax limits” below.

  • The inserter of a pair of parallel arrows f,g:ABf,g:A \;\rightrightarrows\; B is a universal object II equipped with a map i:IAi:I\to A and a 2-cell figif i \to g i.

  • The equifier of a pair of parallel 2-cells α,β:fg:AB\alpha,\beta: f\to g: A\to B is a universal object EE equipped with a map e:EAe:E\to A such that αe=βe\alpha e = \beta e.

  • The inverter of a 2-cell α:fg:AB\alpha:f\to g:A\to B is a universal object VV with a map v:VAv:V\to A such that αv\alpha v is invertible.

  • The power of an object AA by a category CC is a universal object A CA^C equipped with a functor CK(A C,A)C\to K(A^C,A). Of particular importance is the case when CC is the walking arrow 2\mathbf{2}.

Finite 2-Limits

A 2-limit is called finite if its diagram shape and its weight are both “finitely presentable” in a suitable sense (defined in terms of computads; see Street’s article Limits indexed by category-valued 2-functors ). Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category. All finite limits can be constructed from pullbacks, a terminal object, and powers with 2\mathbf{2}.

(2,1)(2,1)-limits

If the ambient 2-category is in fact a (2,1)-category in that all 2-morphisms are invertible then there is a rich set of tools available for handling the 2-limits in this context. We may speak (2,1)(2,1)-limits and (2,1)(2,1)-colimits in this case.

These are then a special case of the more general (∞,1)-limits and (∞,1)-colimits in a (∞,1)-category. A (2,1)-category is a special case of an (∞,1)-category.

Traditionally, (∞,1)-limits are best known in tersm of the presentation of (,1)(\infty,1)-catgeories by categories with weak equivalences in general and model categories in particular. (2,1)-limits can often also be viewed in this way. The corresponding presentation of the (,1)(\infty,1)-limits / (2,1)(2,1)-limits is called homotopy limits and homotopy colimits.

For instance 2-limits in the (2,1)-category Grpd of groupoids, functors and (necessarily) natural isomorphisms. Are equivalently computed as homotopy limits in the model structure on simplicial sets sSet QuillensSet_{Quillen} of diagrams of 1-truncated Kan complexes. (The equivalnce of homotopy limits with (,1)(\infty,1)-limits is discusssed at (∞,1)-limit).

Or for instance, more generally, the 2-limits in any (2,1)-sheaf(=stack) (2,1)-topos may be computed as homotopy limits in a model structure on simplicial presheaves over the given (2,1)-site of diagrams of 1-truncated simplicial presheaves. This includes as examples big (2,1)-toposes such as those over the large sites Top or SmoothMfd where computations with topological groupoids/topological stacks, Lie groupoids/differentiable stacks etc. take place.

Lax limits

A lax limit can be defined like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified transformation, not necessarily an isomorphism. In other words, in place of the 2-category [D,Cat][D,Cat] we use the 2-category [D,Cat] l[D,Cat]_l whose morphisms are lax natural transformations; thus the lax limit LL of a diagram FF weighted by JJ is equipped with a universal lax natural transformation JK(L,F)J\to K(L,F-).

This may look like a different concept, but in fact, for any weight JJ there is another weight Q l(J)Q_l(J) such that lax JJ-weighted limits are the same as Q l(J)Q_l(J)-weighted 2-limits. Here Q lQ_l is the lax morphism classifier for 2-functors. Therefore, lax limits are really a special case of 2-limits. Similarly, oplax limits, in which we use oplax natural transformations, are also a special case of 2-limits.

There is a further simplification of lax limits in the case of “conical” lax limits where the weight J=Δ1J=\Delta 1 is constant at the terminal category. In this case, it is easy to check that a lax natural transformation Δ1K(X,F)\Delta 1 \to K(X,F-) is the same as a lax natural transformation ΔXF\Delta X \to F. Thus, a conical lax limit of FF is a representing object for such lax transformations.

Here are some examples.

  • Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.

  • The lax limit of an arrow f:ABf:A\to B is a universal object LL equipped with projections p:LAp:L\to A and q:LBq:L\to B and a 2-cell fpqf p \to q. Note that this is equivalent to a comma object (f/1 B)(f/1_B).

  • The lax pullback of a cospan AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a universal object PP equipped with projections p:PAp:P\to A, q:PBq:P\to B, r:PCr:P\to C, and 2-cells fprf p \to r and gqrg q \to r.

Note that lax pullbacks are not the same as comma objects. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using “lax pullback” to mean “comma object” can be misleading.

A lax colimit in KK is, of course, a lax limit in K opK^{op}. Thus, it is a representing object for lax natural transformations JK(F,L)J \to K(F-,L). There is a subtlety here, however, because in the case J=Δ1J=\Delta 1, a lax natural transformation Δ1K(F,L)\Delta 1 \to K(F-,L) is the same as an oplax natural transformation FΔLF \to \Delta L. Thus, it is easy to mistakenly say “lax colimit” when one really means “oplax colimit” and vice versa.

Remark

With this in mind, one might be tempted to switch the meanings of “lax colimit” and “oplax colimit”, but there is a good reason not to. Recall that a lax JJ-weighted limit is the same as an ordinary Q l(J)Q_l(J)-weighted limit. Standard terminology in enriched category theory is that a WW-weighted colimit in an enriched category KK is the same as a WW-weighted limit in K opK^{op}, and indeed in that generality there is no other option. Thus, a lax JJ-weighted colimit in KK should be an ordinary Q l(J)Q_l(J)-weighted colimit in KK, hence a Q l(J)Q_l(J)-weighted limit in K opK^{op}, and thus a lax JJ-weighted limit in K opK^{op}.

Here are some examples of lax and oplax colimits:

2-Colimits in CatCat

As shown here, if CC is an ordinary category and F:CCatF \colon C \to Cat is a pseudofunctor, then the oplax colimit of FF is given by the Grothendieck construction F\int F — and its pseudo-colimit is given by formally inverting the opcartesian morphisms in F\int F. This yields a construction of certain pseudo 2-colimits in CatCat.

Moreover, a similar result holds more generally when CC is a bicategory. In this case, F\int F is also a bicategory: a 2-cell from (m:cd,f:m *xy)(m \colon c \to d, f \colon m_*x \to y) to (n:cd,g:n *xy)(n \colon c \to d, g \colon n_*x \to y) is given by a 2-cell μ:mn\mu \colon m \Rightarrow n in CC such that μ *x\mu_* x is a morphism fgf \to g over yy.

Let π *\pi_* denote the functor that sends a bicategory KK to the category whose objects are those of KK and whose hom-sets are the connected components of the hom-categories of KK; let also d *d_* denote the functor that sends a category XX to the corresponding locally discrete bicategory. Then there is an equivalence of categories

[K,d *X][π *K,X] [K, d_* X] \simeq [\pi_* K, X]

It is straightforward to check that the first of the above facts extends to the bicategorical case:

Lax(F,ΔX)[F,d *X] Lax(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]

as does the fact that a lax transformation on the left is pseudo if and only if the corresponding functor on the right inverts the opcartesian morphisms in F\int F. It is almost trivial that the adjunction π *d *\pi_* \dashv d_* holds when restricted to the functor [,] S 1[-, -]_{S^{-1}} that takes two categories or bicategories to the full subcategory of functors that invert the class SS of morphisms. Taking SS to be the opcartesian morphisms in F\int F, then, we have

Ps(F,ΔX)[F,d *X] S 1[π *F,X] S 1[(π *F)[S 1],X] Ps(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]_{S^{-1}} \simeq [\pi_* {\textstyle \int} F, X]_{S^{-1}} \simeq [(\pi_* {\textstyle \int} F)[S^{-1}], X]

Hence the pseudo colimit of FF is got by taking its bicategory of elements, applying the ‘local π 0\pi_0’ functor, and then inverting the (images of the) opcartesian morphisms as usual.

References

  • Ross Street, Limits indexed by category-valued 2-functors

  • Max Kelly, Elementary observations on 2-categorical limits, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, link

  • Ross Street, Fibrations in Bicategories and correction.

Section 6, page 37 in

Chapters 3,4,5 in

Revised on January 31, 2014 09:18:04 by Urs Schreiber (89.204.137.155)