nLab
2-pullback

22-pullbacks

An ordinary pullback is a limit over a diagram of the form ACBA \to C \leftarrow B. Accordingly, a 2-pullback (or 2-fiber product) is a 2-limit over such a diagram.

Definition

Saying that “a 2-pullback is a 2-limit over a cospan” is in fact a sufficient definition, but we can simplify it and make it more explicit.

A 22-pullback in a 2-category is a square

P p A q f B g C\array{P & \overset{p}{\to} &A \\ ^q\downarrow & \cong & \downarrow^f\\ B& \underset{g}{\to} &C }

which commutes up to isomorphism, and which is universal among such squares in a 2-categorical sense. This means that (1) given any other such square

Z v A w f B g C\array{Z & \overset{v}{\to} &A \\ ^w\downarrow & \cong & \downarrow^f\\ B& \underset{g}{\to} &C }

which commutes up to isomorphism, there exists a morphism u:ZPu:Z\to P and isomorphisms puvp u \cong v and quwq u \cong w which are coherent with the given ones above, and (2) given any two morphisms u,t:ZPu,t:Z\to P and 2-cells α:pupt\alpha:p u \to p t and β:quqt\beta:q u \to q t such that fα=gβf \alpha = g \beta (modulo the given isomorphism fpgqf p \cong g q), there exists a unique 2-cell γ:ut\gamma:u\to t such that pγ=αp \gamma = \alpha and qγ=βq \gamma = \beta.

Equivalence of definitions

The simplification in the above explicit definition has to do with the omission of an unnecessary structure map. Note that an ordinary pullback of AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B comes equipped with maps PpAP\overset{p}{\to} A, PqBP\overset{q}{\to} B, and PrCP\overset{r}{\to} C, but since r=fpr = f p and r=gqr = g q, the map rr is superfluous data and is usually omitted. In the 2-categorical case, where identities are replaced by isomorphisms, it is, strictly speaking, different to give merely pp and qq with an isomorphism fpgqf p \cong g q, than to give pp, qq, and rr with isomorphisms rfpr \cong f p and rgqr \cong g q. However, when 2-limits are considered as only defined up to equivalence (as is the default on the nLab), the two resulting notions of “2-pullback” are the same. In much of the 2-categorical literature, the version with rr specified would be called a bipullback and the version with rr not specified would be called a bi-iso-comma-object.

The unsimplified definition would be: a 22-pullback in a 2-category is a diagram

P p A q f B g C\array{P & \overset{p}{\to} &A \\ ^q\downarrow & \searrow & \downarrow^f\\ B& \underset{g}{\to} &C }

in which each triangle commutes up to isomorphism, and which is universal among such squares in a 2-categorical sense. This means that (1) given any other such square

Z r A s f B g C\array{Z & \overset{r}{\to} &A \\ ^s\downarrow & \searrow & \downarrow^f\\ B& \underset{g}{\to} &C }
Z v A w f B g C\array{Z & \overset{v}{\to} &A \\ ^w\downarrow & \cong & \downarrow^f\\ B& \underset{g}{\to} &C }

in which the triangles commute up to isomorphism, there exists a morphism u:ZPu\colon Z \to P and isomorphisms puvp u \cong v and quwq u \cong w which are coherent with the given ones above, and (2) given any two morphisms u,t:ZPu,t\colon Z \to P and 2-cells α:pupt\alpha\colon p u \to p t and β:quqt\beta\colon q u \to q t such that fα=gβf \alpha = g \beta (modulo the given isomorphism fpgqf p \cong g q), there exists a unique 2-cell γ:ut\gamma\colon u \to t such that pγ=αp \gamma = \alpha and qγ=βq \gamma = \beta.

Stephan: I would not write fα=gβf\alpha =g \beta since 1-cells are not composable with 2-cells.

Toby: They are, through the operation of whiskering.

Stephan: Thank you Toby. I inserted this example in horizontal composition

To see that these definitions are equivalent, we observe that both assert the representability of some 2-functor (where “representability” is understood in the 2-categorical “up-to-equivalence” sense), and that the corresponding 2-functors are equivalent.

  • In the simplified case, the functor F 1:K opCatF_1\colon K^{op}\to Cat sends an object ZZ to the category whose
    • objects are squares commuting up to isomorphism, i.e. maps v:ZAv\colon Z\to A and w:ZBw\colon Z\to B equipped with an isomorphism μ:fvgw\mu\colon f v \cong g w, and whose
    • morphisms from (v,w,μ)(v,w,\mu) to (v,w,μ)(v',w',\mu') are pairs ϕ:vv\phi\colon v\to v' and ψ:ww\psi\colon w\to w' such that μ.(fϕ)=(gψ).μ\mu' . (f \phi) = (g \psi) . \mu.
  • In the unsimplified case, the functor F 2:K opCatF_2\colon K^{op}\to Cat sends an object ZZ to the category whose
    • objects consist of maps v:ZAv\colon Z\to A, w:ZBw\colon Z\to B, and x:ZCx\colon Z\to C equipped with isomorphisms κ:fvx\kappa\colon f v \cong x and λ:xgw\lambda\colon x\cong g w, and whose
    • morphisms from (v,w,x,κ,λ)(v,w,x,\kappa,\lambda) to (v,w,x,κ,λ)(v',w',x',\kappa',\lambda') are triples ϕ:vv\phi\colon v\to v', ψ:ww\psi\colon w\to w', and χ:xx\chi\colon x\to x' such that κ.(fϕ)=χ.κ\kappa' . (f \phi) = \chi . \kappa and λ.χ=(gψ).λ\lambda' . \chi = (g \psi) . \lambda.

We have a canonical pseudonatural transformation F 2F 1F_2\to F_1 that forgets xx and sets μ=λ.κ\mu = \lambda . \kappa. This is easily seen to be an equivalence, so that any representing object for F 1F_1 is also a representing object for F 2F_2 and conversely. (Note, though, that in order to define an inverse equivalence F 1F 2F_1\to F_2 we must choose whether to define x=fvx = f v or x=gwx = g w.)

Variations

2-pullbacks can also be identified with homotopy pullbacks, when the latter are interpreted in CatCat-enriched homotopy theory.

Strict 2-pullbacks

If we are in a strict 2-category and all the coherence isomorphisms (μ\mu, κ\kappa, λ\lambda, etc.) are required to be identities, and uu in property (1) is required to be unique, then we obtain the notion of a strict 2-pullback. This is an example of a strict 2-limit. Note that since we must have x=fv=gwx = f v = g w, the two definitions above are still the same. In fact, they are now even isomorphic (and determined up to isomorphism, rather than equivalence).

In literature where “2-limit” means “strict 2-limit,” of course “2-pullback” means “strict 2-pullback.”

Obviously not every 2-pullback is a strict 2-pullback, but also not every strict 2-pullback is a 2-pullback, although the latter is true if either ff or gg is an isofibration (and in particular if either is a Grothendieck fibration). A strict 2-pullback is, in particular, an ordinary pullback in the underlying 1-category of our strict 2-category, but it has a stronger universal property than this, referring to 2-cells as well (namely, part (2) of the explicit definition).

Strict weighted limits

If the coherence isomorphisms μ\mu, κ\kappa, λ\lambda in the squares are retained, but in (1) the isomorphisms purp u \cong r and qusq u \cong s are required to be identities and uu is required to be unique, then the simplified definition becomes that of a strict iso-comma object, while the unsimplified definition becomes that of a strict pseudo-pullback. (Iso-comma objects are so named because if the isomorphisms in the squares are then replaced by mere morphisms, we obtain the notion of (strict) comma object).

Every strict iso-comma object, and every strict pseudo-pullback, is also a (non-strict) 2-pullback. In particular, if strict iso-comma objects and strict pseudo-pullbacks both exist, they are equivalent, but they are not isomorphic. (Note that their strict universal property determines them up to isomorphism, not just equivalence.) In many strict 2-categories, such as Cat, 2-pullbacks can naturally be constructed as either strict iso-comma objects or strict pseudo-pullbacks.

Lax versions

Replacing the isomorphism μ\mu in the simplified definition by a mere transformation results in a comma object, while replacing κ\kappa and λ\lambda in the unsimplified definition by mere transformations results in a lax pullback. In a (2,1)-category, any comma object or lax pullback is also a 2-pullback, but this is not true in a general 2-category. Note that comma objects are often misleadingly called lax pullbacks.

Revised on September 30, 2011 12:57:49 by Stephan (178.192.48.23)