nLab
2-pullback

An ordinary pullback is a limit over a diagram of the form $A\to C\leftarrow B$. 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 $2$-pullback in a 2-category is a square

$$\begin{array}{ccc}P& \stackrel{p}{\to }& A\\ {}^q\downarrow & \cong & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\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

$$\begin{array}{ccc}Z& \stackrel{v}{\to }& A\\ {}^w\downarrow & \cong & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\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 $u:Z\to P$ and isomorphisms $pu\cong v$ and $qu\cong w$ which are coherent with the given ones above, and (2) given any two morphisms $u,t:Z\to P$ and 2-cells $\alpha :pu\to pt$ and $\beta :qu\to qt$ such that $f\alpha =g\beta$ (modulo the given isomorphism $fp\cong gq$), there exists a unique 2-cell $\gamma :u\to t$ such that $p\gamma =\alpha$ and $q\gamma =\beta$.

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 $A\stackrel{f}{\to }C\stackrel{g}{\leftarrow }B$ comes equipped with maps $P\stackrel{p}{\to }A$, $P\stackrel{q}{\to }B$, and $P\stackrel{r}{\to }C$, but since $r=fp$ and $r=gq$, the map $r$ 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 $p$ and $q$ with an isomorphism $fp\cong gq$, than to give $p$, $q$, and $r$ with isomorphisms $r\cong fp$ and $r\cong gq$. 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 $r$ specified would be called a bipullback and the version with $r$ not specified would be called a bi-iso-comma-object.

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

$$\begin{array}{ccc}P& \stackrel{p}{\to }& A\\ {}^q\downarrow & \searrow & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\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

$$\begin{array}{ccc}Z& \stackrel{r}{\to }& A\\ {}^s\downarrow & \searrow & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\array{Z & \overset{r}{\to} &A \\ ^s\downarrow & \searrow & \downarrow^f\\ B& \underset{g}{\to} &C }

$$\begin{array}{ccc}Z& \stackrel{v}{\to }& A\\ {}^w\downarrow & \cong & {\downarrow }^f\\ B& \underset{g}{\to }& C\end{array}$$\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 $u:Z\to P$ and isomorphisms $pu\cong v$ and $qu\cong w$ which are coherent with the given ones above, and (2) given any two morphisms $u,t:Z\to P$ and 2-cells $\alpha :pu\to pt$ and $\beta :qu\to qt$ such that $f\alpha =g\beta$ (modulo the given isomorphism $fp\cong gq$), there exists a unique 2-cell $\gamma :u\to t$ such that $p\gamma =\alpha$ and $q\gamma =\beta$.

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 $F_1:K^{\mathrm{op}}\to \mathrm{Cat}$ sends an object $Z$ to the category whose

  • objects are squares commuting up to isomorphism, i.e. maps $v:Z\to A$ and $w:Z\to B$ equipped with an isomorphism $\mu :fv\cong gw$, and whose
  • morphisms from $(v,w,\mu )$ to $(v\prime ,w\prime ,\mu \prime )$ are pairs $\varphi :v\to v\prime$ and $\psi :w\to w\prime$ such that $\mu \prime .(f\varphi )=(g\psi ).\mu$.

• In the unsimplified case, the functor $F_2:K^{\mathrm{op}}\to \mathrm{Cat}$ sends an object $Z$ to the category whose

  • objects consist of maps $v:Z\to A$, $w:Z\to B$, and $x:Z\to C$ equipped with isomorphisms $\kappa :fv\cong x$ and $\lambda :x\cong gw$, and whose
  • morphisms from $(v,w,x,\kappa ,\lambda )$ to $(v\prime ,w\prime ,x\prime ,\kappa \prime ,\lambda \prime )$ are triples $\varphi :v\to v\prime$, $\psi :w\to w\prime$, and $\chi :x\to x\prime$ such that $\kappa \prime .(f\varphi )=\chi .\kappa$ and $\lambda \prime .\chi =(g\psi ).\lambda$.

We have a canonical pseudonatural transformation $F_2\to F_1$ that forgets $x$ and sets $\mu =\lambda .\kappa$. This is easily seen to be an equivalence, so that any representing object for $F_1$ is also a representing object for $F_2$ and conversely. (Note, though, that in order to define an inverse equivalence $F_1\to F_2$ we must choose whether to define $x=fv$ or $x=gw$.)

Variations

2-pullbacks can also be identified with homotopy pullbacks, when the latter are interpreted in $\mathrm{Cat}$-enriched homotopy theory.

Strict 2-pullbacks

If we are in a strict 2-category and all the coherence isomorphisms ($\mu$, $\kappa$, $\lambda$, etc.) are required to be identities, and $u$ 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 $x=fv=gw$, 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 $f$ or $g$ 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 $\mu$, $\kappa$, $\lambda$ in the squares are retained, but in (1) the isomorphisms $pu\cong r$ and $qu\cong s$ are required to be identities and $u$ 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 $\mu$ in the simplified definition by a mere transformation results in a comma object, while replacing $\kappa$ and $\lambda$ 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.

Discussion

A previous version of this entry prompted the following discussion.

Zoran: I disagree with a second part of the sentence. If it were a 2-limit of THAT diagram strictly speaking we would have from it an arrow to $C$ (which can be skipped in 1-categorical situation as it is superfluous) and several 2-cells in the story. So there is some confusion between sisters like comma objects, 2-pullbacks and alike.

Toby: It seems to me that (without loss of generality) you can take the arrow to $C$ to be (following the picture below) the composite $P\stackrel{p}{\to }A\stackrel{f}{\to }C$ (or the composite $P\stackrel{q}{\to }B\stackrel{g}{\to }C$ if you prefer). But identifying comma objects with lax pullbacks may be trickier.

Mike Shulman: I agree with Toby. The default sense of “2-limit” on the nLab is up to isomorphism everywhere, i.e. what other people call a “bilimit”. In this sense, it is true that a 2-pullback is a 2-limit of a simple cospan; the distinction between iso-comma objects and pseudopullbacks disappears in the world of bilimits, where the limit is only characterized up to equivalence.

Comma objects, however, are never the same as lax pullbacks, except of course in a (2,1)-category.

Zoran: giving in some version of 2-categories the same vertex (of 2-limit) or not, it is principal difference that it is not in the definition of 2-cones of such diagrams to force that the arrow to $C$ is the same as the composition $P\to A\to C$ as Toby states. The arrow to $C$ if it were the 2-limit to that diagram would disagree with $P\to A\to C$ by a 2-cell. Thus the arrow $P\to C$ is a separate datum to include in that case. So the definitions are different. Now, depending on weather we have pseudo, lax, colax, or bilimit this may have or may have not repercussions on the outcome for the vertex of the 2-limit, but this is less important.

Toby: Please note my ‘without loss of generality’. The two definitions (the simplified one below, and the general limit-based one) are equivalent; specifically, each (universal) limit cone is uniquely isomorphic to the one in which the arrow to $C$ is taken to be $P\to A\to C$.