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 $q\gamma =\beta$.

Stephan: I would not write $f\alpha =g\beta$ 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 $F_1:K^{\mathrm{op}}\to \mathrm{Cat}$ 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.