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.

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

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

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

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

which commutes up to isomorphism, there exists a morphism $u:Z\to P$ and isomorphisms $p u \cong v$ and $q u \cong w$ which are coherent with the given ones above, and (2) given any two morphisms $u,t:Z\to P$ and 2-cells $\alpha:p u \to p t$ and $\beta:q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-cell $\gamma:u\to t$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.

The simplification in the above explicit definition has to do with the omission of an unnecessary structure map. Note that an ordinary pullback of $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ comes equipped with maps $P\overset{p}{\to} A$, $P\overset{q}{\to} B$, and $P\overset{r}{\to} C$, but since $r = f p$ and $r = g q$, 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 $f p \cong g q$, than to give $p$, $q$, and $r$ with isomorphisms $r \cong f p$ and $r \cong g q$. 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 $r$ 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

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

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

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

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

in which the triangles commute up to isomorphism, there exists a morphism $u\colon Z \to P$ and isomorphisms $p u \cong v$ and $q u \cong w$ which are coherent with the given ones above, and (2) given any two morphisms $u,t\colon Z \to P$ and 2-cells $\alpha\colon p u \to p t$ and $\beta\colon q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-cell $\gamma\colon u \to t$ 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 $F_1\colon K^{op}\to Cat$ sends an object $Z$ to the category whose
- objects are squares commuting up to isomorphism, i.e. maps $v\colon Z\to A$ and $w\colon Z\to B$ equipped with an isomorphism $\mu\colon f v \cong g w$, and whose
- morphisms from $(v,w,\mu)$ to $(v',w',\mu')$ are pairs $\phi\colon v\to v'$ and $\psi\colon w\to w'$ such that $\mu' . (f \phi) = (g \psi) . \mu$.

- In the unsimplified case, the functor $F_2\colon K^{op}\to Cat$ sends an object $Z$ to the category whose
- objects consist of maps $v\colon Z\to A$, $w\colon Z\to B$, and $x\colon Z\to C$ equipped with isomorphisms $\kappa\colon f v \cong x$ and $\lambda\colon x\cong g w$, and whose
- morphisms from $(v,w,x,\kappa,\lambda)$ to $(v',w',x',\kappa',\lambda')$ are triples $\phi\colon v\to v'$, $\psi\colon w\to w'$, and $\chi\colon x\to x'$ such that $\kappa' . (f \phi) = \chi . \kappa$ and $\lambda' . \chi = (g \psi) . \lambda$.

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 $x = f v$ or $x = g w$.)

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

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 $x = f v = g w$, 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 $f$ 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).

If the coherence isomorphisms $\mu$, $\kappa$, $\lambda$ in the squares are retained, but in (1) the isomorphisms $p u \cong r$ and $q u \cong s$ 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.

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.