# What is the standard terminology regarding the arrows and objects involved in pullbacks?

In Awodey's Category Theory (2006) he discusses the pullback of arrows $f:A \rightarrow C$, $g: B \rightarrow C$ (in a category $\mathbf{C}$) as consisting of arrows $p_1: P \rightarrow A$, $p_2: P \rightarrow B$ such that $f \circ p_1 = g \circ p_2$ and this property is universal (p. 92). So it seems that a pullback has three parts, the object $P$ (which Awodey doesn't put a lot of emphasis on), and the two arrows $p_1,p_2$.

On nLab's page on pullbacks, they refer to the object $P$ as the pullback, which makes the arrows $p_1$ and $p_2$ seem almost incidental, whereas for Awodey they seemed central. (I know the arrows aren't merely "incidental", the point is a matter of emphasis.) But then I don't know what to call the arrows involved ($p_1$ and $p_2$).

Given a setup like the one above, what are some standard ways of talking about $P$, $p_1$, and $p_2$, especially expressions that indicate their relationship to the objects $A$, $B$, and $C$ and arrows $f$ and $g$?

• It is quite common to abuse the notation and refer to $P$ as the pullback, even though the arrows $p_1$, $p_2$ are necessary. People often refer to $p_1$ and $p_2$ as the structural maps of the pullback. The same terminology is also used for general limits and colimits. – Goa'uld May 13 '17 at 22:50
• Call them the pullback object and canoncial projections. – Berci May 13 '17 at 23:14
• @Berci If you wanna post that as an answer, ideally (though not necessarily) with a reference exhibiting that usage I'd be happy to accept it so that this question can fall off of the "unanswered" queue. – Dennis May 13 '17 at 23:27
• Indeed, canonical (morphisms/projections/embeddings) ist the, well, canonical expression used for those morphisms that "belong to" kernels, cokernsle,products, coproducts, limits, colimits, everything defined via universal properties, i.e., as universal objects in another category whose objects are objects of the given category together with certain arrows. Thus for $f\colon A\to B$, we have the canonical (ususally inclusion) homomorphism of the kernel object $\ker f$, or the canonical epimorphism to the cokernel object, or the canonical projection of the product to the individual components – Hagen von Eitzen May 14 '17 at 8:27

## 1 Answer

Let $I$ be a small category; in the case of pullbacks, take $I = (\bullet \rightarrow \bullet \leftarrow \bullet)$. Let $X : I \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$ of shape $I$. Then a limit of $X$ is a final object in the category of cones to $X$ . Thus, it consists of a universal cone, sometimes called the limit cone (or limiting cone) $(p_i : L \to X(i))_{i \in I}$ such that for every cone $(q_i : T \to X(i))_{i \in I}$ there is a unique morphism $f : T \to L$ such that $p_i f = q_i$. It follows that every two limit cones are uniquely isomorphic (in the category of cones, of course), so that one may speak of the limit cone*. The morphisms $p_i : L \to X_i$ are called projections.

The object $L$, considered separately, has no significance, and it does not really make sense to call it a limit of $X$. Quite often mathematicians are just lazy and say "Let $L$ be a limit of $X$" ("with projections $p_i : L \to X(i)$") when they actually mean "Let $(p_i : L \to X(i))_{i \in I}$ be a limit cone" (which is, actually, shorter!). Nevertheless, Mac Lane calls $L$ a limit object and uses, as many others, the notation $L = \varprojlim X$. So in the case of pullbacks, this would be pullback object.

Also notice that $L$ is not unique up to unique isomorphism, so that it is not really unique in a categorical sense. In order to formulate unicity, we really have to remember the projections.

*It is very common to identify objects which are uniquely isomorphic. For example, one speaks of the trivial group, although (in ZFC) for every set $X$ we can endow the set $\{X\}$ with the trivial group structure $X \cdot X = X$. Thus there are as many trivial groups as sets. But they are all isomorphic in a unique way, so that there is no harm to identify them. More generally, one can identify all final objects of a category; and a limit cone of $X$ is just a final object in the category of cones to $X$.