Is the sentence "the diagram is a pullback" wrong?

Question

In the context of category theory I've seen both sentences "the diagram is a pullback" and "the diagram is pullback" being used in the literature to talk about pullback squares. The former sounds better to me, however, I have convinced myself that the latter may be the gramatically correct one, while the first is not.

The argument is the following. When talking about diagrams, the word "pullback" attaches to it to form "pullback diagram" like an adjective, so it would make sense if it were treated as one when describing a diagram. Also, the pullback itself is a limit of another diagram formed by two morphisms, rather than the whole diagram that includes those morphisms, so the diagram can't be said to be a pullback.

So, my question is: which sentence is correct? Or are both usable in the literature?

Answer 1

Generally speaking, the nouns "pullback", "pullback diagram", and "pullback square" are all synonyms. I have never previously seen "pullback" used as an adjective. I can find only two papers (with a shared author) who use the phrase "the diagram is pullback" (and they define the term as an adjective in their paper).

In short, no, "the diagram is a pullback" is correct according to common usage, and "the diagram is pullback" is exceptionally rare (and sounds odd to my ears).

Answer 2

As to your linguistic argument, building on Noah Schweber’s examples, one would say none of these things:

• “This is a dog park. This park is dog.”
• “This is a grain elevator. This elevator is grain.”
• “This is a duck expert. This expert is duck.”

So in English, as opposed e.g. to German, that a word can modify a noun attributively by preceding it doesn’t imply that it’s an adjective that can be used predicatively.

There’s a tendency in math to “predicatize” proper names used as nominal attributes:

• “This is a Hausdorff space. This space is Hausdorff.”
• “This is a Cauchy sequence. This sequence is Cauchy.”

But I can’t think of a case where this is done with nouns other than proper names. (I’d be interested to hear if someone can come up with one.)

As to your mathematical argument, you’re right that “This diagram is a pullback” is strictly speaking not correct because the pullback is a) not a diagram and b) a limit of a different diagram, so strictly speaking it should be something like “This diagram shows a pullback”, but this sort of abuse of language is quite widespread and rarely causes any harm because there’s little potential for misunderstanding. Some authors might include a disclaimer like “If a diagram exhibits a pullback, we will sometimes also refer to the diagram itself as a pullback.”

Even apart from linguistic issues of attributivity and predicativity, this imprecision wouldn’t be resolved by saying “This diagram is pullback” unless you first define “pullback diagram” to refer to this sort of diagram, i.e. a pullback diagram would have to be defined to refer to a different diagram than the one whose limit the pullback is, which seems somewhat confusing. Perhaps moderate imprecision is preferable to this confusing precision.