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I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck.

It is better to have a good category with bad objects than a bad category with good objects.

Question: Does anyone know the source of this quote, or at least when it first appears, or when it was first attributed to Grothendieck?

I wish I were able to properly cite this popular and insightful quote.

Added: On a related topic, another quote I wish I could cite properly, usually attributed to Manin, reads:

Proofs are more important than theorems, definitions are more important than proofs.

Does anyone know a source for this quote?

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    $\begingroup$ What is a bad category? $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Jun 7, 2013 at 2:25
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    $\begingroup$ @Mahdi A category which doesn't possess objects or morphisms which you need in order to make desirable structural arguments work. For instance, depending on which argument you wish to make, a <i>bad</i> category might not be complete, or cocomplete, or cartesian closed, or whatever. $\endgroup$
    – Daniel Moskovich
    Commented Jun 7, 2013 at 2:39
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    $\begingroup$ The category of topological spaces has lots of good objects but is a bad category because it does not have exponentials. $\endgroup$
    – Andrej Bauer
    Commented Jun 7, 2013 at 13:57
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    $\begingroup$ The category of smooth manifolds and smooth maps is another example of a bad category with good objects. $\endgroup$
    – Andrej Bauer
    Commented Jun 7, 2013 at 13:57
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    $\begingroup$ @Owen: Isn't the category of von Neumann Algebras and completely positive maps some kind of poor man's approximation of the bicategory of von Neumann Algebras and bimodules? $\endgroup$
    – André Henriques
    Commented Jun 8, 2013 at 21:52

4 Answers 4

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In response to your second question, this is not precisely what you're looking for, but here is one quote by Yuri Manin along the same lines:

All the other vehicles of mathematical rigor are secondary [to definitions], even that of rigorous proof.

Manin makes this remark in an essay, entitled "Interrelations between Mathematics and Physics" that contains more memorable phrases, such as the final line of this extended quote:

All the other vehicles of mathematical rigor are secondary [to definitions], even that of rigorous proof. In fact, barring direct mistakes, the most crucial difficulty with checking a proof lies usually in the insufficiency of definitions (or lack thereof). In plain words, we are more deeply troubled when we wonder what the author wants to say than when we do not quite see whether what he or she is saying is correct. The flaws in the argument in a strictly defined environment are quite detectable. Good mathematics might well be written down at a stage when proofs are incomplete or missing, but informed guesses can already form a fascinating system: outstanding instances are A. Weil’s conjectures and Langlands’s program, but there are many examples on a lesser scale.

To the contrary, an inexperienced reader of the most interesting physical papers is often left in a vacuum about the precise meaning of the most common terms. Physicists are undoubtedly constrained by their own rules, but these rules are not ours. What is a current algebra, a supersymmetry transformation, a topological field theory, a path integral, finally? They are very open concepts, and it is precisely their openness that makes them so interesting. Here is what the history of our two metiers teaches: we cannot live without each other. At least for some of us, life becomes dull if it goes on for too long without contacts with good physics. In this century romantics comes from physics. Mathematics supplies hygienic habits and headaches.

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    $\begingroup$ This is an interesting quote. Yet the idea that definitions are the most important "vehicles" of mathematical rigor is very old, and has been perfectly expressed already by Pascal in "l'art de persuader". A more original idea is the one of Grothendieck, that definition may have an even higher role in mathematics, as vehicle not only of rigor, but of mathematical creativity itself. That is, the creative work of a mathematicians is not mainly to prove good theorems, it is to "invent" good definitions. His works are a sufficient illustration of this new maxim. $\endgroup$
    – Joël
    Commented Jun 8, 2013 at 15:30
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I would be surprised if the purported Grothendieck quote is really his. He does not lean to the short and sweet. It sounds more like an adaptation of another thing Deligne says in "Quelques idées maitresses de l'oeuvre de Grothendieck" (p. 13): "if the decision to let every commutative ring define a scheme gives standing to bizarre schemes, allowing it gives a category of schemes with nice properties."

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Serre writes

... comme Grothendieck nous l’a appris, les objets d’une catégorie ne jouent pas un grand rôle, ce sont les morphismes qui sont essentiels.

See page 335 in: J-P. Serre, Motifs, Journ. arithm. Luminy 1989 (Ed. G. Lachaud). Asterisque 198-200, Soc. Math. France, 1991, pp. 333-349. Although this quote is from 1991, of course it refers to the 50s and 60s.

Grothendieck's "relative way" of doing mathematics was apparent in most of his work. The first famous example is his generalization of Riemann-Roch.

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    $\begingroup$ This is a great quote- thanks! But it sounds to me like Grothendieck is making a different point here, about how to judge a category, rather than about relative value of "good category" vs. "good objects" or even "good morphisms". $\endgroup$
    – Daniel Moskovich
    Commented Jun 18, 2013 at 3:47
  • $\begingroup$ For me this quote says that a good category has good morphisms, and the objects may be bad. A typical example is the category of schemes with morphisms of finite type. A typical non-example is the category of noetherian schemes with all morphisms of schemes. $\endgroup$
    – Martin Brandenburg
    Commented Jun 18, 2013 at 5:47
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Another quote in the spirit of that given by Martin Brandenburg.

In "Quelques idées maitresses de l'oeuvre de Grothendieck", Deligne writes

On reconnait la patte du Maitre dans l'idee que le probleme n'est pas de definir ce qu'est un motif: le probleme est de definir la categorie des motifs, et de degager les structures qu’elle porte.

which can be translated into

We recognize the touch of the Master in the idea that the problem is not to define what a motive is: the problem is to define the category of motives and to identify its structures.

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