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This is a rather vague question, but perhaps we can talk about it.

There are two types of mathematical objects (which don't exclude each other):

A) There is a good description of morphisms defined on this object.
B) There is a good description of morphisms defined into this object.

Thus A) means that the covariant hom-functor is understood, and B) means that the contravariant hom-functor is understood. This applies most notably to universal objects. Within category theory, the concepts are just dual to each other and so the "theory" of A) is essentially the same as the theory of B). But most categories studied in mathematics don't come together with their dual, so that this categorical argument is not really good. In fact, I have the feeling that in 'daily mathematics', A) appears much more often than B). And that it is easier to work with them. Of course, we could argue about that. For example, I have a better feeling with colimits than with limits. [perhaps I will add examples here]

If you have the same feeling: Can we give reasons for this?

I think that the basic principle of gluing, which appears in many geometric categories, always belongs to A). This could be a reason. What do you think?

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closed as not a real question by S. Carnahan, Mariano Suárez-Alvarez, Qiaochu Yuan, David Speyer, Loop Space Jan 19 '10 at 20:48

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

It's borderline. I think as written it's more appropriate for a blog (and in fact I remember the n-category cafe having a discussion like this). – Qiaochu Yuan Jan 19 '10 at 18:14
It's an interesting question, but I think it would do better at a blog or forum than here. Can someone find the n-cafe link Qiaochu mentions? Those folks don't seem to mind restarting old conversations. – David Speyer Jan 19 '10 at 18:21
Sorry, I'm voting to close. – S. Carnahan Jan 19 '10 at 18:28
I think that this is an interesting question. There is a famous remark, attributed to Deligne, that "all problems in mathematics are psychological". I think that this remark has a very strong grain of truth to it, and I know that I am not alone in doing so. In my view, reflecting on questions like Martin's is one way to build up psychological strength. And, with all due respect to the n-category cafe, he didn't solicit the view of the participants in that forum, but rather, of the participants on this site. – Emerton Jan 19 '10 at 18:47
@Emerton: I agree with all of your points, but they don't seem to support the claim that this site is a good place for this question. My personal opinion is that MathOverflow would be better without moving in the direction of psychological musing. That is not to say that I dislike questions about monads/comonads, limits/colimits and so on. – S. Carnahan Jan 19 '10 at 20:29

Dear Martin,

As Harry points out in his comments, in certain settings (e.g. moduli spaces) an object is characterized by maps in. In others (e.g. the free abelian group on one generator), the characterization is by maps out.

Certainly in algebra, injective objects (characterized by maps in) are typically regarded as more mysterious and black-box like than projectives (where one can typically think of free modules, which are quite concrete). I know several situations in which someone made real progress by judicious use of injective objects, and I'm sure part of the obstruction to previous researchers was just that injectives are not as familiar; in short, there is probably arbitrage to be gained for some (myself, at least) by learning more about injectives in various contexts, and trying to use them as fluently as one uses free objects.

In topology and geometry, perhaps there is more fluidity between the two characterizations. E.g. maps into the circle make it the Eilenberg-Maclane space $K({\mathbb Z},1)$, while maps out define the fundamental group.

You are correct that quotienting by an equivalence relation (gluing) is related to maps out. Perhaps this is one reason why the construction of moduli spaces (e.g. Picard schemes) can be quite involved; they are characterized by maps in, but are often constructed by a gluing procedure, which creates conflict; thus one finds oneself working locally, and is led into sheaf/stack-theoretic issues.

Certainly, the tension between the two characterizations has been a fertile source for good mathematics.

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I'm not sure that your remark addresses all the complications. E.g., it is true that if $C$ is a smooth projective curve over a field $k$, then $Sym^n C$ is the Hilbert scheme of degree $n$ effective divisors on $C$, but I've never found it completely trivial to prove that the symmetric power satisfies the requisite functorial properties. Another e.g.: if $C$ has genus $2$ then blowing down the graph of the hyperelliptic involution in $Sym^2 C$ gives the $Pic^0$ of $C$. I've never found this completely trivial either. Of course, I'm just speaking of my own understanding of these matters. – Emerton Jan 19 '10 at 19:29

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