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for completeness, I created a stub for direct product group. In the course of this I noticed that at Cartesian product no other example than products of sets were mentioned. So I added pointer to direct product group and to product topological space. Much more should be added.
Much more should be added to what? Examples of cartesian products in various categories could be added ad infinitum (Victor Porton had suggested doing something like this once).
The classical examples that you would tell your students about when teaching them the abstract concept of Cartesian product should be added in a list of Examples at Cartesian product. Much like for instance the examples of group objects etc.
The only thing I’d worry about here is adding too many examples of an essentially obvious nature. Some of the examples at group object are good in that they are not “boringly obvious”, and the list doesn’t seem overly long.
If it were me, I’d give maybe just a few examples of an algebraic nature (groups and rings say) before giving the general principle that applies to any concept describable by a Lawvere theory.
Similarly, one or two examples of a topological nature (preorders, topological spaces) should hopefully be enough before giving the general principle that applies to topological categories, or reflective subcategories thereof.
More exotic examples that might be of interest (or at least not boringly obvious) include products of affine schemes, or of cocommutative comonoids, or of locales, or of compactly generated spaces.
Infinite products sometimes have subtleties that are illustrative.
Such as the difference between infinite direct sum and infinite direct products in general, when in the finite case these agree.
Sure.
I’ve never come across a situation where less examples were preferable to more examples when explaining a topic.
Why not just give one example for each major category?
I, for example, would find it personally instructive to compare and contrast how topology views the product versus how logic views a product… or how a group views a product vs. a poset….etc.
There is some bckground story here which not all of you may be aware of. What Todd is objecting to, echoing the memory of a user who wanted to do just that, is the addition of pointless random examples.
Compare to, say, an entry on analytic functions: there are some standard examples of functions already known otherwise, which are usefully mentioned as examples of analytic functions. But you don’t want to list any random power series as an example. Todd is cautioning not to do the analog of that in category-theoretic entries.
On the other hand, until yesterday the entry on Cartesian products named no example besides the archetypical one in Set. This is clearly just as bad an extreme. In #1 I was just suggesting to add more of the useful examples.
And now I declare that we all have wasted enough keystrokes here on talking about what might be added. Everyone considering writing a followup to this thread here please consider using that energy instead to just go ahead and make a useful addition to one of the nLab-entries!
I only advocate canonical examples, not an exhaustive list as it were… sets, groups, hausdorff, logic. I’m sure I don’t have to sell this crowd on how useful it is to view a product from as many different viewpoints as possible.
I just added a slew of examples to cartesian product.
I can easily think of lots of situations in which fewer examples are better when explaining a topic. Once you’ve illustrated a general point with one or two examples of relevance to it, adding more and more examples of the “same sort” just takes up space.
Todd, that’s just excellent. Thank you!
I have merged now cartesian product with product (which also existed and also was lacking substance) and edited a little more.
With regard to the general discussion on examples, I’ve heard a general criticism of the nLab, that there are (often) not enough examples. Qiaochu Yuan has said this for instance. He’s probably right.
But as others have said, obviously it’s possible to overdo examples.
I’ve always liked this quote from the introduction to a book by Gerald Sacks (though obviously it is a bit tongue-in-cheek):
It is no accident that the book suffers from a shortage of examples. As a rule examples are presented by authors in the hope of clarifying universal concepts, but all examples of the universal, since they must of necessity be particular and so partake of the individual, are misleading.
The Tao that can be spoken is not the eternal Tao. The name that can be named is not the eternal name.
Tao Te Ching Chap. 1
Good artists copy; great artists steal
Picasso
Category theorists are dual to ordinary people: they often get more confused when you surround an abstract concept with a lot of distacting specifics. “Could you please not give me an example, to help me understand what you’re saying?” :-)
Kudos! Great quote, Mike!
added this:
Let $G_1, G_2$ be two groups. Then every irreducible representation $\rho \in (G_1 \times G_2) Rep_{irr}$ of their direct product group $G_1 \times G_2$ is the external tensor product of irreducible representations $\rho_i \in G_i Rep_{irr}$ of the two groups separately:
$\rho \;=\; \rho_1 \boxtimes \rho_2 \,.$Here the external tensor product has as underlying vector space the corresponding tensor product of vector spaces, equipped with the evident action
$(g_1, g_2)( (v_1 \otimes v_2) ) \;=\; ( g_1(v_1) \otimes g_2(v_2) ) \,.$By Schur’s lemma see e.g. here.
The converse is false in general. The external tensor product of irreducible representations need not be irreducible itself. For more see Fein 67.
ah, no, there is still a secret assumption of algebraic closure there. It’s fixed in a later version of the same lecture notes (Example 2.7.31 here: pdf). Fixing it now in the entry…
added sentence cross-linking with free product of groups
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