Talk:Coproduct

Please excuse my bad ASCII art. I'll do a real pic as soon as I figure out how. -- Fropuff 06:13, 2004 Jul 10 (UTC)

(Ah, those were the days. Geometry guy 21:36, 16 May 2007 (UTC))

Canonical injections monic?

Following discussion didn't provide a solution; any other input? -- Baarslag 16:57, 2 April 2006 (UTC)

The canonical injections are not necessarily injective, so maybe the link is confusing. Are they always mono though? -- Baarslag 13:36, 1 April 2006 (UTC)

They are not necesarily mono. I'd love to give a counterexample, but I didn't come up with one. Anyone else? -lethe ^{talk +} 15:44, 1 April 2006 (UTC)

Thanks for the speedy reply. I think you are right, but what is wrong with the following proof of i_j : X_j → X being monic?

Take two arrows, say p and q, to X_j with i_j O p = i_j O q and choose f_j : X_j → Y to be the identity. There now exists a unique morphism f from X to Y such that f_j = f O i_j.

Now i_j O p = i_j O q, so

f O i_j O p = f O i_j O q, so

f_j O p = f_j O q, so

p = q. Thus, i_j is monic.

Again, it's probably faulty since I couldn't find this in any textbook. -- Baarslag 17:49, 1 April 2006 (UTC)

Well I don't know if it's faulty, but it's certainly a little fishy; you're supposed to have a collection of maps X_j → Y. You want all those maps to be the identity, which can only happen if each X_j=Y. Thus all your components must be equal. So perhaps what you've proved that if all the components are equal, then the canonical injection is mono. Lemme look at it a little more. -lethe ^{talk +} 18:17, 1 April 2006 (UTC)

Sure take your time, maybe a counterexample will pinpoint the mistake in the 'proof' above. As for some more clarification: the choice of Y and each f_j : X_j → Y is free, and the choice Y = X_j and f_j = id (for just one j) seems to do the trick. Cheers. -- Baarslag 18:56, 1 April 2006 (UTC)

Oh I see, you're only assuming that one of the f_j is an identity map, not all of them. Still, this seems like a loss of generality to me; what if no such collection exists? There is no guarantee that there are morphisms X_k → X_j. Is it so? Anyway, yes, we need a counterexample. I've got two texts that say that it is the case that the canonical injection/projection need not be monic/epic, but neither provides a counterexample. -lethe ^{talk +} 19:05, 1 April 2006 (UTC)

Right, only one. Ofcourse, the proof must be flawed somewhere as you've pointed out, but I still wonder what exactly is wrong. To be sure, my reasoning was the following: the definition states

For any object Y and any collection of morphisms f_j : X_j → Y, there exists a unique morphism f from X to Y such that f_j = f O i_j.

And the 'proof' above says that since the coproduct must handle any f_j : X_j → Y (no constraints here), it must certainly handle f_j = id, and this special case already seemed to force i_j : X_j → X to be monic. --Baarslag 21:16, 1 April 2006 (UTC)

Well like I said, assuming the existence of this family of morphisms entails a loss of generality. Yes, the universal property guarantees the equations to hold for any such family of morphisms. But you choose a particular family of morphisms which may not exist. If this family of morphisms exists, then the equations hold, and therefore the injections are monic. But it may be that there is no such family of morphisms. More explicitly, you choose Y to be equal to one of the X_j. Thus you are assuming that there exist morphisms f_k: X_k→X_j for all k. For the case k=j, we certainly know such a morphism exists, it's the identity morphism. But what about the case i≠j? How do you know you have a morphism X_k→X_j in this case? If you can find one, then the proof goes through, but there's no guarantee of such a morphism. This is the flaw in the proof, or at least, it's the only flaw that I can see. The proof will still go through in some cases, like if all the X_k are equal, but not in general. Do you see my point? Is it correct? -lethe ^{talk +} 03:44, 2 April 2006 (UTC)

Lethe, thanks for your patience with me. I think you're right. Staring at your argument, maybe some specially crafted finite category with carefully chosen arrows might provide the counterexample we are looking for. I will try the most obvious ones. -- Baarslag 16:57, 2 April 2006 (UTC)

Some googling turned up this paper which says on page 19 that in an extensive category, coproduct injections are mono. This suggests not only that the result is not true in general, but gives us a criterion for when it does hold. Now I just have to figure out what an "extensive category" is. Then it should be easy to find a counterexample. -lethe ^{talk +} 19:14, 1 April 2006 (UTC)

The paper of course has a definition of extensive categories, but no nonexamples. -lethe ^{talk +} 19:20, 1 April 2006 (UTC)

I've read the paper, and you're right, it does suggest that coprojections are not mono. Prior to my question here, I've done some googling myself and got the impression that coprojections are mono iff the coproduct is really disjoint. Maybe it rings a bell somewhere. I hope we can crack this one. -- Baarslag 21:01, 1 April 2006 (UTC)

Well I don't know what you mean by "disjoint". Does it mean for concrete categories that the underlying sets of the images of the injection functions are disjoint as sets? Does it have any meaning for generic categories? Certainly I see that for some categories, like groups or pointed spaces, the "components" of the coproduct are not disjoint, so the injections are not injective. But they're still monic, so that doesn't help, and even if it did, I don't know any category-theoretic way to characterize this notion of "disjointness". -lethe ^{talk +} 03:44, 2 April 2006 (UTC)

Oh, now I see that that paper defines "disjointness" for coproducts: the pullback of the injections is the initial object. I think both groups and pointed spaces satisfy this, even though, as sets, they are not disjoint. So OK, nevermind my above comment. -lethe ^{talk +} 04:07, 2 April 2006 (UTC)

OK, so the paper says that if the category is extensive, then its coproducts are disjoint, and its injections are monic. You suggest that perhaps monic injections and disjoint coproducts are equivalent. It sounds believable, I guess. Let's play around with the diagrams. And I'd still really like to know an example of a category that is not extensive. -lethe ^{talk +} 04:10, 2 April 2006 (UTC)

Extensiveness is a little out of my league, but I will try to sketch some finite diagrams (see comment above) today. -- Baarslag 16:57, 2 April 2006 (UTC)

If I'm not mistaken, the following is a counterexample. The coproduct of Z_m and Z_n (integers mod m and n, respectively) in the category of commutative rings is Z_g, where g = gcd(m,n). This is because the equations 1+...+1=0 (m times), which holds in Z_m, and 1+...+1=0 (n times), which holds in Z_n, together imply 1+...+1=0 (g times). Clearly the canonical injections are not injective. I don't think they are monic either by the following argument. Let Z[x] be the free commutative ring on one generator x. This is the ring of polynomials in one variable with integer coefficients. Assume m > 2g. Let g_1, g_2:Z[x] -> Z_m be the unique homomorphisms with g_1(x)=g, g_2(x)=2g. Then g_1 and g_2 are not equal, but f(g_1(x)) = f(g) = 0 and f(g_2(x)) = f(2g) = 0, so f o g_1 = f o g_2. Can someone please confirm? 67.255.11.12 (talk) 03:16, 3 January 2014 (UTC)

Possible solution

Define the following category:

Two unequal arrows p, q: D → A, v: A → C, w: B → C. And define vp = vq. C is a coproduct and v, w the canonical injections, since it's the only limiting cocone on A and B, but v is not mono by definition. Is this correct? -- Baarslag 23:17, 2 April 2006 (UTC)

Yes, that looks correct. Nice work. So I guess you just made a category such that the family of morphisms you use in the proof doesn't exist. I was thinking about looking for something like that in concrete categories. The problem is, in most concrete categories, there are lots of morphisms between most objects. The category of fields has lots of objects with no morphisms between them, but no coproducts. Ho hum. -lethe ^{talk +} 23:44, 2 April 2006 (UTC)

Sorry for putting it on top, I was afraid you'd miss it. You inspired the proof, so it's your 'nice work' as well. I'm still shaky after my forelast 'proof', so I'm not completely sure the current proof is solid. Did you triple-check it? -- Baarslag 00:19, 3 April 2006 (UTC)

The proof, with the assumption of the existence of those morphisms, looks correct to me. I guess this provides a way of understanding why the canonical injections usually are monic. In most categories, we can find such morphisms. Any category with zero morphisms, for example. -lethe ^{talk +} 01:19, 3 April 2006 (UTC)

Dashes arrow

Why is the arrow corresponding to f dashed in the second diagram? Is this common notation? Randomblue (talk) 16:01, 30 September 2008 (UTC)

It should be read as "exists unique f such that the diagram commutes". The notation is famously used by Mac Lane (Categories for the Working Mathematician, Springer, 1971), but probably pre-dates this. 21:47, 4 February 2009 (UTC)

Dual/Opposite !?

I'm having some trouble understanding the Hom_C duality/opposite discussion. I'm trying to reconcile it with a statement from a book on cohomology (R. Bott, differential forms in algebraic topology, page 46) when talking about the non-finite case: "... comes from the fact that the dual of a direct sum is a direct product, but the dual of a direct product is not a direct sum." I can't figure out how to reconcile this statement with what this article states, and thus would like to see a worked example of this. FWIW, since this is cohomology, its about Abelian groups/vector spaces. linas (talk) 15:16, 30 November 2011 (UTC)

Sometimes the dual refers to images of a contravariant functor like Hⁿ, so it may just be the context. ᛭ LokiClock (talk) 19:44, 9 August 2012 (UTC)

"A category with all finite biproducts is known as an additive category."

Doesn't a category need to be enriched in Ab as well as having finite biproducts to be additive? Not clear on that--Daviddwd (talk) 22:47, 27 August 2014 (UTC)

Adjoint?

The adjoint to the product (cartesian product or tensor product) is hom, or the exponential object in the category of sets. See the article on currying for a lurid exposition of this adjunction. Now, the coproduct is opposite of the product. Is there an adjoint, and what can I say about it, either in some small category or in general?

I mean, if you start talking about additive categories and cohomology, then you get lead to tensor-hom adjunction and ext and tor functors, where you can say some abstract stuff. But it seems very domain-specific to cohomology; what can one say, more generally, that makes sense, and is still somehow useful? 67.198.37.16 (talk) 07:13, 22 July 2017 (UTC)

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Improvements required...

This article needs interpretation of the coproduct notation as it appears in 'summation' article for summation sequences. Please Atlantis President (talk) 12:29, 14 March 2019 (UTC)

Existence, or as data?

The definition given on this page, and on the page for products, defines a product and a coproduct as being specific objects for which there exist projection/coprojection maps respectively, with universal properties of factorisation through the product/coproduct respectively. Other sources ask that the projection/coprojection maps are actually part of the data of the product/coproduct. Perhaps one could keep it how it is currently stated, and treat the other perspective about taking product/coproduct objects along with witnesses of their product/coproduct structure? — Preceding unsigned comment added by 128.250.0.120 (talk) 22:21, 15 April 2023 (UTC)

Only one reference?

Is Yuan's work truly enough for this article? Mousefountain (talk) 02:10, 4 January 2024 (UTC)