Talk:Coproduct

message: 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)

message: 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)
replies: text: 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)
replies: text:  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)
replies: text: 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)
thread_title:  Possible solution 

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

message: 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)
replies: text: 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)
thread_title:  Dual/Opposite !? 

message: 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)
thread_title:  "A category with all finite biproducts is known as an additive category."  

message: 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)
thread_title:  Adjoint? 

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thread_title:  External links modified 

message: 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)
thread_title:  Improvements required... 

message: 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)
thread_title:  Existence, or as data? 

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