Talk:Coalgebra

Illustration
The illustration here definitely needs work. As it appears on my browser, all of the blank part of the 8 & 1/2 by 11-inch page below the diagram appears in the article, as does the page number, "1", at the bottom. Maybe I'll get to doing something about it, but if anyone out there can fix it quickly before I get to it, please do. 134.84.86.74 14:36, 14 Oct 2003 (UTC)

This definition of coalgebra is much too narrow for coalgebras as they appear in category theory and computer science. See, for instance, Jacobs and Rutten's "A Tutorial on (Co)Algebras and (Co)Induction".

Coalgebras are formally dual to algebras for a functor. It would be swell if someone included this much broader definition on this page (but I cannot volunteer today. Maybe some later day.).

These are called F-Coalgebras here. A bit strange that the more general concept is named with a longer name. --Tillmo 10:31, 12 March 2006 (UTC)


 * I would not really call F-coalgebras a generalization of coalgebra. Of course they are, but not the correct one. It is like saying a vector space is a generalization of a coalgebra. Of course it is, and one can say that, but it is not very useful. According, people interested in coalgebras are not interested in F-coalgebras. It is the coassociativity and counit conditions that make a coalgebra interesting. The proper categorical generalization of coalgebra is to consider coalgebras in categories besides the category of vector spaces. The choice of terminology for F-coalgebra is a bit sad, much better to say F-comodule, but it is too late for that now sadly. 130.54.16.83 (talk) 05:39, 19 August 2009 (UTC)

What is a "cogebra" and where, if anywhere, has this term been used? 219.117.195.84 (talk) 02:54, 22 May 2009 (UTC)


 * I believe "cogebra" means the same thing as "coalgebra", or possibly something very similar, namely a comonoid in some symmetric monoidal category other than Vect.  It's not terribly common.  John Baez (talk) 03:35, 1 January 2010 (UTC)

Where's the Definition of Cocommutative?
The word cocommutative redirects here. This article uses the word cocommutative, but doesn't define it! Someone fix this, please! John Baez (talk) 03:35, 1 January 2010 (UTC)

cofree?
Would be great if someone can include cofree coalgebras. Thanks! --345Kai (talk) 18:23, 30 April 2010 (UTC)

Could use a contrasting definition
There must be a close relationship between algebras and coalgebras, since the names are similar. Can someone knowledgeable add a clear definition that shows that relationship? For me, "reversing all arrows" doesn't work. I don't see any arrows. How about examples of tiny groups and fields, showing how to change them to cogroups and cofields? For me, that would make this clear, I think. Oh, and some motivation? What can you do with a coalgebra that you can't do just with its algebra alone? David Spector (talk) 12:26, 7 February 2012 (UTC)


 * An arrow is just a morphism. Reversing it just means making it point the other way. Of course, you have to convince yourself that you have, in your posession, some thing for which reversed arrows can be defined (i.e. that reversing them can 'make sense' in some way). Sometimes they can, sometimes they can't. The general setting for all this is category theory, which is simultaneously easy and hard; it can be mind-bending because the easy parts make the hard parts all the more confusing. It is often the case in category theory that arrows can be reversed. It is often the case that some things are easier for the co-case (i.e. reversed), whereas the forward case doesn't exist or doesn't make sense. Perhaps one good example is the dagger compact category which is a finite dimensioinal hilber space for all practical purposes. The usual linear-algebra notions of a "basis of a vector space" becomes a coalgebra when working with that category. Most of this article is really just about ordinary linear algebra, but in an unfamiliar lanuage. User:Linas (talk) 04:25, 27 November 2013 (UTC)

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Informal discussion
Having an informal section before the formal definition is a very good idea for this topic. However the present informal section is likely only easy if you have a rather specific set of prerequisites (quantum mechanics, representation theory), so it could do with a rewrite.

A more elementary approach could be structured along the points:
 * A multiplication operation may often be though of as combining its operands. Conversely a coproduct can be though of as giving you the ways in which an object can be taken apart. (I believe this idea is due to Gian-Carlo Rota.)
 * Because there are in general several ways of taking something apart, one cannot define a coproduct without some sort of additive structure — unlike multiplication, which can be the only operation of e.g. a semigroup.
 * One situation where coalgebras often are implicit is when you have a formula for applying an operation to a composite object, that describes how to do that by applying simpler operations to its pieces. The present "informal" example of the spins is of this type, as is the dual of complex multiplication:
 * $$ \Delta(\mathrm{Re}) = \mathrm{Re} \otimes \mathrm{Re} - \mathrm{Im} \otimes \mathrm{Im},\quad \Delta(\mathrm{Im}) = \mathrm{Im} \otimes \mathrm{Re} + \mathrm{Re} \otimes \mathrm{Im} $$, which is the functional (no explicit arguments) counterpart of $$ (a+ib) \cdot (c+id) = (ac-bd) + i(ad+bc) $$, in the sense that left and right hand sides return the same thing when applied to $$ (a+ib) \otimes (c+id) $$.

130.243.94.123 (talk) 13:08, 23 May 2022 (UTC)