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In most or perhaps all the examples of a co-algebra that I have seen, the properties of sets as the base category was used, like the existence of products and co-product and Cartesian closeness. Does anyone have an example of a co-algebra and a system which makes use of more peculiar categorical properties?

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I'm not sure I understand the question. Are you just asking for an example of a coalgebra internal to a category other than $\text{Set}$ or are you asking for an example of a coalgebra internal to a category which is not concretizable? Examples of the first kind are easy to find (e.g. coalgebras in $(\text{Vect}, \otimes)$). –  Qiaochu Yuan Sep 27 '12 at 20:24
    
@ Qiaochu Yuan Thanks. That was actually enlightening comment. –  Hooman Sep 28 '12 at 17:42
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(Examples of the second kind can be found in the pointed homotopy category of topological spaces, which is known not to be concretizable. For example, $S^1$ is a comonoid in this category.) –  Qiaochu Yuan Sep 28 '12 at 18:10
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You can also learn about corings which are coalgebras in the category of $(A,A)$-bimodules where $A$ is some unital associative algebra over a commutative ring $k$. –  Paul Slevin Dec 4 '12 at 12:36
    
@QiaochuYuan Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer Jun 22 '13 at 8:47

1 Answer 1

up vote 3 down vote accepted

See this MO question and this MO question for several examples of coalgebras in categories other than $(\text{Set}, \times)$.

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