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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 $(\text{Vect}, \otimes)$).2012-09-27
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    @ Qiaochu Yuan Thanks. That was actually enlightening comment.2012-09-28
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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.)2012-09-28
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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 $k$.2012-12-04
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    @QiaochuYuan Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-22

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