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The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is explicit and claims that, given $f,g:C\rightarrow D$ two morphisms of coalgebras, the equalizer $(E,i)$ is given by

$$E=\lbrace{ c\in C/ c_{(1)}\otimes f(c_{(2)})\otimes c_{(3)}=c_{(1)}\otimes g(c_{(2)})\otimes c_{(3)}}\rbrace$$

and $i:E\rightarrow C$ the inclusion.

I understand this is indeed the equalizer when $E,D$ are Hopf algebras, by using the antipode. However, I cannot conclude the same for coalgebras. Any suggestion?, please. Is there a similar description for dg coalgebras?. Thanks.

Victor TC
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    Your $E$ can be equivalently described as $\ker\left(\left(\operatorname{id}C \otimes \left(f-g\right) \otimes \operatorname{id}_C\right) \circ \Delta^{\left(2\right)}\right)$, where $\Delta^{\left(2\right)} : C \to C \otimes C \otimes C$ is the map sending each $c \in C$ to $c{\left(1\right)} \otimes c_{\left(2\right)} \otimes c_{\left(3\right)}$. Thus, your claim about coalgebras is a particular case of Exercise 1.4.30 in Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356v5 (applied ... – darij grinberg May 07 '19 at 04:23
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    ... to $D$ and $f-g$ instead of $U$ and $f$). That $f-g$ is not a coalgebra homomorphism doesn't matter; nor do $f$ and $g$ have to be. Now, what about dg-coalgebras? I don't know, but as a newbie I wouldn't mind a definition of the notion you are using (something is telling me there is more than one). – darij grinberg May 07 '19 at 04:25
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    @darij Thank you for the reference. – Victor TC May 07 '19 at 20:27
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    @darji I got impressed, there are several useful exercises solved there. – Victor TC May 07 '19 at 20:39

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