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  1. Under category

Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ones. I was wondering if the category $C^{/c}$ is also locally presentable ?

  1. Monad

Let $T$ be a monad on a locally presentable category $C$. Under which condition the category of $T$-algebras (the category of algebras over the monad $T$) is a locally presentable? Examples are very welcome.

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    $\begingroup$ Over and under categories of a presentable category are presentable. This is Proposition 1.57 in Locally Presentable and Accessible Categories. If $T : C \to C$ preserves $\lambda$-directed colimits and $C$ is $\lambda$-presentable, then the category of $T$-algebras is also $\lambda$-presentable. This is proved in 2.78 in the same book. A monad preserves $\lambda$-directed colimits if and only if it corresponds to a $\lambda$-infinitary algebraic theory. $\endgroup$ – Valery Isaev Nov 11 '18 at 21:54
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    $\begingroup$ @ValeryIsaev thanks for the references! You should put your comment as an answer. $\endgroup$ – TTip Nov 11 '18 at 22:11
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  1. Over and under categories of a presentable category are presentable. This is Proposition 1.57 in Adámek, Rosický, Locally Presentable and Accessible Categories.

  2. If $T : \mathcal{C} \to \mathcal{C}$ preserves $\lambda$-directed colimits and $\mathcal{C}$ is $\lambda$-presentable, then the category of $T$-algebras is also $\lambda$-presentable. This is proved in 2.78 in the same book. A monad preserves $\lambda$-directed colimits if and only if it corresponds to a $\lambda$-infinitary algebraic theory.

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