# Locally presentable categories

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 ?

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.
• 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. – Valery Isaev Nov 11 '18 at 21:54
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.