finitary functor

According to the usual definition, a functor $F\colon C \to D$ is **finitary** if it preserves all filtered colimits. An important property is that a monadic category (over Set) is finitary (in other words, given by a Lawvere theory) if and only if its forgetful functor is finitary.

A slightly weaker definition is used in *The Joy of Cats*; there, $F$ is **finitary** if it maps every directed colimit in $C$ to a jointly epic diagram (but not necessarily a universal one) in $D$. With this definition, an algebraic category is finitary if and only if its forgetful functor is finitary. The previous result also holds, since the two definitions are equivalent for monads on Set.

Last revised on July 18, 2021 at 04:42:02. See the history of this page for a list of all contributions to it.