# nLab created limit

### Context

#### Limits and colimits

limits and colimits

# Creation of limits

## Definition

Let $F\colon C\to D$ be a functor and $J\colon I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and $F$ both preserves and reflects limits of $J$. This means that, in addition to $J$ having a limit whenever $F \circ J$ does, a cone over $J$ in $C$ is a limiting cone if and only if its image in $D$ is a limiting cone over $F\circ J$.

Of course, a functor $F$ creates a colimit if $F^{op}$ creates the corresponding limit.

If $F$ creates all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ creates that sort of limit (e.g. $F$ creates products, $F$ creates equalizers, etc.).

## Remarks

A monadic functor creates all limits that exist in its codomain, and all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself). Creation of a particular sort of split coequalizer figures prominently in Beck’s monadicity theorem.

One should beware that in Categories Work, a more restrictive notion of “creation” is used which requires the every limit in $D$ to lift to one in $C$ uniquely on the nose, rather than merely up to isomorphism. This corresponds to a version of the monadicity theorem which asserts an isomorphism of categories, rather than merely an equivalence.

Revised on November 2, 2012 23:43:38 by Sridhar Ramesh (98.248.229.177)