# nLab reflected limit

### Context

#### Limits and colimits

limits and colimits

# Reflection of limits

## Definition

Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ reflects limits of $J$ if whenever we have a cone $\eta :{\mathrm{const}}_{x}^{I}\to J$ over $J$ in $C$ such that $F\left(\eta \right)$ is a limit of $F\circ J$ in $D$, then $\eta$ was already a limit of $J$ in $C$.

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

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

## Remarks

Reflection of limits is distinct from preservation of limits, although there are relationships. For instance, a conservative functor reflects any limits which exist in its domain and that it preserves. For if $J$ above has some limit $\theta$ which is preserved by $F$, then there is a unique induced map $\eta \to \theta$ by the universal property of a limit, which becomes an isomorphism in $D$ since $F\left(\eta \right)$ and $F\left(\theta \right)$ are both limits of $F\circ J$; hence if $F$ is conservative then it must already have been an isomorphism in $C$, and so $\eta$ was already also a limit of $J$.

A functor which both reflects and preserves limits, and such that limits exist in its domain whenever they do in its codomain, is said to create them.

Revised on March 6, 2012 21:06:55 by David Corfield (81.155.107.228)