# nLab reflected limit

Reflection of limits

### Context

#### Limits and colimits

limits and colimits

# Reflection of limits

## Idea

A functor is said to reflect limits (colimits) of a given shape if a cone (cocone) is (co-)limiting whenever its image under $F$ is.

## Definition

###### Definition

Let $F\colon C\to D$ be a functor and $J\colon I\to C$ a diagram. We say that $F$ reflects limits of $J$ if whenever we have a cone $\eta\colon const^I_x \to J$ over $J$ in $C$ such that $F(\eta)$ 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 $F^{op}$ 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.).

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.

###### Remark

Reflection of limits is distinct from preservation of limits, although there are relationships, e.g prop. .

## Examples

###### Proposition

A faithful functor reflects epimorphisms and monomorphisms.

(The simple proof is spelled out here at epimorphism.)

###### Example

A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits.

This is evident from inspection of the defining universal property.

###### Proposition

A conservative functor reflects any limits which exist in its domain and that it preserves.

###### Proof

If $J$ in def. 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 $F(\eta)$ and $F(\theta)$ 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 $J$.

Last revised on October 24, 2016 at 08:05:50. See the history of this page for a list of all contributions to it.