nLab reflected limit

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 $J$.

Related pages

• preserved limit

• reflected limit

• created limit

• lifted limit

References

• Emily Riehl, §3.3 in: Category Theory in Context, Dover Publications (2017) [pdf, book website]