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 $F\cdot \eta$ 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.).

Properties

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

Proposition

If there exists at least one cone $\eta$ for $J$ such that $F \cdot \eta$ is a limit for $F \circ J$, and $F$ reflects limits of $J$, then $F$ preserves limits of $J$.

Proof

Reflection implies $\eta$ is a limit for $J$ that is preserved by $F$. By this remark, $F$ necessarily preserves all limits of $J$.

Thus, reflection of limits for a given diagram $J$ either holds vacuously, or holds together with preservation of limits for $J$.

Remark

A functor which both reflects and lifts limits is said to create them.

Let $F_i \colon C \to D_i$ be a family of functors, and $J \colon I \to C$ a diagram. We say that these functors collectively reflect limits of $J$ if and only if a cone $(x, \eta)$ for $J$ that is mapped to a limit cone under all $F_i$ is a limit cone in $C$.

Given such a family, we can consider their “product” $F \colon C \to \prod_i D_i$. We have that:

Proposition

$F \colon C \to \prod_i D_i$ reflects limits of $J$ if and only if the functors $\pi_i \circ F \coloneqq F_i \colon C \to D_i$ collectively reflect limits of $J$.

Proof

For the forwards direction, take a cone $(x, \eta)$ for $J$, and suppose that $(F_i(x), F_i \cdot \eta)$ is a limit cone in $D_i$ for all $i$. Then, by inspection of the universal property, $(F(x), F \cdot \eta)$ is a limit cone in $\prod_i D_i$. Thus, since $F$ reflects limits of $J$, we conclude that $(x, \eta)$ is a limit cone.

For the reverse direction, we exploit limits in product categories are computed componentwise. Take a cone $(x, \eta)$ for $J$, and suppose it is mapped to a limit cone $(F(x), F \cdot \eta)$ for $F \circ J$ in $\prod_i D_i$. This means that $(F_i(x), F_i \cdot \eta)$ is a limit cone for $F_i \circ J$ in each $D_i$. Since the $F_i$ collectively reflect limits, this implies $(x, \eta)$ is a limit cone, as required.

In the following, let $J \colon I \to C$ be a diagram, and $F \colon C \to D, G \colon D \to E$ be functors.

Proposition

If $F$ reflects limits of $J$ and $G$ reflects limits of $F \circ J$, then $G \circ F$ reflects limits of $J$.

Proof

Take a cone $(x, \eta)$ for $J$ such that $((G \circ F)(x), (G \circ F) \cdot \eta)$ is a limit for $(G \circ F) \circ J$. Since $G$ reflects limits of $F \circ J$ we have that $(F(x), F \cdot \eta)$ is a limit for $F \circ J$, and then since $F$ reflects limits of $J$ we conclude $(x, \eta)$ is a limit cone, as required.

Proposition

If $G \circ F$ reflects limits of $J$ and $G$ preserves limits of $F \circ J$, then $F$ reflects limits of $J$.

Proof

Take a cone $(x, \eta)$ for $J$ such that $(F(x), F \cdot \eta)$ is a limit for $F \circ J$. Since $G$ preserves this, $((G \circ F)(x), (G \circ F) \cdot \eta)$ is a limit for $(G \circ F) \circ J$. Then, since $G \circ F$ reflects this, $(x, \eta)$ is a limit for $J$, as required.

Thus, if $G$ both preserves and reflects limits of $F \circ J$, then $F$ reflects limits of $J$ if and only if $G \circ F$ does.

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 $F \cdot \eta$ and $F\cdot \theta$ 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]