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tags: mathematics, category theory
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# Create, Detect, Lift, Preserve, Reflect Limits

(draft)

Consider
$$\mathcal I \stackrel D \longrightarrow \mathcal A \stackrel F \longrightarrow \mathcal B.$$

We denote by $\alpha$ cones of $D$ and by $\beta$ cones over $FD$.

Roughly speaking,

- $F$ preserves $\lim D$ if the existence of the lhs implies the existence of the rhs and
$$F(\lim D)\cong \lim FD,$$

- $F$ lifts $\lim D$ if the existence of the rhs implies the existence of the lhs and
$$F(\lim D)\cong \lim FD.$$

To be more precise we need to take into consideration that limits are not mere objects but limiting cones.

$F$ **[preserves limits](https://ncatlab.org/nlab/show/preserved+limit)** if for all cones $\alpha$ we have that $\alpha$ limiting implies $F\alpha$ limiting.

*Example:* Right-adjoint functors preserve limits.

$F$ **detects limits** if the existence of $\lim FD$ implies the existence of $\lim D$.

*Example:* The inclusion of a full reflective subcategory detects colimits.[^detect1] Regularly monadic functors detect colimits.[^detect2]

$F$ **lifts limits** if $\beta=\lim FD$ implies that there is a limiting $\alpha$ with $F\alpha=\beta$.

$F$ **uniquely lifts limits** if $\beta=\lim FD$ implies that there is a unique limiting cone $\alpha$ with $F\alpha = \beta$.

*Example:* $\sf Top\to Set$ lifts limits uniquely, but does not create nor reflect limits (as defined below).

$F$ **creates limits** if $\beta=\lim FD$ implies that there is a unique cone $\alpha$ with $F\alpha=\beta$ and, moreover, $\alpha$ is limiting.

*Example:* There is only one way to equip the cartesian product of the underlying sets of two groups with a group structure and, moreover, the resulting group is the group-product of the two groups. More generally, limits of algebras are calculated as in Set. More generally still, monadic functors create limits.

$F$ **[reflects limits](https://ncatlab.org/nlab/show/reflected+limit)** if $F\alpha$ limiting implies that $\alpha$ is limiting.

*Example:* Fully faithful functors reflect limits.

**Proposition:** $F$ creates limits iff $F$ lifts limits uniquely and $F$ reflects limits.

In other words, if $F$ reflects limits only limiting cones can map to a limiting cone. On the other hand, if $F$ lifts limits then there is at least one limiting cone that maps to any given limiting cone of the codomain.

*Example:* It is easy to find fully faithful functors that do not lift limits. For example, if $\mathcal B$ has a terminal object and $F:\mathcal A\to \mathcal B$ is a full subcategory without terminal object then there is no cone in $\mathcal A$ over the empty diagram that is limiting in $\mathcal B$. So $F$ reflects terminal objects but does not lift them.

**Proposition:** If $F:\mathcal A\to\mathcal B$ lifts limits and if limits exist in $\mathcal B$, then $\mathcal A$ has limits and $F$ preserves them.

**Remark:** Leinster, p.140, suggests to call the definition above *strictly creating limits* and to defined creation of limits instead as follows. If $FD$ has a limit then there exists a cone on $D$ whose image under $F$ is a limit cone, and that every such cone is itself a limit cone. The [nLab](https://ncatlab.org/nlab/show/created+limit), equivalently, says that $F$ creates the limit of $D$ if whenever the limit of $FD$ exists then the limit of $D$ exists and $F$ preserves and reflects it.

## Reflective subcategories

The next result has many applications. Note that it says something about a left-adjoint preserving certain limits.

**Theorem:** A full reflective subcategory closed under isomorphisms is closed under limits.

## References

Chapter 13 on "Functors and Limits" of [(Adamek, Herrlich, Strecker)](http://katmat.math.uni-bremen.de/acc/acc.pdf) gives a useful overview of the different ways functors $F:\mathcal A\to\mathcal B$ may interact with limits. $F$ preserves limits (p. 223). $F$ creates limits and $F$ lift limits (uniquely) (p. 227). $F$ reflects and detects limits (p. 229).

Leinster, [Basic Category Theory](https://arxiv.org/pdf/1612.09375.pdf).

The [nLab](https://ncatlab.org/nlab/show/created+limit).

A note of [reflecting limits](https://hackmd.io/@alexhkurz/HkILFdYhT).

[^detect1]: This is because the reflector, as a left-adjoint, preserves colimits.

[^detect2]: See AHS 20.33. All monadic functors over sets are regularly monadic.