Create, Detect, Lift, Preserve, Reflect (Co)Limits

(draft)

Consider

$$\mathcal{I}\stackrel{D}{⟶}\mathcal{A}\stackrel{F}{⟶}\mathcal{B}.$$

We denote by

$\alpha$ cones of

$D$ and by

$\beta$ cones over

$FD$.

Roughly speaking,

• $F$ preserves
  $limD$ if the existence of the lhs implies the existence of the rhs and
  
  $$F(limD)\cong limFD,$$

• $F$ lifts
  $limD$ if the existence of the rhs implies the existence of the lhs and
  
  $$F(limD)\cong limFD.$$

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

$F$ preserves limits 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

$limFD$ implies the existence of

$limD$.

Example: The inclusion of a full reflective subcategory detects colimits.^[1] Regularly monadic functors detect colimits.^[2]

$F$ lifts limits if

$\beta =limFD$ implies that there is a limiting

$\alpha$ with

$F\alpha =\beta$.

$F$ uniquely lifts limits if

$\beta =limFD$ implies that there is a unique limiting cone

$\alpha$ with

$F\alpha =\beta$.

Example:

$\mathsf{T}\mathsf{o}\mathsf{p}\to \mathsf{S}\mathsf{e}\mathsf{t}$ lifts limits uniquely, but does not create nor reflect limits (as defined below).

$F$ creates limits if

$\beta =limFD$ 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 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, 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) 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.

The nLab.

A note of reflecting limits.

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1. This is because the reflector, as a left-adjoint, preserves colimits. ↩︎

2. See AHS 20.33. All monadic functors over sets are regularly monadic. ↩︎