(up)

Create, Detect, Lift, Preserve, Reflect Limits (2)

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

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Recall:

$F$ reflects limits if

$F\alpha$ limiting implies that

$\alpha$ is limiting.

Proposition: If

$F$ is fully faithful, then

$F$ reflects limits.

https://ncatlab.org/nlab/show/reflected+limit

Proof: Let

$\alpha$ be a cone in

$\mathcal{A}(A,D)⊲\psi$. ^[1]

$F\alpha$ is cone in

$\mathcal{B}(FA,FD)⊲\psi$.

If

$F\alpha$ is limiting then it corresponds to the identity

$i$ on the lhs of

$\mathcal{B}(FA,\underset{\psi }{lim}FD)\cong \mathcal{B}(FA,FD)⊲\psi$.

That is,

$FA=\underset{\psi }{lim}FD$.

Since

$F$ is fully faithful,

$\alpha$ is the limiting cone corresponding to the identity on the lhs of

$\mathcal{A}(A,A)\cong \mathcal{A}(A,D)⊲\psi$.

QED

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Recall:

• $F$ preserves limits if for all cones
  $\alpha$ we have that
  $\alpha$ limiting implies
  $F\alpha$ limiting.
• $F$ is conservative if
  $Ff$ iso implies
  $f$ iso.

Proposition: If

$F$ is conservative and preserves limits, then

$F$ reflects all limits that exist in the domain.

https://ncatlab.org/nlab/show/reflected+limit

Proof: Let

$\alpha$ be a cone in

$\mathcal{A}(\underset{\psi }{lim}D,D)⊲\psi$.

Let

$a$ be its transpose on the left-hand side of

$\mathcal{A}(\underset{\psi }{lim}D,\underset{\psi }{lim}D)\cong \mathcal{A}(\underset{\psi }{lim}D,D)⊲\psi$.

Then

$F\alpha$ is a cone in

$\mathcal{B}(F\underset{\psi }{lim}D,FD)⊲\psi$.

If

$F\alpha$ is limiting then it arises from the identity arrow

$i$ on the left-hand side of

$\mathcal{B}(F\underset{\psi }{lim}D,F\underset{\psi }{lim}D)\cong \mathcal{B}(F\underset{\psi }{lim}D,FD)⊲\psi$.

$i=Fa$.

Since

$F$ is conservative,

$a$ must be an iso on

$\underset{\psi }{lim}D$, hence

$\alpha$ was limiting.

QED

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1. $\mathcal{A}(A,D)⊲\psi$ stands for set of natural transformations from the object
   $A$ to the functor
   $D$. The notation extends to the case of so-called weighted limits. ↩︎