---
tags: mathematics, category theory
---

([up](https://hackmd.io/@alexhkurz/HJTOwSSjw))

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

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

---

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 $\mathcal A(A,D)\lhd \psi$. [^weighted]

[^weighted]: $\mathcal A(A,D)\lhd \psi$ 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.

$F\alpha$ is cone in $\mathcal B(FA,FD)\lhd\psi$.

If $F\alpha$ is limiting then it corresponds to the identity $i$ on the lhs of $\mathcal B(FA,\lim_\psi FD)\cong\mathcal B(FA,FD)\lhd\psi$.

That is, $FA=\lim_\psi FD$.

Since $F$ is fully faithful, $\alpha$ is the limiting cone corresponding to the identity on the lhs of $\mathcal A(A,A)\cong\mathcal A(A,D)\lhd\psi$.

QED

---

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 $\mathcal A(\lim_\psi D,D)\lhd \psi$.

Let $a$ be its transpose on the left-hand side of $\mathcal A(\lim_\psi D,\lim_\psi D) \cong \mathcal A(\lim_\psi D,D)\lhd \psi$.

Then $F\alpha$ is a cone in $\mathcal B(F\lim_\psi D,FD)\lhd\psi$.

If $F\alpha$ is limiting then it arises from the identity arrow $i$ on the left-hand side of $\mathcal B(F\lim_\psi D,F\lim_\psi D)\cong\mathcal B(F\lim_\psi D,FD)\lhd\psi$.

$i=Fa$.

Since $F$ is conservative, $a$ must be an iso on $\lim_\psi D$, hence $\alpha$ was limiting.

QED