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Definition. A category $\cal K$ is good if there exists a conservative functor $F : {\cal K}\to \bf Set$.

Propositon. Locally small categories are good.

A proof of this statement is contained in [Freyd, §5]: the strategy is to prove it first when $\cal K$ is well powered and every arrow in $\cal K$ is monic. Then, you get rid of these assumptions with a set theoretic swindle. Somewhat hidden in Freyd's comments there is the following:

Conjecture. There exists a good, but non-locally small, category.

Q1: How to find it?

In light of this result, it appears obvious that another question is

Q2: How to characterize good categories? This means we want a necessary and sufficient condition ensuring that $\cal K$ is good: can this property be found?

Freyd attempts to give a partial solution to this question, devising the definition of a proper-conservative functor:

Definition. $F : {\cal K}\to \bf Set$ is proper-conservative if it is conservative and $F^\leftarrow ({\cal S})$ is small whenever $\cal S$ is a small subcategory of $\bf Set$.

Theorem. Locally small categories are precisely those that admit a properly good functor.

This concludes the introduction to the question: what I want to do is a variation on Freyd's theme, along the following lines.


Definition. $\cal K$ is faithfully good if it is good via a functor which is also faithful.

Proposition. $\cal K$ is faithfully good if and only if it is good and concrete.

The proof is fairly simple, and doesn't deserve to be explicated. More interesting is that this situation paves the way to the following further definitions and conjectures:

Definition. A category $\cal E$ is an earth if every locally small category is $\cal E$-good ($\cal E$-goodness is defined in such a way that good = $\bf Set$-good).

In this terminology, Freyd's initial claim is that $\bf Set$ is an earth.

Remarks. Every category of models for a Lawvere theory is an earth. The category of topological spaces is an earth. If $\cal E$ is a $\cal V$-good earth, then $\cal V$ is an earth.

We define a preorder relation on $\bf CAT$ by $$ {\cal J} \preceq {\cal B} = {\cal J} \text{ is $\cal B$-good} $$ We consider the partial order $\le$ on $\bf CAT$ induced by quotienting for ${\cal A} \sim{\cal B} \iff {\cal A}\preceq {\cal B}\preceq \cal A$, and we denote the elements of this poset $(\mathbb{CAT},\le)$ as $\{[{\cal A}] \mid {\cal A} \in\bf CAT\}$.

Remark. Intuitively speaking, $\le$ orders the objects of $\bf CAT$ by complexity: ${\cal A} \le {\cal B}$ means that $\cal B$ contains the structure of $\cal A$ faithfully and in a way that is mindful of its isomorphism classes.

Remark. $[{\bf Set}]$ is the class of earths, and this is the top element of $(\mathbb{CAT},\le)$.

Now we would like to mimic this construction defining faithful complexity:

Definition. We define a preorder relation $\preceq_\text{F}$ on $\bf CAT$ by $$ {\cal J} \preceq_\text{F} {\cal B} = {\cal J} \text{ is faithfully $\cal B$-good} $$

Then we quotient the preorder to make it a partial order and something funny happens: as a corollary of Freyd2, $[\bf Set]$ fails to be a faithful earth (the definition is a straightforward changing as above) because homotopy is not concrete $\Rightarrow [{\bf HTop}] \not\le [{\bf Set}]$. Nevertheless, $[{\bf Set}] \le [{\bf HTop}]$ (as it is easy to prove).

So we get to the final question:

Q3: "Is $\bf HTop$ on the top"? More generally, how do we characterize faithful earths, i.e. maximal elements of $(\mathbb{CAT},\le_\text{F})$ (if they exist, do they?)?

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    $\begingroup$ An easy answer to question 1: let $C$ be the category with two objects $a$ and $b$, whose morphisms consist only of the identities on $a$ and $b$ and a proper class of distinct morphisms $a\rightarrow b$ (but no morphisms $b\rightarrow a$). Then $C$ is not localy small, but has no isos other than the identities so is trivially good. This breaks down if we ask that conservative functors be faithful, as is sometimes done; however, if we demand that conservative functors be faithful, then every good category is clearly locally small. (Now, what about not equivalent to a locally small category?) $\endgroup$ – Noah Schweber May 28 '17 at 17:11
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    $\begingroup$ @NoahSchweber Locally small categories are already closed under equivalence (as long as your small sets are closed under bijections)-it's small ones where that's a loosening. $\endgroup$ – Kevin Carlson May 30 '17 at 17:32

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