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Stream: learning: questions

Topic: internalization via hom

Leopold Schlicht (Mar 23 2022 at 18:17):

Let $C$ be a category with enough "structure".

• $R\hookrightarrow X\times X$ is a model of the theory of equivalence relations in $C$ if and only if for all $Y\in C$, the induced map $\hom(Y, R)\hookrightarrow \hom(Y, X)\times\hom(Y, X)$ is an equivalence relation in $\mathbf{Set}$.
• $(G, \circ, e, (-)^{-1})$ is a model of the theory of groups in $C$ if and only if for all $Y\in C$, the set $\hom(Y, G)$ equipped with the maps $\hom(Y,G)\times\hom(Y,G)\to\hom(Y, G)$, $1\to\hom(Y, G)$, $\hom(Y,G)\to\hom(Y,G)$ induced by $\circ$, $e$, and $(-)^{-1}$ is a group in $\mathbf{Set}$.
• ...

I'm sure there is a general theorem which has all of these statements as special cases. Where can I find it?

Fawzi Hreiki (Mar 23 2022 at 18:45):

(deleted)

Fawzi Hreiki (Mar 23 2022 at 19:00):

The Yoneda embedding preserves and reflects limits, so this is true of any finite limit theory.

Leopold Schlicht (Mar 23 2022 at 19:03):

I asked where I can find a general theorem.

Fawzi Hreiki (Mar 23 2022 at 19:25):

I don’t know of a general theorem, but it’s pretty clear to see that the Yoneda embedding won’t in general preserve any structures which require [[regular logic]] or above since it destroys all colimits/epimorphisms.

Jon Sterling (Mar 23 2022 at 19:48):

I think the "general theorem" is called the Yoneda lemma. (But as @Fawzi Hreiki this generality applies to things like finite limit theories, not beyond)

Reid Barton (Mar 23 2022 at 19:56):

It also works for non-finite limit theories :smile:

Mike Shulman (Mar 23 2022 at 20:26):

And you can make it work for structures in regular/coherent/geometric logic by sheafifying the Yoneda embedding.

Leopold Schlicht (Mar 24 2022 at 11:20):

Jon Sterling said:

I think the "general theorem" is called the Yoneda lemma.

The Yoneda lemma isn't a statement of the form "X is a model of the theory of T if and only if for all Y, hom(Y,X) is a model of T in Set". The general theorem I am searching for is of this form.

Leopold Schlicht (Mar 24 2022 at 11:21):

Is the theory of equivalence relations a limit theory?

Leopold Schlicht (Mar 24 2022 at 11:23):

Fawzi Hreiki said:

The Yoneda embedding preserves and reflects limits, so this is true of any finite limit theory.

Where can I find a proof of the "so"?

Zhen Lin Low (Mar 24 2022 at 12:21):

Leopold Schlicht said:

The Yoneda lemma isn't a statement of the form "X is a model of the theory of T if and only if for all Y, hom(Y,X) is a model of T in Set". The general theorem I am searching for is of this form.

I think you will find that many practicing category theorists call such results "a consequence of the Yoneda lemma" or similar. I try not to but there is no other commonly understood name for it.

Zhen Lin Low (Mar 24 2022 at 12:22):

Leopold Schlicht said:

Is the theory of equivalence relations a limit theory?

Yes, you can express the property of being an equivalence relation using only finite limits.

Jon Sterling (Mar 24 2022 at 13:35):

Leopold Schlicht said:

Jon Sterling said:

I think the "general theorem" is called the Yoneda lemma.

The Yoneda lemma isn't a statement of the form "X is a model of the theory of T if and only if for all Y, hom(Y,X) is a model of T in Set". The general theorem I am searching for is of this form.

I think what I am saying is that it is so close to the Yoneda lemma that you will not find a theorem of this form.

Jon Sterling (Mar 24 2022 at 13:36):

Mike Shulman said:

And you can make it work for structures in regular/coherent/geometric logic by sheafifying the Yoneda embedding.

Oh right, of course.

Fawzi Hreiki (Mar 24 2022 at 14:52):

But then you need to be careful about your use of joins and the existential quantifier. Everything is done locally rather than just component wise.

Leopold Schlicht (Mar 24 2022 at 16:41):

I don't think the Yoneda lemma is enough. We also need a lemma which says something like this: Let $C$ and $D$ be categories with limits, and $F\colon C\to D$ a fully faithful functor which preserves limits. Furthermore, let $T$ be a limit theory with signature $L$ and $A$ be an $L$-structure in $C$. Then $A$ is a model of $T$ in $C$ if and only if $FA$ is a model of $T$ in $D$.

Fawzi Hreiki (Mar 24 2022 at 16:44):

The ‘only if’ is part of your hypothesis. It’s a simple fact that fully faithful functors reflect limits and so you have the ‘if’.

Leopold Schlicht (Mar 24 2022 at 16:52):

Fawzi Hreiki said:

The ‘only if’ is part of your hypothesis.

I don't understand what you mean by that.

Leopold Schlicht (Mar 24 2022 at 16:56):

Don't get me wrong, I see why the "only if" direction is true. Thanks for your hint for the "if" direction.