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Stream: learning: questions

Topic: Kock-Lawvere axiom and Amazing Right Adjoints

Alexander Gietelink Oldenziel (Nov 26 2020 at 14:16):

Does anybody know the relation between the Kock-Lawvere axiom for a ring object $R$ and a Topos having amazing right adjoints for atoms/tiny objects. Is the latter a categorical way of phrasing infinitesimal subspaces $D\subset R$?

Matteo Capucci (he/him) (Nov 26 2020 at 14:35):

To have an amazing right adjoint is the definition of 'infinitesimal object' (for Lawvere, at least). The KC axioms uses an infinitesimal object to say that $R \times R \to R^D$ defined by $(x,y) \mapsto x+\varepsilon y$ has to be an iso

Fawzi Hreiki (Nov 26 2020 at 14:37):

I don't think this is correct actually. The K-L axiom says nothing about amazing right adjoints.

Fawzi Hreiki (Nov 26 2020 at 14:41):

The K-L axiom says that, for our ring object $R$, and our subspace $D = \{d \in R : d^2 = 0\}$, the map $R \times R \rightarrow R^D$, $(x, y) \mapsto (x+y\epsilon$) is an isomorphism. Basically, every map from $D$ to $R$ is affine and has a unique slope.

Eduardo Ochs (Nov 27 2020 at 01:32):

Shouldn't it be $(x,y) \mapsto (\lambda ε{:}D.x + yε)$?...

Fawzi Hreiki (Nov 27 2020 at 02:38):

Yeah I was just being sloppy

Fawzi Hreiki (Nov 27 2020 at 02:40):

With regards to the original question, you should have a look in Kock’s book ‘Synthetic Differential Geometry’. I’ve only read through the first few chapters but he mentions this right adjoint as some point later on.

Ben MacAdam (Nov 27 2020 at 05:55):

The amazing right adjoint implies that $D$ is a small projective, and it holds in a Grothedieck model of synthetic differential geometry whenever $D$ is a representable functor. You can look at 5.5 in Basic Concepts of Enriched Category Theory for some background on how that argument works (I believe the small projective/representable functor thing is due to Bunge).

Jade Master (Jan 28 2021 at 01:25):

I think I understand the intuition behind D being small projective, the idea is that D is small enough that mapping out of it is kind of like a mapping out of a terminal object right? I definitely don't understand the intuition for an amazing right adjoint $R : \mathsf{Set} \to C$ to the functor $C(D,-) : C \to \mathsf{Set}$. For a set X what is the meaning of R(X)?

Fawzi Hreiki (Jan 28 2021 at 01:38):

The amazing right adjoint is defined internal to the category as far as I’ve seen.

Fawzi Hreiki (Jan 28 2021 at 01:38):

See for example https://ncatlab.org/nlab/show/amazing+right+adjoint#definition

Fawzi Hreiki (Jan 28 2021 at 01:43):

There’s a paper by Kock on some of the formal aspects of such objects. It’s in the references to the nLab page I linked. But if you want some geometric intuition, you may want to read this then this then this.

Notification Bot (Jan 28 2021 at 01:48):

This topic was moved here from #general > Kock-Lawvere axiom and Amazing Right Adjoints by Nathanael Arkor

Jade Master (Jan 28 2021 at 01:54):

Oh, so it's a right adjoint to internal him $C(e,-) : C \to C$. Thank you for the links I will read but I was hoping that someone could tell me an informal explanation. I don't know much about geometry, but any intuition anyone has is appreciated.

Todd Trimble (Jan 28 2021 at 02:03):

Jade Master said:

I think I understand the intuition behind D being small projective, the idea is that D is small enough that mapping out of it is kind of like a mapping out of a terminal object right? I definitely don't understand the intuition for an amazing right adjoint $R : \mathsf{Set} \to C$ to the functor $C(D,-) : C \to \mathsf{Set}$. For a set X what is the meaning of R(X)?

In case anyone reading this wants an example: if $E = [C^{op}, \textsf{Set}]$ is a presheaf topos and $c$ is an object of $C$, then

$E(C(-, c), -): E \to \textsf{Set}$

preserves colimits and has a right adjoint, by the Yoneda lemma. It's worthwhile computing this example where the amazing right adjoint exists. (And this example could be beefed up to the internal case, if we assume for example that $C$ has products.)

Jade Master (Jan 28 2021 at 17:58):

Todd Trimble said:

Jade Master said:

I think I understand the intuition behind D being small projective, the idea is that D is small enough that mapping out of it is kind of like a mapping out of a terminal object right? I definitely don't understand the intuition for an amazing right adjoint $R : \mathsf{Set} \to C$ to the functor $C(D,-) : C \to \mathsf{Set}$. For a set X what is the meaning of R(X)?

In case anyone reading this wants an example: if $E = [C^{op}, \textsf{Set}]$ is a presheaf topos and $c$ is an object of $C$, then

$E(C(-, c), -): E \to \textsf{Set}$

preserves colimits and has a right adjoint, by the Yoneda lemma. It's worthwhile computing this example where the amazing right adjoint exists. (And this example could be beefed up to the internal case, if we assume for example that $C$ has products.)

Thanks. To make an example of your example, we could regard $\mathsf{Set}$ as a presheaf category by taking C to be the terminal category. Then the only choice for $C(c,-)$ is the singleton set, and the hom-functor ends up taking the elements of a given set. So then the question is, what is the right adjoint to taking elements...and I'm guessing that the the only thing this could be is the the identity, i.e. the right adjoint sends a set to itself.

Jade Master (Jan 28 2021 at 17:58):

So this was a sort of degenerate example, after my meeting I'll try the case when the presheaf category is Grph.