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Stream: learning: questions

Topic: Kock-Lawvere axiom

Max New (Apr 04 2023 at 17:59):

I have a question about an ambiguity in the statement of the Kock-Lawvere axiom on the nlab (https://ncatlab.org/nlab/show/Kock-Lawvere+axiom).

The Kock-Lawvere axiom says under certain conditions on an algebra $W$, we have that the obvious morphism $W \to R^{\textrm{Spec}_R(W)}$ is an isomorphism.

My question is: does this mean just an isomorphism in the ambient topos or an isomorphism of algebras? I.e., does the axiom require that the inverse is an algebra homomorphism? And if it does not, does it necessarily follow that it is?

Mike Shulman (Apr 04 2023 at 18:20):

If an algebra homomorphism is an isomorphism, then its inverse is automatically an algebra homomorphism.