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Hi, all.
I am confused by the following sentence from the document here
In page 1249, The paper says:
By Theorem 3, $2^{op}\cong 2$
Theorem 3 says “any generator with exactly three endomorphisms is isomorphic to $2$”, therefore, to show $2^{op}\cong 2$, there are two things to check: 1. $2^{op}$ is a generator, 2. there are exactly three functors $2^{op} \to 2^{op}$.
I agree that the axioms $(A^{op})^{op} = A$ and $(F^{op})^{op} = F$ makes sure that there is a one-to-one correspondence from functors $A \to B$ to functors $A^{op} \to B^{op}$. My confusion is to show the first point, that is, why is $2^{op}$ a generator? That is, given functors $f,g:A\to B$ such that $f \neq g$, how can we come up with a functor $a:2^{op} \to A$ such that $f\circ a \neq g \circ a$?
The author does not say anything like “we have an isomorphism $2\to 2^{op}$”, and the paper does not assume any “extensionality of categories” (i.e., we are not able to say two categories are equal iff they have equal collection of objects and a equal collection of arrows), therefore, the assumption $(A^{op})^{op} = A$ does not really say anything about the objects an arrows of $A^{op}$, since the operation is applied on $A$, not on its object (which are defined to be functors $1\to A$), and not on its arrows (which are defined to be functors $2\to A$). Also note that this sentence is stated before assuming $1 = 1^{op}$ and $2 = 2^{op}$, instead, the workflow seems to be: the axioms are sufficient to prove $2\cong 2^{op}$, and since assuming $2 = 2^{op}$ does not break consistency, we can add this assumption.
Or is the author sloppy here with the “minor detail”? If so, I think I am in trouble. Any hint to fix this point?
Thanks to any attempt to help!
(deleted)
Why doesn’t taking the opposite of the arrow $2\to A^{op}$ that shows $f^{op}\neq g^{op}$ work? Since $f\neq g$, we have $f^{op}\neq g^{op}$, since $(-)^{op}$ is an isomorphism on hom-sets, after all.
Why doesn’t taking the opposite of the arrow $2\to A^{op}$ that shows $f^{op}\neq g^{op}$ work? Since $f\neq g$, we have $f^{op}\neq g^{op}$, since $(-)^{op}$ is an isomorphism on hom-sets, after all.
Thank you very much! I thought too much about comprehension, that was my stupid. I have checked that does work so I am out of trouble!
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