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I added a section on Lawvere’s definition to adjoint functor and also made an article for Functorial Semantics of Algebraic Theories.
Also, I am wondering about the connection between the Grothendieck-construction style construction of the Lawvere characterization of adjoints from the hom-set definition and the calculation of comma categories depicted at comma category:
$\array{ (f/g) &\to& E^I \\ \downarrow && \downarrow^{\mathrlap{d_0 \times d_1}} \\ C \times D &\stackrel{f \times g}{\to}& E \times E }$The right side of this square is essentially the image under the 2-sided fibration “category of elements” of the hom functor, and the comma categories arising in Lawvere’s definition can also be seen as arising via the same construction from the profunctors induced by the adjoint functors.
It would be nice if there were a universal 2-sided fibration!
Thanks for starting ETCC. I made elementary theory of the 2-category of categories a redirect to it, since this is what had been requested since long ago from ETCS.
In this vein I think it should say “2-category of categories”, throughout. Shouldn’t it? At least that’s what we used to announce at ETCS. I have edited accordingly, but let me know if that’s not what Lawvere actually axiomatized.
Also I have added the following sentence. Somebody may want to improve on this, but I think we’d need some such sentence on the conceptual relation to ETCS:
This may be thought of as refining or categorifying the elementary theory of the category of sets (ETCS). Where the latter axiomatizes a base topos, ETCS would be thought of as axiomatizing a base 2-topos.
I made some changes to what Urs wrote in at ETCC.
It’s been ages since I last looked at Lawvere’s Category of Categories paper, but I am supposing that cartesian closure (as a 1-category) is one of the axioms, and from that we automatically get 2-category structure, by seeing $Cat$ as enriched in itself. So while it would be correct to focus on 2-categorical notions, I suppose any such could be “folded into” a 1-categorical presentation by taking advantage of cartesian closure.
I have reworked the reference section at ETCC and found a MO discussion.
Given that Mike there comes up with some definite ideas what an ET2CC should look like I would clearly vote for a separate entry for ET2CC or at least a clear conceptual division between ETCC and ET2CC within the entry
Also because from Lawvere’s POV there is a hiatus between ETCC and ETCS whose purpose is to provide discrete objects for undergraduate education within a (cohesive) world that is expressed via ETCC. In particular Lawvere’s conceptual algebra is somewhat orthogonal to a pan-homotopical approach as in n-category theory.
I think that we should distinguish Lawvere's historical proposal for an ETCC, which was groundbreaking but flawed, from a modern construction of an ET2CC.
I agree with the previous two comments.
Sounds fine to me.
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