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added pointer to
I have fixed the edits now. Ben has no associated nForum user (the posts were wrongly associated initially with a banned existing user), hence why the posts now appear as made by the nLab edit announcer.
Link to pseudofunctor.
Okay, I jumped in and added pseudofunctor to the list of “Related concepts”. I don’t think that hurts anything, and that is the first place I looked at originally.
Added the example of monoidal categories.
To put it more explicitly, I suspect there should be one more axiom:
$(\alpha\circ\beta)\circ\gamma = \alpha\circ(\beta\circ\gamma)$for all 0-morphisms $A$, $B$, $C$, $D$, 1-morphisms $A\stackrel{f_3}\to B\stackrel{f_2}\to C\stackrel{f_1}\to D$ and $A\stackrel{g_3}\to B\stackrel{g_2}\to C\stackrel{g_1}\to D$, and finally 2-morphisms $\alpha:f_1\to g_1$, $\beta:f_2\to g_2$, and $\gamma:f_3\to g_3$, where $\circ$ denotes horizontal composition.
This is not accounted for by the weak associativity axiom, since that pertains to the composition of 1-morphisms rather than of 2-morphisms.
I am a little confused about what exactly the coherence theorem for bicategories says, but wouldn’t horizontal associativity of 2-morphisms be something we would expect to have anyway? I can’t imagine not having to require it axiomatically.
hopefully the final edit: I had the details wrong again. Apologies to anyone whose time I wasted. The point stands: I think there should be this extra axiom.
It’s not possible to edit or investigate edits, but comment 17 is mistaken, as one can see by considering identity 2-cells.
Thousands of people have looked over the definition of bicategory, so further edits that are not attested in the literature probably shouldn’t be performed, at least not without prior discussion.
Yes, not to pile on too much but the domains and codomains of the LHS and RHS of the proposed axiom don’t even match, if the composition of 1-arrows is not associative.
I have touched the section “Coherence theorem” (here) adding more and more pertinent hyperlinks to technical terms.
Also cross-linked with Lack’s coherence theorem.
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