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Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.
Thanks! I added some remarks about the way I prefer to think of doctrinal adjunction in terms of double categories. This motivated me to finally create companion pair and conjunction.
Is the statement of prop. 2.2 at doctrinal adjunction as intended:
For the unit and counit of the adjunction $f \dashv u$ to be $T$-transformations, and hence for the adjunction to live in $T$-$Alg_l$, it is necessary and sufficient that $\tilde f$ have an inverse $\bar f$ that makes $(f,\bar f)$ into a lax $T$-morphism, and hence $(f,\bar f)$ into a strong $T$-morphism.
?
The last line repeats the symbls $(f,\bar f)$. If this is what is really meant, it would be more clear to write “into a lax $T$-morphism, which is then necessarily a strong $T$-morphism”.
And in the lines before the proposition, symbols “$\bar f$” refer to 2-morphisms, while here they refer to (inverses of) 1-morphisms. I am not sure if I am parsing this correctly.
Actually, it would be enough to stop at “have an inverse.” The fact that this makes $(f,\bar{f})$ into a lax and a strong $T$-morphism then follows automatically. But $\bar{f}$ is a 2-morphism here too, since it is an inverse of the 2-morphism $\tilde{f}$.
Thanks, I see now where I made a mistake in parsing.
Maybe I’ll find some time to re-arrange the notation of the entry a little. On my system at least, it is somewhat hard to decypher, with different kinds of $n$-morphisms all in the same font and with the difference between small twiddles and small bars hard to make out.
Generally, for better reading experience, I found that
\overline{}
beats
\bar{ }
and it is good to type
\widetilde
even for single symbols to be decorated. But in the entry at hand, maybe one should find altogether different decoration, for readability.
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