I’m enjoying these Lectures, and in particular, the clear explanations and metaphors to deeper understand the meaning of these topics.
I was stunned by seeing existential and universal quantifiers arising from right and left quantifiers. Moreover, the metaphor of ‘generous’ and ‘cautious’ for left and right adjoints, respectively, are powerful. Another pair — I don’t remember if it had already been used — could be ‘collective’ versus ‘individual,’ and a bunch of philosophy can start from there.

I’m wondering if we can also connect elements of partitions with right/left adjoints, through quantifiers. With right adjoints, we tend to exclude elements, leading to finer partitions. However, with left adjoints, we tend to include elements, leading to coarser partitions. If we need to create a finer partition, maybe we need to include ‘more information’: thus, if we go deeper into detail, we exclude elements that do not satisfy more precise requirements. Conversely, with left adjoints, if we ‘ignore’ some information, we can reach a higher level of abstraction; thus, our partition will be more inclusive and coarser. In this way, we could also use the pair ‘specialization’ vs. ‘abstraction,’ and we could even see the philosophical pair ‘deductive reasoning’ vs. ‘inductive reasoning’ as two adjoint movements that in principle may arise from broader categorical concepts. I hope that all of this makes sense.

The concepts of adjunction and approximation, discussed in Lecture 3, in my opinion, are very promising also for interdisciplinary applications. For example, a typical musical process is the compositional process: going from an initial idea to a complete piece. The inverse of this process is almost impossible: nobody exactly knows what the composer did, which precise path he or she did follow, what exactly the initial idea was. Two approximations are possible. From one side, there is an approximation from below, that is, a technical analysis to find recurring elements, connections between melodic fragments, and so on. Such a technical analysis, where higher structures are broken down into small, measurable units, could provide less information than needed, though. On the other side, there is an approximation from above, that could include more information than needed: it is the case of a conceptual/intuitive analysis or an analysis that related elements of a piece with style of the composers, drafts of other pieces, writings and poetics of the composer, some documents about his or her lessons or methods.
In this composition example, the sought ‘inverse’ should be the analysis, but in general, a perfect analysis simply does not exist.
Partitions give equivalence relations: to stay within this example, finer or coarser partitions can characterize in more or less detail elements of the same or more than one musical pieces.
Your suggestions and corrections will be so helpful for me!