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added pointer to
and rewrote the Idea-section to make it clear that these authors require not just existence of left and right adjoints, but in fact an ambidextrous adjoint and satisfying an extract coherence condition.
added pointer to:
Started a section “Modal type theory” (here) meant to show and explain the type theoretic rules for bireflective modalities proposed by Riley, Finster & Licata (2021).
For the moment the section just contains a table of the main inference rules paired with their categorical semantics. The inference rules are those of RFL21, whereas the rendering of the semantics is mine.
I do believe this semantics is equivalent to what they show in their Fig. 6, but unwinding their CwF-rules is tedious (for me, anyways, which is the reason I set out to develop the semantics from scratch). But I have checked in detail that our $\natural$-elim interpretations coincide, which is the main subtlety.
Moreover, in the categorical semantics I show (now in the entry) the respect of the computation rules is manifestly verified (as shown by the diagrams now in the entry), which is really what matters.
Aside from the detailed syntax/semantics tables, the section remains a little telegraphic for the moment. Will expand tomorrow.
Added a reference to
Thanks! I have added also pointer to
and then I further expanded the Idea-section to make it clear that there are three concepts here
essential localizations $\supset$ quintessential localizations $\supset$ bireflective subcategories.
added the observation (here) that a model 1-category theoretic presentation of the “bireflective sub-$\infty$-category” $Grpd_\infty \hookrightarrow T Grpd_\infty$ may be recognized in Hebestreit, Sagave & Schlichtkrull (2020), Lem. 3.19
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