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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023
    • (edited Mar 24th 2023)

    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.

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023

    added pointer to:

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2023

    have spelled out (here) the definition and some properties

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2023
    • (edited Mar 26th 2023)

    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.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorThomas Holder
    • CommentTimeMar 26th 2023

    Added a reference to

    • {#JS96}P. Johnstone, Remarks on Quintessential and Persistent Localizations , TAC 2 no.8 (1996) pp.90-99. (pdf)

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2023
    • (edited Mar 27th 2023)

    Thanks! I have added also pointer to

    • G. M. Kelly, F. W. Lawvere, On the complete lattice of essential localizations, Bull. Soc. Math. Belg. Sér. A 41 (1989) 289–319

    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.

    diff, v10, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2023

    added the observation (here) that a model 1-category theoretic presentation of the “bireflective sub-\infty-category” Grpd TGrpd Grpd_\infty \hookrightarrow T Grpd_\infty may be recognized in Hebestreit, Sagave & Schlichtkrull (2020), Lem. 3.19

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2023
    • (edited Apr 3rd 2023)

    added (here) the observation/derivation that the categorical semantics for (aa̲ ):AA̲(a \mapsto \underline{a}^\natural) \colon A \to \natural \underline{A} is given simply by postcomposition with the naturality square of the unit η \eta^\natural

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2023

    added a remark (here) making explicit some possibly unexpected conventions in the RFL-syntax

    diff, v19, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2023

    added (here) semantic verification for the rule of \natural respecting dependent pairs

    diff, v20, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2023
    • (edited Apr 4th 2023)

    added (here) syntax and semantics for the decomposition of any type AA as the sum over its linear fiber types A x̲A_{\underline{x}}

    diff, v21, current