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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2012

    Since I was being asked I briefly expanded automorphism infinity-group by adding the internal version and the HoTT syntax.

    Mike, what’s the best type theory syntax for the definition of Aut(X) via -image factorization of the name of X?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeSep 26th 2012

    Y:TypeIsInhab(Y=X), or any variant depending on your chosen notation for IsInhab and identity/path types. E.g. Y:Type[IdType(Y,X)] would be another version.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2012
    • (edited Sep 26th 2012)

    Ah, thanks. I should have been able to come up with this myself.

    Can we allow ourselves to write “Y:Type[X=Y]”?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2012
    • (edited Sep 26th 2012)

    I have added a general remark on this to infinity-image in a new section Syntax in homotopy type theory there.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeSep 26th 2012

    Can we allow ourselves to write “Y:Type[X=Y]”?

    Sure, although I would probably mention that [] denotes the support (and, perhaps, = the intensional identity type / path type) whenever I start using this notation on a given page or in a given paper.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeSep 26th 2012

    OOPS! I meant [Y:Type(X=Y)]. I’ve corrected the proof at infinity-image.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2012
    • (edited Sep 26th 2012)

    Hm, I was thinking that

    im(A(idA,b)A×B)A×BB

    is equivalent to

    im(AbB)B.

    Hm…

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeSep 27th 2012

    That’s not even true for 0-truncated objects. In that case (1,b):AA×B is already monic, so the top composite just gives you b:AB rather than its image.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2012
    • (edited Sep 27th 2012)

    Ah, sorry. Right, I am being stupid.

    • CommentRowNumber10.
    • CommentAuthorperezl.alonso
    • CommentTimeJul 2nd 2025

    In Proposition 6.3.49 here, what exactly is what is on top of the 12 that sits at the 0th level of AUT(BU(1))? Is it just BU(1)×B2U(1), or is there a nontrivial action of the first factor on the second? Or, more to the point, suppose I want to describe a flat connection on the bundle controlled by maps BAut(12) (computing Aut(12) inside AUT(BU(1))). Are these just pairs consisting of a flat connection on a principal U(1) bundle along with a flat connection on a bundle gerbe, or can the curvature of the connection on the principal U(1) bundle be more general?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2025

    I think the answer is: Not more general, no.

    Since this happens in low degrees, we can still get a good explicit model by 2-functors:

    So B2U(1) is 2-groupoid with a single object and a single 1-morphism

    Then we are to compute

    (1.) its automorphic endo-2-functors,

    (2.) their natural transformations and

    (3.) their “modifications” (for the latter cf. p. 127 here).

    Inspect the diagrams characterizing the transformations and the modifications, these involve 2-morphisms labeled in U(1) and nothing else.

    Since U(1) is abelian and since there are no non-trivial 1-morphsims which could lead to non-trivial whiskering, the transformations and modications each just contribute a plain copy of U(1).

    • CommentRowNumber12.
    • CommentAuthorperezl.alonso
    • CommentTimeJul 2nd 2025

    Great, thanks, Urs.