Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeJun 11th 2010

    Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010
    • (edited Jun 11th 2010)

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2017

    Is the statement of prop. 2.2 at doctrinal adjunction as intended:

    For the unit and counit of the adjunction fuf \dashv u to be TT-transformations, and hence for the adjunction to live in TT-Alg lAlg_l, it is necessary and sufficient that f˜\tilde f have an inverse f¯\bar f that makes (f,f¯)(f,\bar f) into a lax TT-morphism, and hence (f,f¯)(f,\bar f) into a strong TT-morphism.

    ?

    The last line repeats the symbls (f,f¯)(f,\bar f). If this is what is really meant, it would be more clear to write “into a lax TT-morphism, which is then necessarily a strong TT-morphism”.

    And in the lines before the proposition, symbols “f¯\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.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2017

    Actually, it would be enough to stop at “have an inverse.” The fact that this makes (f,f¯)(f,\bar{f}) into a lax and a strong TT-morphism then follows automatically. But f¯\bar{f} is a 2-morphism here too, since it is an inverse of the 2-morphism f˜\tilde{f}.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2017
    • (edited Jun 21st 2017)

    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 nn-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.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 4th 2018

    I added a sentence to explain the ’doctrine’ part of ’doctrinal adjunction’.

    diff, v16, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)