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-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-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive 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 graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages 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 string-theory 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.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010

    polished and expanded adjoint (infinity,1)-functor

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010
    • (edited Apr 7th 2010)

    added some more stuff,

    am about to write the section on preservation of limits and colimits, but I am wondering:

    in ordinary cat theory, that right(left) adjoints preserve (co)limits follows purely formally from three ingredients:

    1. characterization of adjunction by hom-isomorphism;

    2. preservation of limits by hom-functors in both variables;

    3. the Yoneda lemma.

    It would seem that in the (oo,1)-category context we just check these three ingredients and then deduce the statement that left oo-adjoints preserve oo-colimits formally by direct analogy.

    Now, instead in HTT, section 5.2.3 a rather tedious simp-set gymnastics is performed to show this.

    I am wondering therefore: it seems we essentially do have the three items above in the oo-context:

    1. the first is HTT prop. 5.2.2.8,

    2. the second is essentially HTT A.3.3.12,

    3. and the last one is the combination of statements collected at Yoneda lemma for (infinity,1)-categories .

    So one would hope that we can just argue formally on this basis that left adjoints preserves oo-colimits. But there are probably subtleties in the details...

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010

    added in the details of construction and proofs on how simplicial and simplicial Quillen adjunctions induce oo-adjunctions in Simplicial and derived adjunctions

    Okay, enough (oo,1)-entertainment for today, now I need to focus my thoughts on a talk I will give in Nijmegen tomorrow...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)

    added the proposition on how adjoint (,1)(\infty,1)-functors pass to slices, at adjoint (oo,1)-functor – on over-(oo,1)-categories

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2019

    added publication data for

    diff, v32, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2019

    I seem to recall that recently someone asked a question here on the nForum about characterizing adjoint (,1)(\infty,1)-functors via universal arrows, or equivalently by pointwise representability (that is, if D(X,G)D(X, G -) is representable for all objects XX then these representing objects assemble into a functor left adjoint to GG), and that a reference was given. But after a few minutes of searching I can’t find it, and it doesn’t seem to have made it onto the page adjoint (infinity,1)-functor. Can anyone find the discussion or re-give the reference?

    • CommentRowNumber7.
    • CommentAuthorjweinberger
    • CommentTimeJun 18th 2019

    Possibly this? Having an oo-left adjoint

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2019

    Yes, that’s it; thanks!

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2019

    Added characterization in terms of universal arrows.

    diff, v33, current

  1. added additional reference to HTT for ’universal arrow’ characterization.

    Dylan

    diff, v34, current

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJun 20th 2019

    That statement from HTT is not exactly this proposition. I can see how it contains essentially the same information, but it’s hidden in the phrase “representable right fibration”. In particular, I’m pretty sure that I glanced past that statement in HTT while looking for this result and didn’t notice it. So I edited the attribution here in an attempt to reduce confusion in the reader.

    diff, v35, current

    • CommentRowNumber12.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2019
    “With some work” feels even more misleading though. ‘representable fibration’ is synonymous with “equivalent to an overcategory” so the work required is “know the definition of representable fibration, or guess it” (and also dualize between right and left adjoints, but that seems harmless). It also seems strange to prioritize credit to Rigel-Verity just because they say over/undercategory instead of “something equivalent to an over/undercategory”, doesn’t it?
    • CommentRowNumber13.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2019
    (Sorry the above and this are from Dylan- my phone forgot my login).
    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2019

    I wasn’t thinking of credit, but of clarity for the reader. You also have to realize that the pullback of a representable fibration is the relevant comma category, and that this being equivalent to an overcategory over its codomain is the same as having a terminal object.

  2. Expanded upon a remark about adjunctions on homotopy categories, namely that the existence of a right adjoint can be detected on Ho-enriched homotopy categories.

    Anonymous

    diff, v37, current

  3. I’m not certain about the caution I gave, so I’ve removed it.

    Anonymous

    diff, v37, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2020
    • (edited Jul 14th 2020)

    I have added the definition (here) of adjoint (,1)(\infty,1)-functors as an adjunction in the homotopy 2-category of (infinity,1)-categories.

    I had wanted to next add pointers to theorems proving the equivalence of this to other definitions, but I am getting a bit lost in the text. Maybe somebody can help:

    In Riehl-Verity 16, p. 3 it says that the equivalence to Lurie’s definition is

    Now 4.4.5 in Riehl-Verity 15 is a Remark, and it mainly points to the previous Observation 4.4.4. The relevant bit of Observation 4.4.4, in turn, seems to say that another Observation 3.4.4 implies that the homotopy 2-categorical definition of adjunction implies the expected hom-equivalence.

    This is all very plausible from a broad-brush perspective, of course, but it remains unclear to me if/where a claim is made concerning the crucial fine-print (of which there is a fair bit, already notation-wise). Observation 4.4.4 speaks of an “equivalence of hom-spaces” which is probably really “natural equivalence”. The next sentence has “This should be thought of as [[the expected statement]]” . I can see how it is probably all clear to the authors at this point, but it does make it hard to cite a result here.

    Then, Remark 4.4.5 speaks of a definition of Lurie’s “more complicated” than the authors “prefer to recall”. This probably alludes to Lurie’s definition via Cartesian fibrations. But their pointer is actually to Lurie’s Proposition 5.2.2.8, which is just the proposition where Lurie proves that this is equivalent to the expected natural hom-equivalence, hence to what, presumably, the previous Observation 4.4.4 did mean to recall.

    Similarly, I am not sure yet where exactly in Riehl-Verity 17 “the converse” is proven, and in which form.

    Anyway, that just to explain where I get lost. If anyone here can state with authority a crisp form of the expected statement, such as

    An adjunction in the homotopy 2-category of (,1)Cat(\infty,1)Cat is equivalently a natural hom-equivalence as in …

    then it would be great to add that to our entry here.

    diff, v39, 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)