Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorThomas Holder
    • CommentTimeOct 10th 2020

    Added a reference to

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2020

    I have hyperlinked combinatorial functors, since it seems wrong not to.

    Since the entry does’t exist yet, I’ll create a stub. Best if you touch it afterwards.

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeOct 10th 2020

    I must confess that I’ve picked up the terminology from Lawvere without seeing much connection to combinatorics whose enumeration problems seem more naturally connected to bijections à la Joyal than injections.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2020

    So the “combinatorial functors” are not combinatorial functors?

    In that case the link should be removed again and instead some clarification added.

    • CommentRowNumber5.
    • CommentAuthorThomas Holder
    • CommentTimeOct 10th 2020

    Ah, interesting reference you dug up there! That might in fact be the concept that Lawvere had in mind. I’ll check it out when occasion arises.

    • CommentRowNumber6.
    • CommentAuthorThomas Holder
    • CommentTimeOct 11th 2020
    • (edited Oct 11th 2020)

    I shuffled the link to combinatorial functor downwards and motivated the terminology with a quote from Lawvere. It looks to me that their strict combinatorial functors Set monoSet monoSet_{mono}\to Set_{mono} apparently studied by Myhill might correpond to objects in the Schanuel topos but we have to wait for an energetic model theorist to sort out the connection to the Crossley-Nerode concept.

    diff, v17, current

    • CommentRowNumber7.
    • CommentAuthorThomas Holder
    • CommentTimeOct 13th 2020

    Added a further description of the objects bringing them closer to what might rightfully be called combinatorial functor.

    diff, v18, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2020
    • (edited Oct 13th 2020)

    Thanks for looking into it. Though I’ll say that I find there remains room to clarify the remark on combinatorial functors (I admit it remains unclear to me, without digging into the references).

    Further in the vein of hyperlinking all technical terms (that’s what eventually constitutes the power of the wiki), I have added double square brackets to binomial coefficient and to name binding.

    diff, v19, current

    • CommentRowNumber9.
    • CommentAuthorThomas Holder
    • CommentTimeOct 13th 2020
    • (edited Oct 13th 2020)

    I replaced your link with a (hopefully in this context more suggestive) link to falling factorial where I terminologically highlighted the binomial coefficients.

    To sort out the messy details of the Crossley-Nerode reference I count on the energetic model theorist.

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 14th 2020

    Slightly off-topic The category FinSet monoFinSet_{mono} appears under the name FIFI in the work of algebraists/representation theorists. Apparently it does a lot of cool things. I do wonder how any of these relate to the sheaves on it, though not terribly seriously.

  1. Re #10: Thanks for raising that! I think it would be good to make a page for this category, linked to from Schanuel topos, and mentioning this line of work in representation theory. No times myself just now though…!

    • CommentRowNumber12.
    • CommentAuthorThomas Holder
    • CommentTimeOct 16th 2020
    • (edited Oct 16th 2020)

    Minor clarification concerning [FinSet mono,Set][FinSet_mono,Set] added.

    How about generalizing to (,1)(\infty,1)-toposes !?

    I guess it still makes sense to define Sh ((Grpd fin) mono op,J at)Sh_\infty((\infty Grpd_{fin})_{mono}^{op},J_{at}) where (Grpd fin) mono(\infty Grpd_{fin})_{mono} is the (,1)(\infty, 1)-category of finite homotopy types with monomorphisms (aka (-1)-truncated morphisms) as morphisms and J atJ_{at} is generated by singletons.

    diff, v21, current

    • CommentRowNumber13.
    • CommentAuthormaxsnew
    • CommentTimeSep 13th 2022
    • CommentRowNumber14.
    • CommentAuthorvarkor
    • CommentTimeJul 21st 2023
    • (edited Jul 21st 2023)

    Add a cross-reference to FinInj.

    diff, v24, current