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.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 1st 2022
    • (edited Aug 1st 2022)

    The new notion of weak **-autonomous category is a very simple generalization of the one of **-autonomous category, and a specialization of the one of closed monoidal category, which allows to speak about duality in more situations. For example, the category Vec 𝕂Vec_{\mathbb{K}} is a weak **-autonomous category but is not a **-autonomous category.

    Any return/collaboration would be very much appreciated.

    v1, current

    • CommentRowNumber2.
    • CommentAuthormaxsnew
    • CommentTimeAug 1st 2022
    • (edited Aug 1st 2022)

    The morphism d:A(A)d : A \multimap (A \multimap \bot) \multimap \bot is the unit for the continuation monad, and so this is asking that the unit of this monad is a mono. I’m not sure this has much in common with *-autonomous categories, but it is typical in programming language semantics that the unit of the monad is mono, or even stronger that the unit of the monad is the equalizer of ηT\eta T and Tη:TT 2T\eta : T \to T^2 (see Moggi ’88, Computational lambda-calculus and monads)

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 1st 2022
    • (edited Aug 1st 2022)

    The idea comes from models of differential linear logic which are **-autonomous categories but this is often too restrictive. For instance a lot of categories of topological vector spaces must verify this definition. So this is already linked with programming language semantics. I’m going to take a look at what you say! It seems very to the point in fact.

    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 1st 2022
    • (edited Aug 1st 2022)

    I’m wondering if the unit of the monad is this equalizer in Vec 𝕂Vec_{\mathbb{K}}.

    I don’t know what are the multiplication and the unit of the continuation monad?

    Also, it seems to be related to **-autonomous categories by the paper “Linear continuation and duality” (Paul-André Melliès, Nicolas Tabareau), 2008 on the nlab page.