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 beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration 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 lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string 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.
    • CommentAuthorMike Shulman
    • CommentTimeSep 22nd 2017

    Finally created funny tensor product. This is not really a very good name for a serious mathematical concept, but I don’t know of a better one.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 22nd 2017

    Seems like some useful characterisation in section 2 of Mark Weber’s Free Products of Higher Operad Algebras, arXiv:0909.4722, so I’ll add that in.

    How common is the use of “white” rather than “funny” as mentioned at Gray tensor product?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 22nd 2017

    I also added that this constitutes of one of the two symmetric monoidal closed structures on CatCat.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 22nd 2017

    I added to David’s addition, commenting that the other was of course the cartesian closed structure, and that both products are semicartesian.

    • CommentRowNumber5.
    • CommentAuthormaxsnew
    • CommentTimeSep 22nd 2017

    I added an explicit description as a pushout which I found in the Weber paper and I added it as an example to semicartesian category

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 22nd 2017

    I thought I saw somewhere once that the cartesian product, if it exists in a category, is terminal among all semicartesian monoidal structures? Probably easy if true, but I’ve not sat down to work it out.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeSep 22nd 2017

    Re #6 that sounds plausible, since if we have projections ABAA\otimes B \to A and ABBA\otimes B\to B they induce a map ABA×BA\otimes B \to A\times B

    Re #2 I’ve never heard “white” in the context of 1-categories, only in the context of 2-categories where it’s being compared to the “gray” as well as the “black” one, and even that I think I’ve only heard as a joke.

    • CommentRowNumber8.
    • CommentAuthormaxsnew
    • CommentTimeMay 4th 2018

    Add a section about “separate functoriality”. Terminology is a little awkward here: should we say “separately functorial bifunctor” or “separately functorial functor of many arguments”? I stuck with the awkward but at least brief “separately functorial functor” vs “jointly functorial functor”.

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 4th 2018

    I think I’d prefer “separately functorial map”.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2018

    Or “separately functorial operation”.

    • CommentRowNumber11.
    • CommentAuthorJohn Baez
    • CommentTimeMar 31st 2021

    Added reference to the paper showing Cat has just two symmetric monoidal closed structures.

    diff, v7, current

    • CommentRowNumber12.
    • CommentAuthorJohn Baez
    • CommentTimeMar 31st 2021

    Pointed out that premonoidal categories are monoids in Cat with its funny tensor product.

    diff, v7, 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)