Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 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 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 nforum 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 sheaves 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

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 8th 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded missing parentheses. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/11">v11</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   Added missing parentheses.

   Anonymous

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeMay 25th 2022
   • PermaLink
   Author: varkor
   Format: MarkdownItexMention left- and right-strengths. <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/12">v12</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   Mention left- and right-strengths.

   diff, v12, current

3. • CommentRowNumber3.
   • CommentAuthorvarkor
   • CommentTimeMay 3rd 2023
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdd some cross-references. <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/15">v15</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   Add some cross-references.

   diff, v15, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadjusted the Idea-paragraph in particular I added a line that $V$-strong functors make sense between $V$-tensored $V$-enriched categories and hence deleted this sentence further down: > The first thing to notice about (covariant) **tensorial strengths** is that they attach to a [[functor]] from a [[monoidal category]] to itself, say $T: V \to V$. (The concept doesn't make much immediate sense if $T$ is a functor between different monoidal categories.) added pointer to: * [[Kruna S. Ratkovic]], *Tensorial Strength*, Section 3 of: *Morita theory in enriched context* (2012) &lbrack;[arXiv:1302.2774](https://arxiv.org/abs/1302.2774), [hal:tel-00785301](https://theses.hal.science/tel-00785301/document)&rbrack; * [[Tarmo Uustalu]], *Strong functors, Strong monads*, in: *[Containers for effects and contexts](https://www.ioc.ee/~tarmo/ssgep15/)*, Lecture notes (2015) &lbrack;[pdf](https://www.ioc.ee/~tarmo/ssgep15/ssgep-1a.pdf)&rbrack; * [[Dylan McDermott]], [[Tarmo Uustalu]], Section 3 of: *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 &lbrack;[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/17">v17</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   adjusted the Idea-paragraph

   in particular I added a line that $V$-strong functors make sense between $V$-tensored $V$-enriched categories

   and hence deleted this sentence further down:

   The first thing to notice about (covariant) tensorial strengths is that they attach to a functor from a monoidal category to itself, say $T: V \to V$. (The concept doesn’t make much immediate sense if $T$ is a functor between different monoidal categories.)

   added pointer to:

   • Kruna S. Ratkovic, Tensorial Strength, Section 3 of: Morita theory in enriched context (2012) [arXiv:1302.2774, hal:tel-00785301]

   • Tarmo Uustalu, Strong functors, Strong monads, in: Containers for effects and contexts, Lecture notes (2015) [pdf]

   • Dylan McDermott, Tarmo Uustalu, Section 3 of: What Makes a Strong Monad?, EPTCS 360 (2022) 113-133 [arXiv:2207.00851, doi:10.4204/EPTCS.360.6]

   diff, v17, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded the warning ([here](https://ncatlab.org/nlab/show/tensorial+strength#WarningOnTerminology)) that [Kock 1972](https://ncatlab.org/nlab/show/tensorial+strength#Kock72) said "strong functor" for "enriched functor" <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/18">v18</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   Added the warning (here) that Kock 1972 said “strong functor” for “enriched functor”

   diff, v18, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have given the paragraphs alluding to some work by C. S. Peirce an Examples-environment ([here](https://ncatlab.org/nlab/show/tensorial+strength#ForHeytingAlgebras)) but it remains a little mysterious without further details or references. In the course of this, I have taken the liberty of deleting this lead-in paragraph: > There's rather a lot more one could say about strengths, and I may come back to more of that later, but I would like to say that strengths are kind of a trade secret. The first mathematician I know of who intuitively grasped strength was C.S. Peirce! <a href="https://ncatlab.org/nlab/revision/diff/tensorial+strength/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensorial+strength/18">v18</a>, <a href="https://ncatlab.org/nlab/show/tensorial+strength">current</a>

   I have given the paragraphs alluding to some work by C. S. Peirce an Examples-environment (here) but it remains a little mysterious without further details or references.

   In the course of this, I have taken the liberty of deleting this lead-in paragraph:

   There’s rather a lot more one could say about strengths, and I may come back to more of that later, but I would like to say that strengths are kind of a trade secret. The first mathematician I know of who intuitively grasped strength was C.S. Peirce!

   diff, v18, current