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.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 9th 2010
   • (edited Jun 9th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI'm reading the nLab page and Kelly's book, and I'm unfortunately unable to tell what the issue is with just extending the definition of naturality in the naïve way. Is extranaturality a stronger, weaker, or incomparable condition to honest naturality?

   I’m reading the nLab page and Kelly’s book, and I’m unfortunately unable to tell what the issue is with just extending the definition of naturality in the naïve way. Is extranaturality a stronger, weaker, or incomparable condition to honest naturality?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 9th 2010
   • (edited Jun 9th 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDid you look at the nLab page [[extranatural transformation]] (which I recommended to you elsewhere)? You could say that the notions are "incomparable" but closely related. The archetype is the evaluation map $[x, y] \otimes x \to y$ which is natural in $y$ and extranatural in $x$. A general rule is that in a closed monoidal category, in which $(- \otimes x) \dashv [x, -]$, any transformation of the form $F(x) \to [G(x), a]$, natural in $x$, gives rise to an extranatural transformation $F(x) \otimes G(x) \to a$, and conversely. And any transformation of the form $a \otimes F(x) \to G(x)$ gives rise to an extranatural transformation of the form $a \to [F(x), G(x)]$, and conversely. An intuitive way of grasping extranaturality is through string diagrams. Instances of naturality correspond to strings or wires running vertically, and instances of extranaturality correspond to cups and caps. In a closed monoidal context, as for example in compact closed categories, the closure allows one to bend "natural wires" into extranatural ones, or unbend extranatural ones into natural ones, if you catch my drift. There are a few lemmas or propositions at [[extranatural transformation]] which are usefully visualized in terms of string diagrams, including the "yanking" lemma, of which triangular equations in compact closed categories are a famous instance. Try reading the nLab page, and ask here again if you have any questions.

   Did you look at the nLab page extranatural transformation (which I recommended to you elsewhere)?

   You could say that the notions are “incomparable” but closely related. The archetype is the evaluation map $[x, y] \otimes x \to y$ which is natural in $y$ and extranatural in $x$. A general rule is that in a closed monoidal category, in which $(- \otimes x) \dashv [x, -]$, any transformation of the form $F(x) \to [G(x), a]$, natural in $x$, gives rise to an extranatural transformation $F(x) \otimes G(x) \to a$, and conversely. And any transformation of the form $a \otimes F(x) \to G(x)$ gives rise to an extranatural transformation of the form $a \to [F(x), G(x)]$, and conversely.

   An intuitive way of grasping extranaturality is through string diagrams. Instances of naturality correspond to strings or wires running vertically, and instances of extranaturality correspond to cups and caps. In a closed monoidal context, as for example in compact closed categories, the closure allows one to bend “natural wires” into extranatural ones, or unbend extranatural ones into natural ones, if you catch my drift. There are a few lemmas or propositions at extranatural transformation which are usefully visualized in terms of string diagrams, including the “yanking” lemma, of which triangular equations in compact closed categories are a famous instance.

   Try reading the nLab page, and ask here again if you have any questions.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 9th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI think I've got it now, but you should probably include an explanation of the string diagrams. Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c?

   I think I’ve got it now, but you should probably include an explanation of the string diagrams. Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 9th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex> you should probably include an explanation of the string diagrams. Sadly, I don't know how to draw up or deal with pictures like that, and the pictures needed for a full explanation aren't really there. Maybe you can teach me how to draw. I really need someone to tutor me so that I can continue with my page on surface diagrams. :-) > Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c? (Yes, or say something like that.) Good question; I don't know. We could try something like that.

   you should probably include an explanation of the string diagrams.

   Sadly, I don’t know how to draw up or deal with pictures like that, and the pictures needed for a full explanation aren’t really there. Maybe you can teach me how to draw. I really need someone to tutor me so that I can continue with my page on surface diagrams. :-)

   Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c?

   (Yes, or say something like that.) Good question; I don’t know. We could try something like that.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 9th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWell, if I say that a transformation $F(b,b,c) \to c$ is extranatural in $b$, it seems to me pretty clear that I am referring to the source, since $b$ doesn't appear in the target.

   Well, if I say that a transformation $F(b,b,c) \to c$ is extranatural in $b$, it seems to me pretty clear that I am referring to the source, since $b$ doesn’t appear in the target.

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 9th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexWhen I talk about something has property x in a certain variable, I use it to mean "in the coordinate(s) filled by that letter".

   When I talk about something has property x in a certain variable, I use it to mean “in the coordinate(s) filled by that letter”.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 10th 2010
   • (edited Jun 10th 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI agree with Mike that it would be pointlessly redundant to say "$F(b, b, c) \to G(c)$ is source-extranatural in $b$", and I can't think of a natural situation where you'd need to speak of extranaturality of a transformation where it wasn't clear which type of extranaturality was involved. So I currently think it's is an unnecessarily fussy distinction after all. (Should that situation ever arise, we can revisit the issue.) Max Kelly went further: he would just say "natural" to cover any case, whether ordinary natural or extraordinary natural. And again for exactly the same reason as above: if "natural" means either, then there is an unambiguous interpretation of what "$F(b, b, c) \to G(c)$ is natural in $b$ and $c$" means for either.

   I agree with Mike that it would be pointlessly redundant to say “$F(b, b, c) \to G(c)$ is source-extranatural in $b$”, and I can’t think of a natural situation where you’d need to speak of extranaturality of a transformation where it wasn’t clear which type of extranaturality was involved. So I currently think it’s is an unnecessarily fussy distinction after all. (Should that situation ever arise, we can revisit the issue.)

   Max Kelly went further: he would just say “natural” to cover any case, whether ordinary natural or extraordinary natural. And again for exactly the same reason as above: if “natural” means either, then there is an unambiguous interpretation of what “$F(b, b, c) \to G(c)$ is natural in $b$ and $c$” means for either.