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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexI added a few links to [[pasting diagram]] that will be happy to be filled in with content someday... In other words, I added double brackets to a few places where "pasting diagram" occurred.

   I added a few links to pasting diagram that will be happy to be filled in with content someday…

   In other words, I added double brackets to a few places where “pasting diagram” occurred.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI added some quick Idea-section and a reference to [[pasting diagram]], but not really good yet. Hopefully it inspires somebody to improve it.

   I added some quick Idea-section and a reference to pasting diagram, but not really good yet. Hopefully it inspires somebody to improve it.

3. • CommentRowNumber3.
   • CommentAuthorEric
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks Urs. Those references will provide some good train reading material :) I plan to spend some more time on Street's AOS for now though. It is cool to relate the 1-cocycle condition $\partial C = 0$ to a triangle of morphisms. Ditto for 2-cocycle. it made me think of "lengths" and "areas". If the area of a triangle is zero, the sum of the length of the two shorter edges is equal to the length of the third longer leg, i.e. $$L_{12} + L_{01} = L_{02}.$$ If the volume of a tetrahedron is zero, the area of the opposite quadrilaterals are equal, i.e. $$A_{123} + A_{013} = A_{023} + A_{012}.$$ So if you squash an $n$-simplex in one-dimension, the opposite $(n-1)$-complexes have the same $(n-1)$-volume.

   Thanks Urs. Those references will provide some good train reading material :)

   I plan to spend some more time on Street’s AOS for now though. It is cool to relate the 1-cocycle condition $\partial C = 0$ to a triangle of morphisms. Ditto for 2-cocycle. it made me think of “lengths” and “areas”. If the area of a triangle is zero, the sum of the length of the two shorter edges is equal to the length of the third longer leg, i.e.

   $$L_{12} + L_{01} = L_{02}.$$

   If the volume of a tetrahedron is zero, the area of the opposite quadrilaterals are equal, i.e.

   $$A_{123} + A_{013} = A_{023} + A_{012}.$$

   So if you squash an $n$-simplex in one-dimension, the opposite $(n-1)$-complexes have the same $(n-1)$-volume.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> It is cool to relate the 1-cocycle condition ∂C=0 to a triangle of morphisms. This I had tried to spell out in some detail at [[group cohomology]], by the way.

   It is cool to relate the 1-cocycle condition ∂C=0 to a triangle of morphisms.

   This I had tried to spell out in some detail at group cohomology, by the way.

5. • CommentRowNumber5.
   • CommentAuthorEric
   • CommentTimeApr 16th 2010
   • (edited Apr 16th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks for the pointer, but [[group cohomology]] is still a bit over my head. It re-enforces a thought that has crept into my head lately though. I wonder if category theory is really what I thought it was. You know in most of what I do I am interesting in cochains "on the nose". Passing to cohomology is interesting for some, but not for me. I like to work directly with cochains. In the stuff we did, if you pass to cohomology, then even we'd have $ab = (-1)^{|a||b|} ba$, but we don't pass to cohomology and that is what gives the interesting stuff. I'm beginning to get the sense that category theory is about "passing to cohomology" and what I want is some kind of "pre-category theory" or something. When I look at things like the exchange law, they don't make "geometric" sense to me but make "homotopic" sense. For example, if you have two solid circles in a plane intersecting at a single point of their boundary, then I would call that "geometrically" different (i.e. it has two disjoint interiors) than a single circle although the two are topologically the same (kind of). The exchange law seems to say something like "a solid figure 8 is the same as a solid figure 0". Or something....

   Thanks for the pointer, but group cohomology is still a bit over my head. It re-enforces a thought that has crept into my head lately though. I wonder if category theory is really what I thought it was.

   You know in most of what I do I am interesting in cochains “on the nose”. Passing to cohomology is interesting for some, but not for me. I like to work directly with cochains. In the stuff we did, if you pass to cohomology, then even we’d have $ab = (-1)^{|a||b|} ba$, but we don’t pass to cohomology and that is what gives the interesting stuff.

   I’m beginning to get the sense that category theory is about “passing to cohomology” and what I want is some kind of “pre-category theory” or something. When I look at things like the exchange law, they don’t make “geometric” sense to me but make “homotopic” sense. For example, if you have two solid circles in a plane intersecting at a single point of their boundary, then I would call that “geometrically” different (i.e. it has two disjoint interiors) than a single circle although the two are topologically the same (kind of).

   The exchange law seems to say something like “a solid figure 8 is the same as a solid figure 0”. Or something….

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2010
   • (edited Apr 16th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I'm beginning to get the sense that category theory is about "passing to cohomology" an No, the opposite is true! Passing to cohomology is [[decategorification]] of hom-$\infty$-categories. The true point of category theory is not to do that. At least not mandatorily. Instead, category theory comes and says: "See all the things you all have been studing for centuries are really decategorifications of this and that. If you realize this, your life will become much happier."

   I’m beginning to get the sense that category theory is about “passing to cohomology” an

   No, the opposite is true!

   Passing to cohomology is decategorification of hom-$\infty$-categories.

   The true point of category theory is not to do that. At least not mandatorily. Instead, category theory comes and says: “See all the things you all have been studing for centuries are really decategorifications of this and that. If you realize this, your life will become much happier.”

7. • CommentRowNumber7.
   • CommentAuthorEric
   • CommentTimeApr 16th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexOk ok! :) By the way, did my comment about "lengths" and "areas" make sense? That is pretty imho.

   Ok ok! :)

   By the way, did my comment about “lengths” and “areas” make sense? That is pretty imho.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded more literature to the References-section at [[pasting diagram]]

   added more literature to the References-section at pasting diagram

9. • CommentRowNumber9.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 16th 2017
   • (edited Jul 16th 2017)
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItexAdded a reference to Schommer-Pries' thesis to [[pasting diagram]], since this is a recommendable introduction.

   Added a reference to Schommer-Pries’ thesis to pasting diagram, since this is a recommendable introduction.

10. • CommentRowNumber10.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 18th 2017
    • (edited Jul 18th 2017)
    • PermaLink
    Author: Peter Heinig
    Format: MarkdownItexRelated to [a discussion on string diagrams](https://nforum.ncatlab.org/discussion/1255/string-diagrams-for-closed-categories/#Comment_64103), [here is the revised version of Verity's thesis](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html) and, since Verity in "Appendix A" writes > For our purposes the approach of [40] seems most appropriate and we assume that the reader is familiar with that paper. here [is the article of Power's in question](http://www.sciencedirect.com/science/article/pii/002186939090229H) In Verity's thesis, starting from Definition A.0.7, _definitions_ of pasting schemes in a bicategorical context are given. In introductory writings, like the (very useful, to me at least) appendix-section of Schommer-Pries' thesis, or on the (current) nLab page on pasting diagrams, the latter are usually intuitively defined, saying things like "polygonal arrangement". [Terminological and nLab-technical notes: do others agree that, by and large, it is better to use, like Power and Verity, "pasting scheme" than "pasting diagram", since pasting "diagrams" are not diagrams in the usual technical sense? The clash with the the "scheme"s of algebraic geometry seems harmless, the concepts being even somewhat similar in spirit (and both these technical terms deriving from a more general sense of "scheme"). Should the, already existing, but lying-fallow-for-over-four-years [[pasting scheme]] be made "the" nLab entry on the matter, with [[pasting diagram]] being merged _into it_? Personally, I would favour this, i.e., pasting diagram -> pasting scheme. If there is general agreement, I would do so in the next few days. ]

    Related to a discussion on string diagrams,

    here is the revised version of Verity’s thesis

    and, since Verity in “Appendix A” writes

    For our purposes the approach of [40] seems most appropriate and we assume that the reader is familiar with that paper.

    here

    is the article of Power’s in question

    In Verity’s thesis, starting from Definition A.0.7, definitions of pasting schemes in a bicategorical context are given. In introductory writings, like the (very useful, to me at least) appendix-section of Schommer-Pries’ thesis, or on the (current) nLab page on pasting diagrams, the latter are usually intuitively defined, saying things like “polygonal arrangement”.

    [Terminological and nLab-technical notes: do others agree that, by and large, it is better to use, like Power and Verity, “pasting scheme” than “pasting diagram”, since pasting “diagrams” are not diagrams in the usual technical sense? The clash with the the “scheme”s of algebraic geometry seems harmless, the concepts being even somewhat similar in spirit (and both these technical terms deriving from a more general sense of “scheme”).

    Should the, already existing, but lying-fallow-for-over-four-years pasting scheme be made “the” nLab entry on the matter, with pasting diagram being merged into it? Personally, I would favour this, i.e., pasting diagram -> pasting scheme. If there is general agreement, I would do so in the next few days. ]

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 18th 2017
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexNo, pasting schemes are different from pasting diagrams. They should not be merged. > do others agree that, by and large, it is better to use, like Power and Verity, “pasting scheme” than “pasting diagram” No, I think that's a matter of opinion. Both should be written up properly.

    No, pasting schemes are different from pasting diagrams. They should not be merged.

    do others agree that, by and large, it is better to use, like Power and Verity, “pasting scheme” than “pasting diagram”

    No, I think that’s a matter of opinion. Both should be written up properly.

12. • CommentRowNumber12.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 18th 2017
    • PermaLink
    Author: Peter Heinig
    Format: MarkdownItexThanks for the response. To me it seems that Power and Verity on the whole avoid "pasting diagram". But I do not have much experience with the literature on this.

    Thanks for the response. To me it seems that Power and Verity on the whole avoid “pasting diagram”. But I do not have much experience with the literature on this.