Timeline for Effects of "weak" vs. "strict" categories in Eckmann-Hilton arguments
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10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 13, 2010 at 5:36 | vote | accept | Sridhar Ramesh | ||
| Feb 13, 2010 at 1:35 | comment | added | Mike Shulman | I think the answer to your question is "yes." Regarding the theory of categories as a multicategory, perhaps you are looking for something like fc-multicategories / virtual double categories? ncatlab.org/nlab/show/fc-multicategory | |
| Feb 12, 2010 at 2:38 | comment | added | Sridhar Ramesh | (Though I'm still curious about the question I asked in the above comments. But I can understand if perhaps I phrased things so incoherently as that it's difficult to answer) | |
| Feb 11, 2010 at 13:03 | comment | added | Tom Leinster | Sridhar, the diagrams in the Eckmann-Hilton clock live in a bicategory, and the 2-cells marked $\ell$ are unit coherence isomorphisms. The setup is this: we've got some object A of some bicategory, and we've got 2-cells alpha and beta from 1_A to 1_A. The 2-cell $\ell$ is the coherence isomorphism $1_A \circ 1_A \to 1_A$. | |
| Feb 11, 2010 at 7:13 | comment | added | Sridhar Ramesh | Is it something like a higher-dimensional analogue of that same phenomenon? (Man, these character limits are brutal... this and the last two posts were meant to be one single comment) | |
| Feb 11, 2010 at 7:12 | comment | added | Sridhar Ramesh | (e.g., if A, B, and C happened to all be identical, in addition to composing the left and right morphisms, there would also be simply projection the left morphism, projecting the right morphism, producing the square of the left morphism, etc.)? That is, the theory of categories doesn't concern itself with such cases as where the three objects in binary composition all happen to accidentally line up; it can't see all these functions from Hom(A, A) x Hom(A, A) to Hom(A, A) for objects A, and so it doesn't impose coherence equalities between them. | |
| Feb 11, 2010 at 7:12 | comment | added | Sridhar Ramesh | Is the idea here analogous to the fact that, in some sense, the theory of ordinary categories (which I want to describe using a multicategory of some sort, but can't quite, but suppose I could if I had the multicategory live in some fancy category other than Set) sees one and only one function from Hom(A, B) x Hom(B, C) to Hom(A, C) for objects A, B, and C [binary composition], even though, for any particular actual objects in an actual category, there may be many non-equivalent functions of this type which can be built out of the structure of a category | |
| Feb 11, 2010 at 6:56 | comment | added | Sridhar Ramesh | I'm not sure how to read the diagram you linked to. It looks like a diagram of a plain-vanilla Eckmann-Hilton argument, except for the $l$s and $l^{-1}$ pieces, and I'm not sure what those are. That having been said, I <i>think</i> I understand anyway. At least, the last paragraph above seems like what I was thinking the answer was anyway (as for why there's no demand for a coherence isomorphism between the path all the way around and the identity). But let me make sure I understand: (Question coming in a followup comment with more characters left...) | |
| Feb 11, 2010 at 5:54 | history | answered | Mike Shulman | CC BY-SA 2.5 |