Timeline for Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$
Current License: CC BY-SA 4.0
12 events
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| Jun 4, 2019 at 13:14 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jun 4, 2019 at 12:56 | history | edited | Ali Taghavi |
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| Jun 4, 2019 at 7:45 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jun 3, 2019 at 2:56 | answer | added | Tim Campion | timeline score: 9 | |
| Jun 3, 2019 at 1:14 | comment | added | Todd Trimble | Naturality in the arguments $G$ and $H$ follows the well-known definition (compatibility between the isomorphism $\phi_{G, H}: F(G \times H) \to F(G) \otimes F(H)$ and morphisms $G \to G', H \to H'$ as expressed in the form of commutative squares). For further "coherence conditions", see ncatlab.org/nlab/show/monoidal+functor and ncatlab.org/nlab/show/symmetric+monoidal+functor, which include consideration of not just the monoidal products but also the monoidal units $1, \mathbb{Z}$. By the way, don't you mean to replace "on" with "to" in the question? | |
| Jun 3, 2019 at 0:44 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jun 3, 2019 at 0:39 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jun 3, 2019 at 0:33 | comment | added | Ali Taghavi | @ToddTrimble could you please suggest some naturality properties which I can add to my question? | |
| Jun 3, 2019 at 0:32 | comment | added | Ali Taghavi | @ToddTrimble to be honest the Leray Hirsch theorem, the Kunneth formula and the functor $X\mapsto C(X)$ on topological space were an indirect motivation for this question. | |
| Jun 3, 2019 at 0:28 | comment | added | Ali Taghavi | @ToddTrimble Thank you for your very helpful comment. Yes you are right. We should consider isomorphisms rather than equality. I revise the question. But I was not thinking to some naturalitiy properties. But the naturality you pointed out make the question more meaningfull. | |
| Jun 3, 2019 at 0:22 | comment | added | Todd Trimble | I feel this question could stand more structure. First, let's use an isomorphism $\cong$ in place of an equality; otherwise it's hard to make sense of the question. Then: should the isomorphism be natural? Should we be demanding more than mere naturality: should the isomorphism get along (be compatible) with associativity or symmetry isomorphisms? Do you care only about existence of an isomorphism (that satisfies to-be-specified properties), or should the choice of isomorphism be part of the structure considered? Etc. | |
| Jun 3, 2019 at 0:09 | history | asked | Ali Taghavi | CC BY-SA 4.0 |