Timeline for Examples of categorification
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| May 17 at 1:58 | comment | added | Todd Trimble | @SidharthGhoshal Oh, absolutely. I recommend Joyal's theory of species to your attention. (A species is a functor from the groupoid of finite sets and bijections to a suitable receiving category like the category of sets. The derivative $F'$ of a species $F$ is defined by $F'[S] = F[S+1]$. The free commutative monoid construction is the "analytic functor" induced from the terminal species; it is manifestly true that the derivative of the terminal species is again the terminal species.) But this comment box is too small to give such a hurried introduction. | |
| May 17 at 1:34 | comment | added | Sidharth Ghoshal | This example makes me wonder if there’s a categorification of the derivative such that the free commutative monoid is invariant under it. | |
| Oct 17, 2016 at 0:01 | comment | added | Todd Trimble | @geodude I'm having trouble thinking of a definitive example, but see for example here arxiv.org/pdf/math/0004133v1.pdf and maybe also Joyal's article in Springer Lecture Notes in Mathematics 1234, which can be seen as a deep meditation on categorifying $-\log(1-x)$ as the species of free Lie algebras. | |
| Oct 16, 2016 at 23:46 | comment | added | geodude | So I'm six years late but... who can give me a reference on this? | |
| Nov 8, 2013 at 16:30 | comment | added | Todd Trimble | @ColinTan Feel free to email me if you want to continue this discussion. I've had some ideas on this, but I much prefer not dragging the discussion out in comment boxes. | |
| Nov 8, 2013 at 15:38 | comment | added | user2529 | We could start by defining the exponential function on nonnegative integers, by trying to complete $FinSet$ under applications of the free monoid object $exp$. | |
| Nov 5, 2013 at 12:03 | comment | added | Todd Trimble | @ColinTan I suppose it might be possible, but I don't know of any natural or systematic way of producing such a monoidal category with coproducts (over which the monoidal product distributes), such that isomorphism classes of objects with these operations gives what you want. Interesting question. | |
| Nov 5, 2013 at 2:21 | comment | added | user2529 | What if we restrict the exponential function to the nonnegative real numbers? | |
| Nov 1, 2013 at 16:11 | comment | added | Todd Trimble | @ColinTan It's not so simple as that; there is no such monoidal category. We would want coproducts in the underlying category to play the role of addition, and we would also want additive inverses. But as an exercise, one can show that if a coproduct $x + y \cong 0$, then $x \cong 0 \cong y$. | |
| Nov 1, 2013 at 15:55 | comment | added | user2529 | To recover the exponential function on the real numbers, one chooses which monoidal category $C$? | |
| Jan 13, 2011 at 0:14 | comment | added | Todd Trimble | @Martin: thank you very much! The mathematics is indeed beautiful to contemplate; one of my favorite applications is to the structure of the Lie algebra operad as a categorified logarithm valued in superspaces, via the PBW theorem. I tried to say something about this a long time ago at math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf , but I'm afraid I didn't do justice to it... | |
| Jan 12, 2011 at 23:20 | comment | added | Martin Brandenburg | @Todd: What a fantastic example!! Really, this is one of my favorite MO answers so far. | |
| Oct 26, 2010 at 21:34 | comment | added | Jan Weidner | Thanks Todd for the additional explanation, this example is really nice! Thanks Tom for the additional information! | |
| Oct 26, 2010 at 16:25 | comment | added | Tom Goodwillie | And if you do your symmetrizing in a derived way, you get the free infinite loopspace, or $E_\infty$ monoid -- stable homotopy instead of homology -- spectrum objects instead of abelian group objects -- ... | |
| Oct 26, 2010 at 11:43 | history | edited | Todd Trimble | CC BY-SA 2.5 |
added 1309 characters in body
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| Oct 26, 2010 at 6:19 | comment | added | Jan Weidner | Can you explain that a bit? | |
| Oct 25, 2010 at 22:07 | history | answered | Todd Trimble | CC BY-SA 2.5 |