Timeline for How can you unitalize a higher category?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2017 at 21:13 | comment | added | მამუკა ჯიბლაძე | It would correspond to declaring $f\in\hom_{C'}(x,y)$ to be a pair$$\left\langle f\cdot:\hom_C(\_,x)\to\hom_C(\_,y),\cdot f:\hom_C(y,\_)\to\hom_C(x,\_)\right\rangle$$such that$$(\alpha\cdot f)\circ\beta=\alpha\circ(f\cdot\beta)$$for any $\alpha\in\hom_C(y,y')$, $\beta\in\hom_C(x',x)$. | |
| Mar 11, 2017 at 21:11 | comment | added | Theo Johnson-Freyd | @YonatanHarpaz That paper is very interesting. I have begun reading it. In the example I care about, it is possible (I go back and forth about how likely it is) that every object has an identity morphism, at least in some naive sense. So I think your paper might in fact be just what I need for the actual project... | |
| Mar 11, 2017 at 21:09 | comment | added | Theo Johnson-Freyd | @მამუკაჯიბლაძე Very cool. I did not know that construcion. | |
| Mar 11, 2017 at 20:32 | comment | added | მამუკა ჯიბლაძე | There also is the bimultiplication algebra $M_A$ (see e. g. Extensions and obstructions for rings by Mac Lane). In a sense it is "even more interesting" than the multiplier algebra: it gives a crossed bimodule$$0\to Z_A\to A\to M_A\to P_A\to0$$which seems to be the proper analog of$$0\to Z(G)\to G\to\operatorname{Aut}(G)\to\operatorname{Out}(G)\to0.$$Might be more useful as the kernel is smaller than with the one-sided multipliers. | |
| Mar 11, 2017 at 20:25 | comment | added | Theo Johnson-Freyd | it in general comes from abstract nonsense of adjunctions. I certainly enjoy abstract nonsense, but any individual left adjoint isn't too rich. On the other hand, construction of the multiplier algebra is not a functor, and yet nevertheless it is "natural". This I find very rich. | |
| Mar 11, 2017 at 20:23 | comment | added | Theo Johnson-Freyd | @PaceNielsen With Heinrich's comments, I have removed the language you found disparaging. (I didn't intend "entire" disparagement, and do apologize.) I do want to clarify the difference between procedures and their outputs. I make no claim about whether the outputs of these unitalizations are interesting or not for a given input. Simply that (free) unitalization, qua procedure, is more straightforward and regular-behaving than construction of the multiplier algebra. For instance, free unitalization is a left adjoint to a forgetful functor, and pretty much everything I think one can say about | |
| Mar 11, 2017 at 20:15 | comment | added | Theo Johnson-Freyd | @HeinrichD These are good names, thanks! I have updated the question. | |
| Mar 11, 2017 at 20:14 | history | edited | Theo Johnson-Freyd | CC BY-SA 3.0 |
Changed vocabulary
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| Mar 10, 2017 at 11:35 | comment | added | HeinrichD | Concerning the pre-question, I suggest the names "unitalization" and "multiplier algebra". | |
| Mar 9, 2017 at 21:33 | comment | added | Yonatan Harpaz | It's not exactly what you asked for, but maybe you will find arxiv.org/abs/1210.0212 relevant. | |
| Mar 9, 2017 at 11:13 | comment | added | fosco | You might be interested in the conceptual framework proposed in this paper, although there are no higher categories in it. Read in particular S7.3 | |
| Mar 8, 2017 at 21:51 | comment | added | Benjamin Steinberg | BTW the "interesting Unitilization" can produce uninteresting results. Let X be your favorite set and define a product by xy=x. Then this gives a semigroup whose Unitilization is trivial. | |
| Mar 8, 2017 at 20:13 | comment | added | Theo Johnson-Freyd | @PaceNielsen I agree the one I find more interesting is not always an embedding, although there is always a map. I don't apologize for this choice of valuative word --- of course it is intended to (slightly) provoke, but mostly I think that by clearly broadcasting my own biases, I can get answers that are more useful for my application. In my case, I have some objects which have identities and some which (as far as I can tell) do not, and it is important that I unitalize without destroying the identities I already have. | |
| Mar 8, 2017 at 19:20 | comment | added | Benjamin Steinberg | The same problem Pace points out already comes up for embedding semigroups in monoids (so the one-object case). If the semigroup does not act faithfully on the right of itself, then taking the endomorphisms of the right regular representation do not give an embedding. | |
| Mar 8, 2017 at 19:05 | comment | added | Pace Nielsen | The first construction goes by the name "Dorroh extension by $\mathbb{Z}$." [In ring theory this is actually a very nice, natural, and interesting construction. So in the future, you might be careful about using judgemental words.] The second construction [which is also nice, natural, and interesting] does not embed $A$ in a unital ring in general. Consider the case where $A=\{0,x\}$ has trivial multiplication and $x+x=0$. | |
| Mar 8, 2017 at 18:43 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |