Timeline for Dennis trace map for stable $\infty$-category, naively
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16 events
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| Nov 17, 2020 at 16:06 | comment | added | Denis Nardin | I believe one can extract an argument of the type you're looking for from the proof of lemma 3.2.10 in Madsen's Algebraic K-theory and traces. I don't have time to do it right now, but I plan to go back to this and write an answer following these lines if I ever have the time to breathe again :). (essentially you're asking why $THH_0(S_2C)=THH_0(C)×THH_0(C)$) | |
| Apr 9, 2020 at 20:13 | comment | added | D.-C. Cisinski | The argument about the universal property of $K$-theory is diagram chasing (if you look at the proof). In fact, if you restrict to $K_0$ (i.e. gorget about higher coherence), this becomes completely elementary. Otherwise, you may be interested by the notion of shadow introduced by Shulman and Ponto in their paper "Duality and traces for indexed monoidal categories". The case of $THH$ is discussed explicitly in this paper of Campbell and Ponto: arXiv:1803.01284.The later is the point of view of $THH$ as a trace. | |
| Apr 8, 2020 at 15:42 | comment | added | Simon Henry | I'm aware of the characterization of THH(A) as a trace in the monoidal category of presentable stable $\infty$-categories, but I do not quite know how to use this to connect my question to the papers you mentioned, which apply to stable monoidal categories. I'm also aware of the construction of the trace map using the universal property of K-theory, in fact as mentioned in the question I know how to directly deduce the additivity of the THH(A) valued trace from the additivity of THH(A) itself, but I'm looking for something more direct/elementary (only involving diagram chasing in $A$). | |
| Apr 8, 2020 at 15:33 | comment | added | D.-C. Cisinski | Otherwise, the existence of Dennis trace map is more about the universal property of $K$-theory: since $THH$ is additive (it takes Verdier quotients to fiber sequences of spectra) the spectrum of maps $K\to THH$ is equivalent to $THH(finite spectra)$, the $\pi_0$ of which being $\mathbf Z$ with the canonical element $1$. And if you take $\pi_0(THH(A))$, the Dennis trace map really is the very usual trace of the identity. | |
| Apr 8, 2020 at 15:31 | comment | added | D.-C. Cisinski | It is the category of stable $\infty$-categories which is monoidal here, not $A$... | |
| Apr 8, 2020 at 15:30 | comment | added | Simon Henry | @Denis-Charles: Well the "trace" I'm considering here is the 'universal one' which sends an arrow $f:a \rightarrow a \in A$ to its image in THH(A) (defined as the coend in the question). To construct Dennis trace map, you need to show that this trace is additive (and other higher coherence condition), which is equivalent to proving that all traces functions are additive. I do not know how to define this 'trace' with value in THH(A) in terms of strong duality, especially that here A is not assumed to be monoidal. | |
| Apr 8, 2020 at 15:20 | comment | added | D.-C. Cisinski | Aren't all traces define like this? Also, I am not sure I understand your question, since the Dennis trace maps goes to THH, as opposed as to going from it. Another way to see THH consists in saying that there is a symmetric monoidal bicategory of stable $\infty$-categories (with bimodules as 1-cells) and that the trace map defines a functor from stable $\infty$-categories to spectra (since spectra form endomorphisms of the unit). The even more general framework where to look might be the one of higher categories in which every cell not in the top dimension has adjoints. | |
| Apr 8, 2020 at 12:47 | comment | added | Simon Henry | @Denis-Charles : Thank you very much, that's very interesting! I haven't finish reading them yet (I definitely will) but from what I gathered so far, I'm not sure they answer the question however: they all prove additivity when the trace is defined as $Tr(f) = ev \circ (f \otimes Id) \circ coev$ for $f$ an endomorphism of a dualizable object. It does not seems to prove the results for a general "trace function" in the sense of a map out of THH(A) for a stable but not necessarily monoidal category A. But I'll keep reading, maybe the proof can be adapted... | |
| Apr 8, 2020 at 8:00 | comment | added | D.-C. Cisinski | But, historically, the first conceptual approach to this is this paper by Peter May math.uchicago.edu/~may/PAPERS/AddJan01.pdf | |
| Apr 8, 2020 at 7:59 | comment | added | D.-C. Cisinski | This may be complemented by this paper of Galauer arxiv.org/abs/1303.0153 | |
| Apr 8, 2020 at 7:58 | comment | added | D.-C. Cisinski | This paper of Groth, Ponto and Shulman seems to be an answer, doesn’t it? arxiv.org/abs/1212.3277 | |
| Apr 7, 2020 at 21:25 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Apr 7, 2020 at 18:55 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Apr 7, 2020 at 18:51 | comment | added | Denis Nardin | As much as I'd love to take credit, it's the Dennis trace map :) | |
| Apr 7, 2020 at 18:47 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Apr 7, 2020 at 15:49 | history | asked | Simon Henry | CC BY-SA 4.0 |