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bio website math.northwestern.edu/…
location Seattle, WA
age 21
visits member for 3 years, 10 months
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Graduate student at Northwestern. Feel free to email me at dwilson [at] u. northwestern. edu :)

14h
comment “abstract” description of geometric fixed points functor
(and monoidality for those follows from the same claim for spaces)
14h
comment “abstract” description of geometric fixed points functor
Well I guess you also need to know it deloops well. Then you'd get monoidality because smashing commutes with hocolims and everything is a hocolim of (deloopings of) suspension spectra.
14h
comment “abstract” description of geometric fixed points functor
Ah, I may have to add monoidality... I haven't thought about it.
15h
comment “abstract” description of geometric fixed points functor
I don't know if this is right, so I'll leave it as a comment. But my guess is that it is characterized by the properties: (i) it preserves homotopy colimits, and (ii) it makes the diagram commute between G-spaces, spaces, G-spectra, and spectra- i.e. geometric fixed points of a suspension spectrum are suspension spectra of geometric fixed points
Apr
6
comment Finite limits in the category of smooth manifolds?
We also have pullbacks for transverse maps, so- more generally- if you write the canonical presentation of a limit as a reflexive equalizer of some products, one could present the equalizer as a pullback and ask if the given maps are transverse. But I don't think this is the most general thing one can say...
Mar
27
comment Pushout of categories along embeddings gives homotopy pushout?
@DanRamras: Maybe it's worth noting that Thomason's notion is a homotopy pushout with respect to a different set of equivalences (namely, those maps of categories inducing an equivalence on their groupoid completions.)
Mar
23
comment Pushout of categories along embeddings gives homotopy pushout?
@AaronMazel-Gee: In the 1-category case cofibrations just need to be injective on objects, there's no condition on morphisms.
Mar
23
comment Pushout of categories along embeddings gives homotopy pushout?
@AaronMazel-Gee: I don't understand what you're saying. The types of maps the OP calls 'embeddings' are in particular cofibrations in the canonical model structure on Cat. The given diagram is a homotopy pushout in Cat, it's just that the nerve functor does not preserve pushouts (homotopy or otherwise).
Mar
23
comment Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Ah! Thank you, that answers my question.
Mar
23
comment Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Probably I'm confused by `maximal substring'. Shouldn't your example $X \rightarrow A \rightarrow X$ have only one $X$-component because every string of just $X$s is contained in $X \rightarrow X$ (the composite)? How are you ordering strings?
Mar
13
comment A question about the proof of Quillen's Theorem A
The nerve of a category and it's opposite are always weakly equivalent. Taking geometric realization inverts all your arrows.
Mar
6
comment Why is the derived tensor product only defined for bounded above derived categories?
It should be pointed out that life isn't perfect: you lose some theorems when you extend your complexes to be unbounded and use Spaltenstein's methods. For example, the proper base change theorem no longer holds in the same generality as in the bounded case. (See Higher Topos Theory Remark 6.5.4.3.) That said, if you're working with nice $X$ you should be okay.
Feb
9
comment Motivic derived algebraic geometry
I have no idea what I'm talking about... but if such a thing existed where your generalized ring objects lived in motivic spectra, then perhaps you could build a motivic tmf a la Lurie?
Feb
4
awarded  Popular Question
Feb
1
comment origin of spectral sequences in algebraic topology
@TomGoodwillie: I suppose that's true but I feel like those examples usually (i) factor through the stable category or (ii) end up not quite being spectral sequences towards the fringe (e.g. Bousfield-Kan sseqs). Do you know of an example which is neither of those?
Jan
31
comment origin of spectral sequences in algebraic topology
Every spectral sequence I've ever encountered has arisen from filtering some object in a stable homotopy theory (like a derived category, or spectra, etc. etc.) and applying a homological functor (like taking homotopy groups/homology groups, etc.). In good cases this yields a spectral sequence starting with the homological functor applied to the associated graded object, converging to the colimit of the filtered object. If you can stomach infty-categories, Lurie's Higher Algebra 1.2.2 has a nice exposition, otherwise just think "what I know from Weibel but nonabelian" and you'll be ok.
Jan
3
comment Small objects in categories
I think the fact that we had the initial/final object was incidental. That a point is a compact generator is what we're after. In the category of chain complexes of $R$-modules you'd start with $R$; closing under finite homotopy colimits would give finite complexes of free modules. Again, idempotent completing gives perfect complexes.
Jan
3
answered Small objects in categories
Jan
3
comment Inflection points on elliptic curves over a field of characteristic 2
@MP: that's neat! does this always work? why?
Dec
19
comment Reconstructing the Chow ring from the derived category
@Sasha: It doesn't look like Adeel wants to use the tensor structure, otherwise the result was known a long time ago by Thomason in more generality that we can reconstruct the whole scheme, and thus the Chow groups.