# November 24, 2009 {#week284}

A couple of weeks ago there was a meeting of the American Mathematical
Society here at UC Riverside. Mathematicians flooded in from across the
western US and even further. They gave hundreds of 20-minute talks,
drank lots of coffee, ate a few too many pastries, and chatted with each
other. Julie Bergner and I ran a session at this conference. My student
Chris Rogers took videos of the talks in our session, and you can see
them here:

1) Special session on homotopy theory and higher algebraic structures,
AMS Western Section Meeting, November 7-8, 2009. Talks available as
Quicktime videos at `http://math.ucr.edu/~jbergner/amsriverside09.htm`

These talks add up to a nice look at recent work on homotopy theory,
$n$-categories, and categorification - some of my favorite subjects. So,
I'd like to quickly summarize each talk and give some links to related
papers.

But first: something a bit less technical!

Last week I asked you to provide a nice proof of Proposition 10 from the
last book of Euclid - the one where he constructs the Platonic solids.
Euclid uses this proposition to construct the icosahedron, but it's
appealing in its own right. In modern language it says:

> Inscribe a regular pentagon, hexagon and decagon in a circle, and call
> their side lengths P, H and D. Then
>
> ::: {align="center"}
> P^2 = H^2 + D^2.
> :::

I find this fascinating. One reason is that it's simple but far from
obvious. Another is that it's secretly all about the golden ratio and
its role in 5-fold symmetry. And another is that Euclid's proof is
ingenious but not very intuitive - so it seems there should be a better
proof. For example, a proof that uses the icosahedron!

(Last week I gave a proof using algebra and trigonometry, but it wasn't
terribly interesting.)

The science fiction writer Greg Egan said that Euclid's proof was
"really dazzling, but it made me feel like he'd pulled a coin out from
behind my ear." Egan wrote a modernized version of this proof, which
you can see in the Addenda to ["Week 283"](#week283). But he then
went on to give a number of other proofs, including two that I like a
lot better:

2) nLab, Pentagon-decagon-hexagon identity,
`http://www.ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity`

One of these proofs uses the icosahedron. As I'd dreamt in
["Week 283"](#week283), it proceeds by showing that two right
triangles hiding in the icosahedron are congruent. Namely, the triangles
ABC and AB'C' shown here:

::: {align="center"}
![](icosahedron_with_right_triangles.gif)
:::

The other proof is purely $2$-dimensional. For this, Egan starts by
recalling Proposition 9 from the same book of Euclid. This result states
the main property of the "golden triangle".

What's a golden triangle? Well, if you draw a regular pentagon and
connect each vertex to every other, you'll get a pentagram in a
pentagon - but you'll also see lots of tall skinny isosceles triangles:

::: {align="center"}
![](dodecahedron/pentagram.png){width="400"}
:::

These are "golden triangles". They have angles of 36 degrees, 72
degrees, and 72 degrees. How many are there in this picture?

In Proposition 9, Euclid shows that for a golden triangle, the ratio of
the edge lengths is the golden ratio. Actually he shows something
equivalent, roughly this:

> Inscribe a regular hexagon and decagon in a circle. Then the ratio of
> their side lengths is the golden ratio:
>
> ::: {align="center"}
> D/H = \varphi = (-1 + √ 5)/2
>
> Why is this equivalent? Well, if you inscribe a regular decagon in a
> circle and draw lines from its center to its vertices, you get ten
> golden triangles. The long edge of each triangle is H, since the
> radius of the circle equals the edge of an inscribed circle. The short
> edge is D.
>
> Oddly, Euclid does not use Proposition 9 to prove Proposition 10, even
> though it's relevant and it comes right before! But Egan's proof
> uses it. Check out the [nLab
> entry](http://www.ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity)
> for details and pretty pictures. I think it's great that 21st-century
> technology is being used to improve a proof dating back to 300 BC.
>
> Now... on to the talks on homotopy theory and higher algebraic
> structures! I'm afraid the length of my summaries will be
> proportional to how much I understood. You can click on the talk
> titles to see the videos.
>
> Bright and early at 8 am on Saturday morning, Aaron Lauda kicked off
> the special session with a talk on "[Categorifying quantum
> groups](ams_2009/Lauda_AMS_UCR_2009.mov)". Luckily he'd come from
> the east coast, so he was wide awake, and his energy was contagious.
>
> From the very beginning of This Week's Finds you can see that I was
> interested in Crane and Frenkel's dream of categorified [quantum
> groups](http://en.wikipedia.org/wiki/Quantum_group). Now this dream is
> coming true! Aaron gave a great series of talks on this subject in
> Kyoto this February, and you can see them all here:
>
> 3) Aaron Lauda, Kyoto lectures:
>
> -   Categorification of quantum \mathfrak{sl}(2),
> -   Categorification of one half of the quantum group,
> -   Categorification of quantum \mathfrak{sl}(n),
> -   Cyclotomic quotients of the rings R(\nu).
>
> Available at `http://www.math.columbia.edu/~lauda/talks/kyoto/`
>
> At Riverside he explained what people often call the "Khovanov-Lauda
> algebra" R associated to a simply laced Dynkin diagram. This gives a
> way of categorifying the "positive part" of the corresponding
> quantum group.
>
> 4) Mikhail Khovanov and Aaron Lauda, A diagrammatic approach to
> categorification of quantum groups I-III, available as
> [`arXiv:0803.4121`](http://arxiv.org/abs/0803.4121),
> [`arXiv:0804.2080`](http://arxiv.org/abs/0804.2080), and
> [`arXiv:0803.3250`](http://arxiv.org/abs/0807.3250).
>
> Huh? Well, just as the group of matrices has a subgroup consisting of
> upper triangular matrices and a subgroup consisting of lower
> triangular matrices, so any quantum group breaks into a "positive
> part" and a "negative part", with a bit of overlap. It's easier to
> categorify either of these parts than the whole thing, because when
> you deal with the whole thing you get formulas involving negative
> numbers, which are harder to categorify.
>
> How does the algebra R help us categorify the positive part of a
> quantum group? Or, for that matter, the negative part? - the two parts
> look alike, so we randomly choose to work with the positive part.
>
> The answer is simple: we just form the category of representations of
> R!
>
> But how do we get back from this category to the positive part of the
> quantum group? In other words, how do we "decategorify"? Again, the
> answer is simple: just take its [Grothendieck
> group](http://ncatlab.org/nlab/show/Grothendieck+group)! A bit more
> precisely: we take the category of [finitely
> generated](http://en.wikipedia.org/wiki/Finitely_generated_module)
> [projective](http://en.wikipedia.org/wiki/Projective_module)
> R-modules, and look at isomorphism classes of these, and let these
> generate an abelian group with relations saying that direct sums
> should act like sums:
>
> \[M \oplus  N\] = \[M\] + \[N\]
>
> This gives a certain "integral form" of the positive part of the
> quantum group. If we tensor this with the complex numbers, we get the
> more familiar complex version of the quantum group.
>
> One of the great virtues of the Khovanov-Lauda algebra is that it has
> a nice presentation, with generators and relations given in terms of
> pretty pictures. This is great for computations. However, the
> presentation is a bit complicated, so I can't help but wonder where
> it came from. Maybe there's some nice geometry underlying the whole
> story?
>
> Indeed, Aaron has also worked on more geometrical approaches to
> categorifying quantum groups, at least for the simplest of simple Lie
> algebras, namely \mathfrak{sl}(2):
>
> 5) Aaron Lauda, A categorification of quantum \mathfrak{sl}(2), available as
> [`arXiv:0803.3652`](http://arxiv.org/abs/0803.3652).
>
> 6) Aaron Lauda, Categorified quantum \mathfrak{sl}(2) and equivariant cohomology
> of iterated flag varieties, available as
> [`arXiv:0803.3848`](http://arxiv.org/abs/0803.3848).
>
> But there are also lots of other people tackling the geometrical side
> of the story. One of these is Anthony Licata of Stanford University!
> Right after Aaron, he gave a talk on "[Categorification via quiver
> varieties](ams_2009/Licata_AMS_UCR_2009.mov)", based on these papers:
>
> 6) Sabin Cautis, Joel Kamnitzer, and Anthony Licata, Coherent sheaves
> and categorical \mathfrak{sl}(2) actions, available as
> [`arXiv:0902.1796`](http://arxiv.org/abs/0902.1796).
>
> 7) Sabin Cautis, Joel Kamnitzer, and Anthony Licata, Derived
> equivalences for cotangent bundles of Grassmannians via categorical
> \mathfrak{sl}(2) actions, available as
> [`arXiv:0902.1797`](http://arxiv.org/abs/0902.1797).
>
> The first paper studies various notions of a categorified
> representation of \mathfrak{sl}(2). The second studies an example coming from
> cotangent bundles of Grassmannians. That's a lot of math to compress
> into a 20-minute talk! Luckily Licata was able to do it, by leaving
> out all but the most fundamental concepts.
>
> His work follows the philosophy that "geometrization leads to
> categorification". This is based on a branch of math called
> "geometric representation theory".
>
> The name here is a bit misleading, since a lot of group representation
> theory is geometrical in nature. For example, if we have a group G
> acting as symmetries of a space X, we get a representation of G on the
> vector space of functions on X. And there are many sophisticated
> refinements of this idea. But "geometric representation theory" is
> different. It gets representations in unexpected new ways, often
> starting from the cohomology of a space X on which G does *not* act!
>
> I think this is the best place to start learning geometric
> representation theory:
>
> 8) Neil Chriss and Victor Ginzburg, Representation Theory and Complex
> Geometry, Birkhauser, Boston, 1997.
>
> I've spent some time on this book, but not enough. The results still
> seem strange to me. They're like an outcropping of unfamiliar rocks
> poking through the strata of mathematics that make sense to me. I'd
> need to dig deeper to get a sense of what's going on down there. Just
> thinking about this makes me itch to understand geometric
> representation theory better. I know specific results, but not the
> overall pattern!
>
> You expect, for example, to get representations of \mathfrak{sl}(2) whenever you
> build vector spaces starting from C^2. Why? Because the group \mathrm{SL}(2)
> acts as symmetries of C^2, and thus on any vector space functorially
> constructed from it. But Ginzburg found some unexpected new ways of
> getting representations of \mathfrak{sl}(2)... and Licata sketched how this lets
> you categorify these representations.
>
> Here's the example Licata explained. The group \mathrm{SL}(2) acts on C^2 and
> thus on its $n$th tensor power. Everyone knows that. But we can also get
> this representation in an unexpected way. Start with the space of all
> k-dimensional subspaces of C^n. This is called the "Grassmannian"
> \mathrm{Gr}(k,n). Form a vector space by taking the cohomology of the cotangent
> bundle T*\mathrm{Gr}(k,n). Then take the direct sum of these vector spaces as
> k goes from 0 to n.
>
> We get a big fat vector space. But here's the cool part: Ginzburg
> figured out how to make this big fat space into a representation of
> \mathfrak{sl}(2)! And this representation is isomorphic to the $n$th tensor power
> of C^2.
>
> The trick is to get operators on cohomology groups that satisfy the
> relations for \mathfrak{sl}(2). As usual in geometric representations theory, we
> build these using "spans". These are setups where you have three
> spaces and two maps like this:
>
>                          S
>                         / \
>                        /   \
>                      P/     \Q
>                      /       \
>                     v         v 
>                    X           Y
>
> We can pull back cohomology classes along P, and then if we're lucky
> we can push them forward along Q, getting an operator from the
> cohomology of X to the cohomology of Y. I explained why spans are
> geometrically interesting back in ["Week 254"](#week254).
>
> Anyway, so Ginzburg got a representation of \mathfrak{sl}(2) using this trick. To
> categorify this representation, Licata replaced the cohomology of
> T*\mathrm{Gr}(k,n) by a category called the "bounded [derived
> category](http://en.wikipedia.org/wiki/Derived_category) of [coherent
> sheaves](http://en.wikipedia.org/wiki/Coherent_sheaf)" on this space.
> That's a plausible strategy, because it's known quite generally that
> for any smooth variety X you can take the Grothendieck group of this
> category and get back the cohomology of X.
>
> In fact, if you have no idea what a "bounded derived category of
> coherent sheaves" is, this should make you want to know! It's a
> categorification of cohomology. Here's a good place to start learning
> more:
>
> 9) Andrei Caldararu, Derived categories of sheaves: a skimming.
> Available at
> `http://www.math.wisc.edu/~andreic/publications/lnPoland.pdf`
>
> Next came two talks on another approach to categorification, called
> "groupoidification". This involves replacing vector spaces by
> groupoids and linear operators by spans of groupoids. The reverse
> process, "degroupoidification", is an entirely systematic procedure
> for squashing groupoids into vector spaces and spans of groupoids into
> linear operators. I explained how this works back in
> ["Week 256"](#week256).
>
> First Alex Hoffnung spoke about "[A categorification of the Hecke
> algebra](ams_2009/Hoffnung_AMS_UCR_2009.mov)". The idea here is to
> see the [Hecke algebras](http://en.wikipedia.org/wiki/Hecke_algebra)
> associated to [Dynkin
> diagrams](http://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram)
> as a special case of a much more general construction: the Hecke
> bicategory.
>
> Given a finite group G, the Hecke bicategory Hecke(G) is a gadget
> where:
>
> -   objects are finite G-sets;
> -   the groupoid of morphisms from X to Y is the weak quotient (X x
>     Y)//G.
>
> Here the "weak quotient" is a bit like the ordinary quotient of a
> set by a group action - but instead making elements *equal* when
> there's a group element mapping one to another, we make them
> *isomorphic*. So, it's a groupoid. (For more details, see
> ["Week 249"](#week249).)
>
> Using a systematic procedure for turning groupoids into vector spaces,
> we can squash Hecke(G) down into a category that has a mere vector
> space of morphisms from X to Y.
>
> Now, a category where the set of morphisms between any two objects is
> a *vector space*, and composition is linear in each argument, is
> sometimes called an
> "[algebroid](http://planetmath.org/encyclopedia/Algebroids.html)".
> Why? Because an algebroid with one object is an algebra - in the same
> way that a groupoid with an object is a group.
>
> So, the Hecke bicategory gets squashed down into something that
> deserves to be called the "Hecke algebroid" of G.
>
> Now pick a finite field and a Dynkin diagram. This gives a simple
> algebraic group G and a very important G-set X, called the "flag
> variety" of G. Take the Hecke algebroid of G and concentrate your
> attention on the morphisms from X to X. By what I've said, these form
> an algebra. And this is the famous "Hecke algebra" associated to our
> Dynkin diagram! The usual parameter q that appears in the definition
> of a Hecke algebra is just the number of elements in our finite field.
>
> Alex Hoffnung illustrated his talk with a picture of a cow jumping
> over the moon, wearing a bowtie, and getting killed by a lightning
> bolt. You'll have to watch his talk to see how this is relevant. The
> otherwise excellent slides do not explain this joke:
>
> 8) Alex Hoffnung, A categorification of the Hecke algebra,
> `http://math.ucr.edu/~alex/hecke.pdf`
>
> Then Christopher Walker gave a talk on "[A categorification of Hall
> algebras](ams_2009/CWalker_AMS_UCR_2009.mov)". Unfortunately, the
> cameraman showed up a little late, so the video of his talk starts
> after a couple of minutes have gone by. Fortunately, the next week he
> passed his oral exam at UCR with a longer version of the same talk!
> So, check out the slides for that:
>
> 9) Christopher Walker, A categorification of Hall algebras,
> `http://math.ucr.edu/~cwalker66/Oral_Exam_talk_11_10.pdf`
>
> But here's the idea in a nutshell. Take a simply-laced Dynkin
> diagram. Draw arrows on the edges to get a directed graph. Let this
> graph freely generate a category, say Q. There's a groupoid of
> "quiver representations", where:
>
> -   objects are functors from Q to the category of vector spaces over
>     some fixed finite field;
> -   morphisms are natural isomorphisms.
>
> Next, apply our systematic procedure for turning groupoids into vector
> spaces! In the case at hand, we get the positive part of the quantum
> group associated to our Dynkin diagram. The usual parameter q that
> appears in the definition of a quantum group is just the number of
> elements in our finite field.
>
> (Here we see a difference from the Khovanov-Lauda approach, where q is
> a formal variable.)
>
> So far, this is actually an old theorem of Ringel. The trick is to use
> it to systematically "groupoidify" quantum groups - or at least
> their positive parts - and then work with them at the groupoidified
> level. And that's what Christopher is doing now!
>
> His talk explains more, and you can learn more about groupoidification
> and its applications to Hecke and Hall algebras here:
>
> 10) John Baez, Alex Hoffnung and Christopher Walker,
> Higher-dimensional algebra VII: groupoidification.
> [`arXiv:0908.4305`](http://arxiv.org/abs/0908.4305).
>
> Next came three talks on homotopy theory.
>
> Jonathan Lee of Stanford University spoke on "[Homotopy colimits and
> the space of square-zero upper-triangular
> matrices](ams_2009/Lee_AMS_UCR_2009.mov)". You can see slides of his
> talk here:
>
> 11) Jonathan Lee, Homotopy colimits and the space of square-zero
> upper-triangular matrices,
> `http://math.stanford.edu/~jlee/homotopy-talk.pdf`
>
> He talked about his work on a conjecture of Halperin and Carlsson.
> There are different ways to formulate it, but here's a nice
> topological way. Suppose the torus T^n acts freely on a finite CW
> complex X. Then the sum of the Betti numbers of X is at least 2^n.
> There's also a nice purely algebraic way!
>
> Nitu Kitchloo of UC San Diego spoke on "Universal Bott-Samelson
> resolutions". As a warmup for this, I should just tell you what a
> Bott-Samelson resolution is.
>
> I spoke quite a bit about Schubert cells in
> ["Week 184"](#week184) and subsequent Weeks. The idea is that if
> you have a Grassmannian, or more generally any space of the form G/P
> where G is a simple Lie group and P is a parabolic subgroup, it comes
> equipped with a decomposition into cells. These are the "[Schubert
> cells](http://en.wikipedia.org/wiki/Schubert_variety)". They're
> packed with fascinating algebra, geometry, and combinatorics. They
> are, in fact, algebraic varieties! But, they're not smooth - they're
> singular.
>
> And so, if you were an algebraic geometer, you might be tempted to
> "resolve" their singularities: that is, find a smooth variety that
> maps onto them in a nice way. Bott and Samelson figured out a way to
> do this... but not just one way. So, you might want to find a
> "best" - or more technically, a "universal" - Bott-Samelson
> resolution. And that's what Nitu Kitchloo talked about.
>
> After lunch, Maia Averett of Mills College started the show with a
> talk on "[Real Johnson-Wilson
> theories](ams_2009/Averett_AMS_UCR_2009.mov)", based on work with
> Nitu Kitchloo and Steve Wilson. This was heavy-duty homotopy theory of
> the sort I can only gape at in awe. It's part of a big network of
> ideas which include [elliptic
> cohomology](http://www.math.harvard.edu/~lurie/papers/survey.pdf) and
> higher steps in the "chromatic filtration" - topics I discussed back
> in ["Week 197"](#week197) and ["Week 255"](#week255).
>
> You can see some slides here:
>
> 12) Maia Averett, Real Johnson-Wilson theories,
> `http://www.math.uchicago.edu/~fiore/1/Averett.pdf`
>
> Real Johnson-Wilson theories are certain generalized cohomology
> theories (see ["Week 149"](#week149)). They can be thought of as
> "higher" versions of real K-theory. Thanks to complex conjugation,
> the group Z/2 acts on the complex K-theory spectrum KU, and if we take
> the homotopy fixed points we get the real K-theory spectrum KO. But
> complex K-theory is just the first of the Johnson-Wilson theories!
>
> To get the others, you do something roughly like this. (I'm reading
> some stuff to figure this out, and I could be getting it wrong.) The
> [spectrum](http://en.wikipedia.org/wiki/Spectrum_%28homotopy_theory%29)
> for [complex cobordism
> theory](http://en.wikipedia.org/wiki/Complex_cobordism) is called MU.
> If you localize this at 2 you get something called the [Brown-Peterson
> spectrum](http://en.wikipedia.org/wiki/Complex_cobordism#Brown-Peterson_cohomology),
> BP. The generalized cohomology for this, applied to a one-point space,
> is a ring on infinitely many generators. If you do some trick to kill
> off all the generators above the nth, you get the $n$th Johnson-Wilson
> theory. And since this was built starting from complex cobordism
> theory, complex conjugation acts on it. So, we can take the homotopy
> fixed points, you get the $n$th "real" Johnson-Wilson theory.
>
> Emin Tatar of Florida State University spoke on "[Abelian sheaves and
> Picard stacks](ams_2009/Tatar_AMS_UCR_2009.mov)":
>
> 13) A. Emin Tatar, Abelian sheaves and Picard stacks,
> `http://www.math.ucr.edu/~jbergner/tatar_slides.pdf`
>
> This talk assumed a fair amount of background, so let me just sketch a
> bit of that background. For more details, try this:
>
> 14) A. Emin Tatar, Length 3 complexes of abelian sheaves and Picard
> 2-stacks, available as
> [`arXiv:0906.2393`](http://arxiv.org/abs/0906.2393).
>
> You've probably heard me talk about
> [2-groups](http://arxiv.org/abs/math/0307200). These are categorified
> groups. More precisely, they're categories with a tensor product,
> where every morphism has an inverse and every object x has an inverse
> with respect to the tensor product: that is, an object x* such that
>
> x \otimes x* \cong 1
>
> and
>
> x* \otimes x \cong 1
>
> 2-groups are a great way to dip your toe in vast ocean of $n$-category
> theory. They're one step to the right of groups in the $n$-groupoid
> version of the periodic table:
>
>                        k-tuply groupal $n$-groupoids
>
>                   n = 0           n = 1             n = 2
>
>     k = 0         sets           groupoids        2-groupoids
>
>     k = 1        groups          2-groups          3-groups
>
>     k = 2        abelian         braided           braided
>                  groups          2-groups          3-groups
>
>     k = 3         " "            symmetric         sylleptic
>                                  2-groups          3-groups
>
>     k = 4         " "             " "              symmetric
>                                                    3-groups
>
>     k = 5         " "             " "                "  "
>
> Just as abelian groups are especially simple and nice, so are
> symmetric 2-groups. Where an abelian group obeys the equation
>
> xy = yx
>
> a symmetric 2-group instead has an isomorphism
>
> S~x,y~\colon x \otimes y \to y \otimes x
>
> with the property that doing it twice gives the identity:
>
> S~y,x~ S~x,y~ = 1
>
> Lately people have been generalizing a lot of math from abelian groups
> to symmetric 2-groups. See ["Week 266"](#week266) for more, and
> especially this:
>
> 15) Mathieu Dupont, Abelian categories in dimension 2, Ph.D thesis,
> l'Universite Catholique de Louvain, 2008. Available as
> [`arXiv:0809.1760`](http://arxiv.org/abs/0809.1760). Original available
> in French at `http://hdl.handle.net/2078.1/12735`
>
> But the simplest symmetric 2-groups are those with this extra
> property:
>
> S~x,x~ = 1
>
> Emin Tatar calls these "Picard categories", following Deligne.
>
> (I would like to call these "Picard 2-groups", but that might be
> confusing, since "Picard group" already means something quite
> different. To add to the confusion, it seems that Dupont and others
> use "Picard category" as a synonym for symmetric 2-group!)
>
> Anyway, there's a nice description of Picard categories. They're all
> equivalent to the 2-groups that you get from 2-term chain complexes of
> abelian groups!
>
> It's nice to see how this works. Take a 2-term chain complex of
> abelian groups:
>
>         d
>     A <--- B
>
> Then there's a category where the objects are elements of A, and the
> morphisms from a to a' are elements b of B with
>
> a' = a + db
>
> Addition lets you compose morphisms - but it also lets you add
> objects, making this category into a 2-group. And the abelianness
> makes this not just a symmetric 2-group, but even a Picard category!
>
> But the cool fact is that every Picard category is equivalent to one
> arising this way.
>
> In fact, Deligne went a lot further. There's a general principle that
> anything really important that you can do with sets, you can also do
> with [sheaves](http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29)
> of sets. So, you might guess that anything really important you can do
> with categories, you can do with sheaves of categories.
>
> That's morally correct - but not quite technically correct, because
> we need to take the definition of "sheaf" and replace some equations
> by isomorphisms to make it applicable to categories. If we do this, we
> get the concept of a "[stack](http://arxiv.org/abs/math/0412512)".
>
> Then everything works great. Just as we can talk about sheaves of
> abelian groups, we can talk about stacks of Picard categories - or
> "Picard stacks", for short. And the cool fact I mentioned
> generalizes to these! Every Picard stack is equivalent to one that
> comes from a 2-term complex of sheaves of abelian groups. This was
> proved by Deligne quite a while ago - it's Lemma 1.4.13 here:
>
> 15) Pierre Deligne, La formule de dualité globale, Sem. Geom.
> Algébrique Bois-Marie 1963/64, SGA 4 III, No. XVIII, Springer Lecture
> Notes in Mathematics 305, 1973, pp. 481-587. Also available at
> `http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-3/sga43.pdf`
>
> But you can also see a different proof in Proposition 8.3.2 of this
> paper by Tatar's advisor and Behrang Noohi:
>
> 16) Ettore Aldrovandi and Behrang Noohi, Butterflies I: morphisms of
> 2-group stacks, Adv. Math. 221 (2009), 687-773. Also available as
> [`arXiv:0808.3627`](http://arxiv.org/abs/0808.3627).
>
> Now, what did Tatar do? He categorified all this stuff once more! In
> other words, he defined Picard 2-stacks, and proved that every Picard
> 2-stack is equivalent to one coming from a 3-term chain complex of
> sheaves of abelian groups!
>
> Next, David Spivak of the University of Oregon spoke on "[Mapping
> spaces in quasi-categories](ams_2009/Spivak_AMS_UCR_2009.mov)".
> [Quasicategories](http://arxiv.org/abs/math/0608040) are a nice way to
> formalize the idea of an (\infty,1)-category - that is, an \infty-category where
> all the morphisms above the $1$-morphisms are weakly invertible.
> Technically, quasicategories they're just [simplicial
> sets](http://en.wikipedia.org/wiki/Simplicial_set) with a special
> property. So, one can study them using all the simplicial machinery
> that homotopy theorists have been developing over the years.
>
> However, there are many other ways to formalize (\infty,1)-categories. A
> classic one is "simplicial categories". These are just categories
> "enriched over simplicial sets". In other words, they have a
> simplicial set of morphisms from any object to any other object, and
> composition is a map of simplicial sets.
>
> (If I'd been willing to use this jargon earlier, I could have defined
> an algebroid to be a category "enriched over vector spaces".
> However, I didn't want to scare away all my readers - at least, not
> so soon! By this point I figure all the wimps are gone.)
>
> A while back, Jacob Lurie described a way to turn any quasicategory
> into a simplicial category - see for example Remark 1.1.5.18 here:
>
> 16) Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies
> 170, Princeton University Press, Princeton, NJ, 2009. Also available
> as [`arXiv:math/0608040`](http://arXiv.org/abs/math/0608040).
>
> This involves taking two vertices of our quasicategory - which,
> remember, is just a simplicial set with some properties - and cooking
> up a simplicial set of "morphisms" from one to the other. Recently
> Daniel Dugger and David Spivak have come up with another way:
>
> 17) Daniel Dugger and David I. Spivak, Rigidification of
> quasi-categories, available as
> [`arXiv:0910.0814`](http://arxiv.org/abs/0910.0814).
>
> 18) Daniel Dugger and David I. Spivak, Mapping spaces in
> quasi-categories, available as
> [`arXiv:0911.0469`](http://arxiv.org/abs/0911.0469).
>
> And that's what David explained in his talk!
>
> The day concluded with two talks of a somewhat more concrete nature.
> Ben Williams of Stanford University spoke on "[An application of
> A^1-homotopy theory to problems in commutative
> algebra](ams_2009/Williams_AMS_UCR_2009.mov)". Like Jonathan Lee, the
> problems he was considering included the conjecture of Halperin and
> Carlsson that I mentioned before. But, he used ideas from
> A^1-homotopy theory. So, let me say a word about that.
>
> I actually tried my hand at explaining A^1-homotopy theory near the
> end of ["Week 255"](#week255). It's an attempt to do homotopy
> theory for algebraic varieties, where homotopies are parametrized not
> by the interval but by the line - since the line is an algebraic
> variety. Algebraic geometers call the line A^1, just to make the rest
> of us feel dumb.
>
> In his work on A^1-homotopy theory, Voevodsky studied certain
> cohomology groups for a variety X, called "[motivic cohomology
> groups](http://en.wikipedia.org/wiki/Motivic_cohomology)". The
> curious thing is that they're bigraded instead of just graded.
> Instead of getting cohomology groups H^p^(X,A) with coefficients in an
> abelian group A, we get cohomology groups H^p,q^(X,A).
>
> Why is this? I wish I understood it better... but I think it's
> basically because we could already define cohomology groups for
> varieties without this extra notion of homotopies parametrized by the
> line... but now we can also define them *with* that notion, as well.
> The old cohomology groups were defined using sheaves; the new one is
> defined using simplicial sheaves, and the *simplicial* aspect of these
> sheaves gives a new grading.
>
> And indeed, Voevodsky was able to relate motivic cohomology to another
> bigraded gadget: the "higher Chow groups" of the variety X. These
> are a lot easier to define, so let me describe those. Consider the
> free abelian group generated by irreducible subvarieties of
> codimension k in
>
> X \times \Delta^n
>
> where \Delta^n is the $n$-simplex. (Actually, we should only use
> subvarieties that hit the faces of the simplex "properly".) As we
> let n vary, we get a simplicial abelian group. But a simplicial
> abelian group is just a chain complex in disguise! - I explained how
> in item H of ["Week 116"](#week116).
>
> So, define the higher Chow groups to be the homology groups of this
> chain complex. They depend on two parameters: the "simplicial"
> dimension n, but also the "geometrical" codimension k.
>
> Obviously it would take me a few years of hard work to get from this
> to the point of actually understanding Ben William's talk!
>
> Finally, Christian Haesemeyer of UCLA wrapped up the day with a talk
> "[On the K-theory of toric
> varieties](ams_2009/Haesemeyer_AMS_UCR_2009.mov)". For quite a while
> I've been meaning to explain toric varieties, which are a marvelous
> playground for exploring algebraic geometry. Roughly: just as an
> [algebraic variety](http://en.wikipedia.org/wiki/Algebraic_variety)
> looks locally like the solution set of a bunch of polynomial
> equations, a [toric
> variety](http://en.wikipedia.org/wiki/Toric_geometry) looks like the
> solution set of a bunch of polynomial equations *where you're not
> allowed to add, only multiply!*
>
> This restriction makes them marvelously tractable - you can easily
> describe them using pictures called "fans". Here's a nice informal
> explanation of how this works:
>
> 19) David Speyer, Toric varieties and polytopes,
> `http://sbseminar.wordpress.com/2009/02/09/toric-varieties-and-polytopes/`
>
> Toric varieties and fans,
> `http://sbseminar.wordpress.com/2009/02/18/toric-varieties-and-fans/`
>
> Once you become a fan of fans - and it's easy to do - you can't
> resist wanting to take all your favorite invariants of algebraic
> varieties and see what they look like for toric varieties. Like
> [K-theory](http://en.wikipedia.org/wiki/K-theory)!
>
> Hmm. I'm only described the first day's worth of talks, and it's
> taken more than one day. And I'm left with a lot of questions. For
> example:
>
> -   Aaron Lauda wrote: "It turns out, at least in the simply-laced
>     case, that our algebras are also isomorphic to the Ext algebras
>     between simple perverse sheaves on the Lusztig quiver variety.
>     Lusztig's bilinear form can be seen as taking the graded
>     dimension of this Ext algebra, so it is natural that there is a
>     relationship between the two constructions." Can someone say more
>     about what's going on here? Please *don't* assume I understand
>     what Aaron told me!
> -   How does the representation Licata describes, involving the
>     cohomology of the cotangent bundle of the Grassmannians \mathrm{Gr}(n,k)
>     for k between 0 and n, fit into a more general story? I think the
>     disjoint union of these Grassmannians should be thought of as the
>     space of 1-stage "Springer flags" in n dimensions - where an
>     m-stage Springer flag is a chain of m subspaces of C^n. I vaguely
>     recall that it's interesting to generalize by letting m be
>     arbitrary. And I think that an even more general story - where we
>     pass from \mathfrak{sl}(2) to \mathfrak{sl}(N) - involves Springer flags in the category
>     of quiver representations. Is this right? What's the big picture?
> -   Is my account of Johnson-Wilson theories accurate? What are the
>     most important things that I left out here?
> -   What's "motivic" about Voevodsky's motivic cohomology? Does he
>     propose a definition of motives? How is it related to
>     Grothendieck's conception of motives? How, from this viewpoint,
>     can we see that motivic cohomology should be bigraded?
> -   What other things should I have said, but didn't?
>
> If you have answers, or just other questions, please visit the
> [$n$-Category
> Café](http://golem.ph.utexas.edu/category/2009/11/this_weeks_finds_in_mathematic_45.html).
>
> Happy Thanksgiving!
>
> ------------------------------------------------------------------------
>
> **Addenda:** I thank Toby Bartels yet again for catching a messed-up
> link, and David Corfield for catching some typos.
>
> For more discussion visit the [$n$-Category
> Café](http://golem.ph.utexas.edu/category/2009/11/this_weeks_finds_in_mathematic_45.html).
> Please try to answer my questions above!
>
> ------------------------------------------------------------------------
>
> *There are two fundamental and completely different examples in group
> theory: the "symmetric group" of permutations of n objects, and the
> "linear group" of n by n matrices over a field. Lusztig says the
> linear group is a quantum version of the symmetric group, with the
> value of Planck's constant telling you which field you're looking
> at. He has made that idea precise in a thousand beautiful ways for the
> past 30 years.* - David Vogan
> :::