Cohomology and Computation (Week 26)

June 18, 2007

Cohomology and Computation (Week 26)

Posted by John Baez

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This week in our seminar on Cohomology and Computation we continued discussing the bar construction, and drew some pictures of a classic example:

Week 26 (May 31) - The bar construction, continued. Comonads as comonoids. Given adjoint functors $L: C \to D$ and $R: D \to C$ , the bar construction turns an object $d$ in $D$ into a simplicial object $\overline{d}$ . Example: the cohomology of groups. Given a group $G$ , the adjunction $L: Set \to Grp$ , $R: Grp \to Set$ lets us turn any $G$ -set $X$ into a simplicial $G$ -set $\overline{X}$ . This is a "puffed-up" version of $X$ in which all equations $g x = y$ have been replaced by edges, all equations between equations (syzygies) have been replaced by triangles, and so on. When $X$ is a single point, $\overline{X}$ is called $E G$ . It’s a contractible space on which $G$ acts freely. The "group cohomology" of $G$ is the cohomology of the space $B G = E G/G$ .

Last week’s notes are here; next week’s notes are here.

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I’m running behind on putting up these course notes… but maybe that’s good: you have a bit more to chew on even though classes are actually over here at UCR!

I seem to be vacillating a lot between denoting the result of the bar construction with an underline and denoting it with an overline, as here:

$the bar construction turns objects d \in D into simplicial objects \overline{d}: \Delta^{op} \to D$

In this particular environment I’m having more trouble drawing underlines than overlines! But don’t let this confuse you: there’s no difference.

Posted at June 18, 2007 6:45 AM UTC

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Read the post Cohomology and Computation (Week 27)
Weblog: The n-Category Café
Excerpt: Defining the cohomology of algebraic gadgets using the bar construction.
Tracked: June 19, 2007 6:41 AM

Re: Cohomology and Computation (Week 26)

You’re saying this bar construction works for any adjunction?

If so, then we get a cohomology for the Giry monad.

Posted by: David Corfield on March 25, 2009 10:05 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

minor quibble
the bar construction usually refers to some way of assembling the simplicial object
into an object in some other category
e.g. algebra –> dg coalgebra

Posted by: jim stasheff on March 25, 2009 2:13 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

I think a lot of people (category theorists?) use the term differently, the way I do.

Posted by: John Baez on March 25, 2009 5:17 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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David wrote:

You’re saying this bar construction works for any adjunction?

Yes, in that any adjunction gives a simplicial object. It works best when the adjunction is monadic: in this case the resulting simplicial object satisfies a nice universal property. This is the case that gives most of the well-known cohomology theories for algebraic gadgets.

An adjunction $L : C \to D$ , $R: D \to C$ is monadic if $D$ is the category of algebras of the monad $R L : C \to C$ .

For the details in concentrated form, see Todd Trimble’s notes on bar constructions. This seminar is a far more leisurely, gentle introduction to these ideas.

If so, then we get a cohomology for the Giry monad.

Indeed! If the Giry monad lets us see probability measure spaces as ‘algebraic gadgets’ (algebras of a monad), we get a cohomology theory for probability measure spaces.

Posted by: John Baez on March 25, 2009 5:16 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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I discussed algebras for the Giry monad here.

Might it be easier to work with your simplified monads:

$M: Set \to Set$

sending any set $X$ to the set $M X$ whose elements are certain very nice measures on $X$ : finite positive linear combinations of Dirac delta measures, and

$P: Set \to Set$

sending any set $X$ to the set of probability measures on $X$ that are finite linear combinations of Dirac delta measures?

But what are the algebras here? Any other than free ones? At least in the Polish space case one had bounded, closed and convex subsets of $\mathbb{R}^n$ , with probability measures being sent to their mean, as algebras.

Posted by: David Corfield on March 25, 2009 5:40 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Oh, is that perhaps why Voevodsky is talking about ‘(pre-)ordered topological vector spaces’?

Wouldn’t a bounded, convex and closed subset of a topological vector space be an algebra for the Giry monad?

Is measure homology in the air?

Posted by: David Corfield on March 25, 2009 5:52 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

I’m not up to speed on the details of the Giry monad, but isn’t it analogous to the monad generated by the adjunction between sets and Banach spaces? Which isn’t monadic… though that may not be germane here.

On a general waffly note: l^1-flavoured homology (of spaces or groups) has a tendency to be rather hard to compute. To my knowledge no-one’s ever got anything out of cotriple cohomology for the Set-Ban adjunction, for instance. But perhaps in this setting enough freedom exists to choose good simplicial resolutions for cases of interest.

(And as regards measure homology, I have to indulge my customary harrumph that quite a few papers on bounded cohomology always cite Ivanov and Brooks, but not the more general constructions of Johnson which predate them. Chalk it up to the bitterness of the long-distance CIBAist, I guess.)

Posted by: Yemon Choi on March 25, 2009 10:37 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

So we have measure homology, and now also bounded and volume (or is that just a form of measure?) cohomology.

For all their faults of ignoring Johnson (who was he and what is CIBA?), at least the bounded cohomologists invite you to their subject.

Monod isn’t guilty in any case,

I would like to point out that the group algebra case (that is, bounded cohomology) was indeed prominent in B. Johnson’s memoir. One can find therein several aspects that became intensively studied later, such as quasimorphisms, amenability and the problem of the existence of outer derivations. Nevertheless, it is M. Gromov’s paper (which also refers to ideas of W. Thurston) that gave all its impetus to the theory.

So the relevant memoir is

Johnson, B. E., Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972).

There’s even an audio file of Monod’s invitation.

Posted by: David Corfield on March 26, 2009 9:54 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Yemon wrote:

I’m not up to speed on the details of the Giry monad, but isn’t it analogous to the monad generated by the adjunction between sets and Banach spaces? Which isn’t monadic… though that may not be germane here.

Yes, they should be similar: I bet in both cases the monad assigns to a set $X$ something like the set of ‘convergent infinite linear combinations’ of elements of $X$ .

In the case of the Giry monad the original set $X$ is equipped with some extra structure that makes the quoted phrase meaningful: this is the structure of a ‘standard Borel space’, which is a very nice sort of measurable space. (For details regarding standard Borel spaces, see the end of week272.)

I don’t know any details about the monad generated by the adjunction between sets and Banach spaces.

I would be very interested if the cohomology associated to any monad of this general flavor gave something useful!

Posted by: John Baez on March 25, 2009 11:35 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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David wrote:

Might it be easier to work with your simplified monad…

Here’s a simplified monad that could be even better for your purposes: it sends any set $X$ to the set of finite formal linear combinations $\sum_i a_i x_i$ with $x_i \in X$ , where the coefficients $a_i \in [0,1]$ sum to 1.

The algebras of this include any convex subset of $\mathbb{R}^n$ .

Posted by: John Baez on March 25, 2009 8:18 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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So we can give a free resolution of a general convex subset of $\mathbb{R}^n$ .

For finite $X$ , $|X| = n$ , the free algebra is the $(n - 1)$ -simplex.

Is there another characterisation of the free algebra for infinite $X$ , other than as the application of your monad to $X$ ?

Then we Hom the free resolution into other algebras of the monad?

Posted by: David Corfield on March 26, 2009 12:48 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Here’s a curiosity that I noticed the other day. I don’t know if it’s useful. The interior of $\Delta^{n-1}$ , which is the set of $n$ -ary operations of your theory, is naturally a vector space over $\mathbb{R}$ .

You can say this explicitly as follows. Given $\mathbf{p}, \mathbf{q} \in \Delta^{n-1}$ , their ‘sum’ in this vector space is $\frac{1}{p_1 q_1 + \cdots + p_n q_n} (p_1 q_1, \ldots, p_n q_n).$ Given $\mathbf{p} \in \Delta^{n-1}$ and $\lambda \in \mathbb{R}$ , the ‘scalar product’ of $\lambda$ with $\mathbf{p}$ is $\frac{1}{p_1^\lambda + \cdots + p_n^\lambda} (p_1^\lambda, \ldots, p_n^\lambda).$ The ‘zero’ is the uniform distribution $(1/n, \ldots, 1/n)$ .

You can of course check the vector space axioms directly. But perhaps it’s more illuminating to reason as follows.

There is an isomorphism of monoids $\exp: (\mathbb{R}, +, 0) \to (\mathbb{R}_{> 0}, \cdot, 1).$ (It’s essentially the only isomorphism of monoids with this domain and codomain.) Simply because it’s a bijection, the vector space structure on $\mathbb{R}^n$ induces a vector space structure on $\mathbb{R}_{> 0}^n$ , for any $n$ . The ‘addition’ and ‘scalar multiplication’ in $\mathbb{R}_{> 0}^n$ are pointwise multiplication and powers.

Next, $\mathbb{R}^n$ has a one-dimensional subspace $\langle (1, \ldots, 1) \rangle$ . The corresponding subspace $W$ of $\mathbb{R}_{> 0}^n$ consists of the elements of the form $(t, \ldots, t)$ ( $t > 0$ ). An element of the quotient space $\mathbb{R}_{> 0}^n/W$ is therefore an equivalence class of elements $(t_1, \ldots, t_n) \in \mathbb{R}_{> 0}^n$ where $\mathbf{t}$ is equivalent to $\mathbf{u}$ if and only if $t_i/u_i = t_j/u_j$ for all $i, j$ . Thinking of $\mathbf{t}$ and $\mathbf{u}$ as measures on $\{1, \ldots, n\}$ , this says that the corresponding probability distributions are equal. So, there is a bijection $\mathbb{R}_{> 0}^n/W \cong int(\Delta^{n-1}),$ giving $int(\Delta^{n-1})$ the structure of a vector space.

(Technical aside: I’m assuming that $n\geq 1$ . By the interior of $\Delta^{n-1}$ I mean the set of $\mathbf{p} \in \Delta^{n-1}$ such that $p_i > 0$ for all $i$ . When $n \geq 2$ this is the same as the interior in the usual sense; but when $n = 1$ it gives the one-point space, not the empty space.)

Posted by: Tom Leinster on March 27, 2009 3:38 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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When you said that exp is

essentially the only isomorphism of monoids with this domain and codomain,

the ‘essentially’ presumably means $exp(a x)$ for nonzero real $a$ covers all cases.

We’ve talked a lot about turning energies into probabilities over the years, and the special case at zero temperature, e.g., here.

But I don’t think I’d ever seen that normalisation you’re doing as quotienting out a subspace. Interesting.

Posted by: David Corfield on March 27, 2009 11:51 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Looking at that discussion I just mentioned, I see it got curtailed somewhat. It continued:

JB: David writes

So if there is a categorifying chain of values for paths between $x$ and $y$ which runs: truth values, set, …, n-groupoid,… are there other chains like cost, cost of passing between paths,… or probability, probability of passing between paths,….. or amplitude, amplitude of passing between paths….

Good point! You’re shooting ahead of me here, and it’s a bit embarrassing, because as you note:

this just seems to be pointing to things like your higher-gauge theory.

Part of what’s been bugging me a lot about higher gauge theory is that I don’t understand how Lagrangians fit into it. Usually people write down a Lagrangian as a function of some fields, which lets you compute an action, and minimizing the action give you equations of motion. You can do this in higher gauge theory too. BUT, usually the action for an ordinary gauge theory is required to be INVARIANT UNDER THE GROUP OF GAUGE TRANSFORMATIONS, since then gauge transformations will map solutions of the equations of motion to solutions. In higher gauge theory we have a 2-GROUP of gauge transformations. What does it mean for an action to be “invariant” under a 2-group? 2-groups really want to act not on a mere set, but on a CATEGORY - and the proper notion of “invariance” is “weak invariance”, i.e. invariance up to a specified isomorphism satisfying some (understood) coherence laws. This suggests that actions in higher gauge theory should really take values in a category. And so, presumably, should Lagrangians. But, what category or categories??? Some categorification of the real numbers, maybe.

The problem is, I don’t see the physics pointing me towards any particular choice. Probably I’m just being dumb. It’s especially galling because I already think I know what one *result* of path-integral quantizing a higher gauge theory mightbe: a 2-Hilbert space of states! I wrote a paper on 2-Hilbert spaces once….

Hmm, this suggests that the appropriate “categorified transition amplitudes” lie not in C but in Hilb!!!

Maybe it would be good to think how the fundamental groupoid arises through Lagrangian reasoning. In the path connectedness case, we have a space $X$ , and a Lagangian map from $X$ to truth values, i.e. a subset of $X$ . Then for any path in $X$ , there is an action formed by integrating $L$ along it. This tells you whether the path lies wholly in $X$ . Now you form the integral over all paths with the same endpoints. This just sees whether there is any path in $X$ between those two points.

Up a dimension, we’re looking for a set of homotopy classes of paths…

Let me postpone thinking about this, even though it sounds really cool.

Hmm. I have a feeling this ought to be slicker. Also the first ‘integrations’ in each case were over truth values and yet were ANDs rather than ORs.

I just want to say something about *this*. This actually seems right to me. In physics, a path integral is an integral over paths of the EXPONENTIAL of the action, which in turn is obtained by integrating a Lagrangian along the path. The exponential turns addition into multiplication. In fact, it’s often good to think of the “exponential of the action”, as more fundamental than the action. It has a clearer meaning. In quantum physics, the exponentiated action $exp(i S/\hbar)$ tells you the RELATIVE AMPLITUDE for taking that path. In statistical mechanics there’s a version where you get the RELATIVE PROBABILITY.

In the situation you’re talking about, the exponentiated action is a truth value saying whether the path is continuous - i.e., the POSSIBILITY of following that path. Now about that “first ‘integration’” that’s bugging you. The exponentiated action is given by a “product integral” along the path. I don’t know if you’ve thought about product integrals, but they’re just like integrals with + replaced by times. I reinvented them when I was a kid so I have a certain fondness for them. Normally you get them by multiplying lots of numbers that are really close to 1, instead of adding lots of numbers that are really close to zero… but normally you can reduce them to ordinary integrals using “exp” and “ln”.

You however are doing product integrals in the rig of truth values: you are computing the possibility of a certain path as a product of possibilities of lots of little paths! And in this case there’s no “exp” and “ln” to save us - unless there’s some logical operation nobody every told me about, that converts “or” to “and”. It’s also neat to think about product integrals in the rig of costs: the rig $R^{min} = (R \union + \{\infty\}, min, +\infty, +, 0)$ . Here we compute the cost of a path as an ORDINARY integral along the path…but the ordinary integral uses +, which is really MULTIPLICATION in the rig of costs. So, it’s again a case of a product integral. And again there’s no “exp” and “ln” to save us.

Presumably homotopy theory is treating a space as though it’s infinitely cheap to go through the space, and infinitely expensive to go outside. So is there a cheap way to get from $x$ to $y$ ? Yes, so long as there’s a path [OR] along which [AND] all points are cheap.

Right! I like to think of truth values as a funny version of the rig of costs where the only two prices are “free” and “you can’t afford it”. Anyway, now I should go back to your categorified version of the whole setup:

Up a dimension, we’re looking for a set of homotopy classes of paths: We have a space $X$ , paths and paths between paths. The Lagrangian takes paths to truth values. When we integrate the Lagrangian along a path between paths it tells us whether we can do this all within $X$ . Then for a given path $f$ , we integrate [form set union] over all paths with the same end points, collecting all those homotopic to $f$ . Now we integrate [form set union] over all paths $f$ , forming the union of homotopy classes.

and think of the final result as an ordinary integral of a product integral. Btw, the “state sum models” in TQFT are all done by multiplying an amplitude for each labelled simplex and then summing over labellings, so it’s the same sort of deal.

About the transformation between quantum and classical, the trouble I’m having is that according to your lectures Sets and Relations are already on the quantum side.

Yeah! But that’s GOOD.

In today’s talk I explained how for any rig $R$ there’s a PROP whose morphisms from $x^n$ to $x^m$ are $n \times m$ matrices with entries in $R$ , with the “tensor product” of morphisms being direct sum of matrices. In other words, “finitely generated free $R$ -modules, made into a symmetric monoidal category using direct sum”. This lets us do “matrix mechanics” in the manner we’ve been discussing, and when $R$ is the rig of truth values we get “finite sets and relations, made into symmetric monoidal category using disjoint union” But we can also use the rig of costs….

Oh, I see that I’d already got the point. But isn’t that all the same odd to call anything matrix-like ‘quantum’?

Well, FinRel is a symmetric monoidal category where the product is not cartesian, and it’s a *-category, so in many ways it more closely resembles Hilb than Set or FinSet. The superposition principle is a bit stunted given that the rig of truth values has just one nonzero element, but don’t let that fool you.

Putting it naively, the things in the sets of Rel are perfectly classical. If the mere fact that things are related is enough to make them quantum, isn’t that a sign that my twins entanglement idea is right - that a chunk of the weirdness of entanglement is little more exciting than that a twin marrying 12000 miles away makes you instantly an in-law. Or is your worry here that this is just about information? But then Fuchs, Cave et al. want to say this is really all EPR experiments are doing.

Well, the sexier features of QM probably require a more interesting rig.

I’m not sure this is the right analogy; in quantum mechanics entanglement is about tensor products of vector spaces, so in FinRel we should be looking at tensor products of modules over the rig of truth values….

Hmm, it might turn out that you’re right, and that the analog of a “entangled state” boils down to a pair of sets with a relation between them. But this is something one just needs to calculate. For example, maybe a relation $f: S \to T$ can be dualized to give a relation $g: 1 \to S* \otimes T$ just like a linear operator $f: V \to W$ gives a linear operator $g: C \to V^{*} \otimes W$ - which is the same as a state in $V^{*} \otimes W$ . The identity operator $f: V \to V$ gives a maximally entangled state $g: C \to V^{*} \otimes V$ , so maybe we can do the same thing with relations.

But, first I’d need to figure out if there really is a tensor product of “modules over the rig of truth values” (probably), and what it is.

Presumably a module over the rig of truth values has to look something like a vector of truth values. Imagine the vector answering the two questions Are you a parent? Are you an aunt/uncle? Any individual can be in one of 4 states. For any two unrelated people as far as you know they could be in any one of 16 states. Finding out about one of them doesn’t help you with the other. But for two siblings (with no other siblings) they can only be in 4 states, and finding out about the state of one tells you about the other.

Okay, very sensible. Now I can translate what you said into math lingo. We only need (for now) to think about FREE modules of the rig $R$ , namely those of the form $R^n$ . And, with any luck, the tensor product of $R^n$ with $R^m$ is always $R^{n m}$ , where you tensor two vectors by multiplying them entrywise to get a rectangular array. And, “entangled states” are those that aren’t expressible as a tensor product of two vectors. For $R$ the rig of truth values, you’ve got entanglement whenever you’ve got a rectangular array $A_{i j}$ of truth values that’s not of the form ( $B_i$ and $C_j$ ). There are lots of these, but as usual, they’re always expressible as a sum - an “or” - of unentangled states.

Posted by: David Corfield on March 27, 2009 12:23 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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unless there’s some logical operation nobody every told me about, that converts “or” to “and”

You mean negation?

Posted by: Toby Bartels on March 27, 2009 10:03 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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David started working out the free algebras of the monad that sends any set $X$ to the set of formal ‘finite convex linear combinations’ of elements of $X$ , as defined more precisely here:

David wrote approximately:

For finite $X$ with $|X| = n$ , the free algebra is the $(n−1)$ -simplex.

Right.

Is there another characterisation of the free algebra for infinite $X$ , other than as the application of your monad to $X$ ?

I don’t know, but it’s not hard to love: it’s an ‘infinite-dimensional simplex’ with one vertex for each element of $X$ , where every point in the interior is a finite convex linear combination of the vertices.

I guess we can think of this as a topological space, namely what topologists often call an inductive limit (really colimit) of $n$ -simplices. When $X$ is countable, it probably deserves to be called $\Delta^\infty$ . It’s a relative of the topologists’ infinite-dimensional sphere, $S^\infty$ , or the infinite-dimensional projective spaces $\mathbb{R}P^\infty$ and $\mathbb{C}P^\infty$ .

Posted by: John Baez on March 27, 2009 5:43 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Is there another characterisation of the free algebra for infinite $X$ , other than as the application of your monad to $X$ ?

The description John just gave works very generally, to describe free algebras $T X$ whenever the monad $T$ comes from a finitary Lawvere theory.

First, without any finitary assumption on $T$ , we know that the free algebra functor

$T: Set \to T-Alg,$

being left adjoint to the forgetful functor, preserves general colimits. In particular, any set $X$ is the colimit of the diagram consisting of finite subsets $S$ of $X$ and inclusions between them, and so we have

$T X = colim_{fin. S \subseteq X} T S.$

But if $T$ comes from a finitary Lawvere theory, like the theory of affine sets we’re talking about here, then the forgetful functor

$T-Alg \to Set$

creates and preserves filtered colimits, for example the colimit displayed above. In other words, the colimit

$T X = colim_{fin. S \subseteq X} T S$

may be computed in $Set$ , and this filtered colimit is nothing but the set-theoretic union of the $T S$ .

That’s just what John was saying: $T X$ is the union of finite-dimensional simplices with vertices labeled in $X$ . But the underlying principle is very general.

Posted by: Todd Trimble on March 27, 2009 11:06 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Is there anything kind to say about a resolution of a convex subset of $\mathbb{R}^n$ according to this monad?

What does Yemon mean by good simplicial resolutions - good for what?

Posted by: David Corfield on March 30, 2009 3:57 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Sorry, didn’t mean to be gnomic. I meant resolutions for which one can *compute* some kind of (nonabelian?) cohomology. Finite length would be a start, which ought to be possible for polytopes/polyhedra – I’ve been meaning for about three years now to see if Fourier-Motzkin elimination is relevant here, but it keeps slipping down the list of things to do.

I probably need to explain the cause for any Eeyorish tone that may have come across. The bar resolution/construction for Banach modules gives you Hochschild cohomology as studied by some masochistic Banach algebraists, which sits there in its resplendent naturality of definition but mocks one’s pitiful attempts to calculate any of the cohomology groups in degrees 2 and above. So my ingrained reaction when bar resolutions arise in (complete) normed contexts is one of pessimism, however if we are just looking at polytopes in finite dimensions things ought to be rosier.

Oh, and I forgot to say thanks for the link to the audio file of Monod. (Interestingly, while his admirable Springer Lecture Notes on Cts Bdd Coho demonstrate awareness of the BEJ work – and Monod is absolutely right to say that the *significance* of the topic should largely be credited to Gromov – he seems not to have been aware when he wrote those notes, that for discrete groups there is a Cartan-Eilenberg style homological approach – via the apparatus developed by Helemskii’s school in Moscow. Guess you can’t please all of the people all of the time…)

Posted by: Yemon Choi on March 30, 2009 8:40 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

In functional analysis, indeed there are homological methods of A. Ya. Helemskii’s school
( book ), depending on the usage of particular resolutions; we do not deal with an abelian category hence it is not clear a prirori what kind of chain complexes are admissible resolutions, without having some additional device to make some choices. However, the case for non-resolution dependent definition is not being lost. Namely one can choose what admissible epimorphisms are and refine the theory to get a nonabelian homological algebra of the sort. Such frameworks exist(look at the book by Inassaridze, or fundamental works by Bourn, Janelidze and others); there is also an independent framework in Alexander L. Rosenberg,
Homological algebra of noncommutative ‘spaces’ I (warning: 199 pages) which could in principle give a solid derived functor foundations for the works of Helemskii school, though the author does not seem to be interested to proceed to do it (his motivation for the machinery being mainly different). The basic notion there is right ‘exact’ category what is a category equipped with a choice of singleton pretopology whose members are strict epimorphisms.

Mariusz Wodzicki has developed a rich story with many nontrivial unpublished results on homological algebra of functional spaces (not functional algebras). I hope somebody persuades him to publish those one day; it is a pity they are burried in his personal notebooks so far. I heard from him an exposition of a part of the story in 2006 – very nontrivial and beautiful matter.

Posted by: Zoran Skoda on April 7, 2009 10:39 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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The way I see it, the main thing about the bar resolution is that it means: whenever you have a monadic adjunction, there’s a cohomology theory waiting for you to study it, with a bunch of nice formal properties. If you start wanting to do computations, then you can seek more ‘useful’ resolutions: that is, ones that make the computations easier. But first you have see why you’d care.

Posted by: John Baez on March 30, 2009 9:09 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Oh, absolutely. I agree, and think that the full force or “meaning” of the construction hasn’t been exploited when it comes to cohomology of Banach widgets. But then again, I would say that, especially if I’m trying in vain to write proposals.

One problem in Banach world is that IIRC the adjunction which gives Hochschild cohomology via the bar resolution, goes from Banach spaces to Banach modules, which means the base category of your adjunction is not abelian (it’s not Barr-exact, cokernel problems I think). This seems to underly some reasons why not everything works (everything doesn’t work?) as well as in usual commutative algebra.

But we’re getting off topic, if other work permits I’ll try to find time to revisit the Giry monad and some of the links people have suggested here.

Posted by: Yemon Choi on March 30, 2009 10:26 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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…whenever you have a monadic adjunction, there’s a cohomology theory waiting for you to study it…

So if we form the category of monads, and I see you can do this for the category of monads in any 2-category, do we get interesting relationships between cohomologies?

Before that, though, what new cohomologies will appear by working with a 2-category different from $Cat$ ?

Posted by: David Corfield on April 1, 2009 12:25 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Actually, checking whether there is any reasonable (co)homology theory for convex spaces is one of many things on my todo-list… here’s what I noticed:

1) Resolutions of polytopes by simplices may be nontrivial. For example one may consider the kernel pair of the projection (3-simplex) —>> square. This is a polytope which is itself not a simplex.

2) Instead of studying convex spaces, it is in many circumstances technically easier to study conical spaces, i.e. replace probability measures for the Giry monad by arbitrary finite measures. These conical spaces coincide with semimodules over the positive reals. Hence one should expect the category of conical spaces to share many properties of an abelian category, except that negatives don’t exist. There seem to be people studying homology theories for semimodules, but I haven’t looked into this in detail yet. Is there a generalization of derived categories to this setting?

Posted by: Tobias Fritz on April 2, 2009 12:27 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

More on algebras for the Giry monad and a discrete version, although these are subprobability measures.

Re: Cohomology and Computation (Week 26)

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David wrote:

Does homotopy have a ‘monadic’ face, as cohomology does?

Once you understand how Quillen generalized ‘homological algebra’ (derived categories) to ‘homotopical algebra’ (model categories), and how all this stuff is secretly a branch of $\infty$ -category theory, it becomes a bit clearer what’s going on.

In homological algebra you typically start with an abelian category $A$ . But then, perhaps unconsciously, you want to ‘ $\infty$ -categorify’ it. So, you switch to working with $Ch(A)$ , the category of chain complexes in $A$ . Remember, a chain complex in $A$ is the same as a strict $\infty$ -category in $A$ , and a chain map between these is just a strict $\infty$ -functor in $A$ .

But soon you realize that strict $\infty$ -functors are annoyingly… strict! Unfortunately weak $\infty$ -functors are a bit scary, even between strict $\infty$ -categories. So, you pull a clever stunt: you define a weak $\infty$ -functor from $x$ to $y$ to be a strict $\infty$ -functor from a ‘resolution’ of $x$ to $y$ . The idea is that taking a resolution ‘puffs up’ $x$ , replacing all equations by isomorphisms, ad infinitum. After we’ve puffed up $x$ , there are no interesting equations to preserve, so strict $\infty$ -functors aren’t really evil.

When $A$ comes from a category of algebraic gadgets — i.e., gadgets defined by a monad — this ‘resolution’ process can be done in a beautiful, using what in these notes I call the bar construction.

So far everything has been quite ‘abelian’ in nature. Quillen wanted to go beyond this — to generalize homological algebra so that it actually included homotopy theory. So, he invented a formal characterization of what it meant to ‘puff up’ an object in a larger class of categories, which he called ‘model categories’.

Instead of ‘puff up’, he said ‘cofibrant replacement’.

I believe there are plenty of situations where cofibrant replacement in a model category can be done using the bar construction. If so, the experts will soon remind us, and we’ll see the monadic face of homotopy theory.

Posted by: John Baez on April 3, 2009 6:50 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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I believe there are plenty of situations where cofibrant replacement in a model category can be done using the bar construction. If so, the experts will soon remind us, and we’ll see the monadic face of homotopy theory.

I am busy typing this stuff into the $n$ Lab.

On the train back home today I started for instance abelian sheaf cohomology from an $\infty$ -categorical perspective, indicating how it since inside general cohomology and higher topos theory.

Posted by: Urs Schreiber on April 3, 2009 10:13 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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In algebraic situations, like chain complexes, the bar construction frequently does give you cofibrant replacements (aka projective resolutions). In more topological situations, the bar construction isn’t usually “cofibrant” in the model-categorical sense, but it’s often “cofibrant enough” for the purpose of defining derived functors. In the language of DHKS, it is a deformation; see in particular HLC&EHT.

Posted by: Mike Shulman on April 6, 2009 9:02 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Is it that whenever you see a cohomology theory, you expect an adjunction to generate it via the bar construction?

So, the cohomology of dynamic systems, discussed here, arises from an adjunction?

Posted by: David Corfield on March 27, 2009 10:56 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Is it that whenever you see a cohomology theory, you expect an adjunction to generate it via the bar construction?

I was thinking about this problem a bit without having a good full answer yet.

There are these two superficially different notions of general cohomology theories:

one is that of “generalized spaces” (sheaves of spaces/ $\infty$ -groupoids), “generalized spectra” (sheaves of spectra/stable $\infty$ -groupoids), etc. where cohomology $H(X,A)$ is the connected components of the Hom-space $Hom(X,A)$ in the corresponding $(\infty,1)$ -category.

This captures a large range of cohomology theories, group cohomology, groupoid cohomology, nonabelian cohomolohy, sheaf cohomology, nonabelian sheaf cohomology, etc.

Also that cohomology of dynamical systems which you mention seems to be just the special case of this for sheaves of groupoids on measure spaces.

Now, there is the “mondaic” cohomology, which is being discussed here.

In concrete constructions it is clear how this fits into to the “homotopical” cohomology notion above:

namely one way to compute that Hom-space $H(X,A)$ of $\infty$ -groupoid valued sheaves is to model them as simplicial sheaves, resolve the simplicial sheaf $X$ to something bigger, then apply $A$ to that resolution and take a codiagonal.

This process of “resolvin $X$ and applying $A$ ” produces those structures appearing in bar resolutions, etc.

But I am still lacking a general crisp statement that would nicely relate “monadic cohomology” with “homotopical cohomology” in an insightful way.

Probably the general abstract nonsense answer is given by derived algebraic geometry with its $\infty$ Barr-Beck theorem, which probably tells us how we can think of the monad appearing in monadic cohomology as a generalized cover of generalized $\infty$ -stacks, thus connecting this to the geometrical/homotopical picture.

I wish I were quicker with understanding this stuff.

(I should mention that it was Zoran Škoda a while ago who started alerting me to the necessity of understanding this relation better.)

Posted by: Urs Schreiber on March 27, 2009 12:03 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Urs wrote: I wish I were quicker with understanding this stuff.

You’re leaving me in the dust as it is!

Posted by: jim stasheff on March 27, 2009 1:23 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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You’re leaving me in the dust as it is!

I should say the same again less hastily and better worded, then it shouldn’t sound very mysterious.

The thing is somehow to think of each adjunction as being the adjunction

pullback-to-cover-space $\leftrightarrow$ pushdown-from-cover-to-base-space

of the “coefficient things” in question. That should relate the geometric notions of descent/cohomology with the algebraic notions.

Posted by: Urs Schreiber on March 27, 2009 2:07 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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While I doubt that I have seen a really good account of “monadic descent” and its relation to (nonabelian) cohomology, the kind of story that is relevant here is told for in low degree for instance in

F. Borceux, S. Caenepeel, G. Janelidze, Monadic approach to Galois descent and cohomology

Posted by: Urs Schreiber on March 27, 2009 2:30 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Can I make a plea for care in the use of the two expressions `bar construction’ and `bar resolution’. Historically they were not the same. The bar resolution is a monadic resolution typically using a generalisation of the nerve construction derived from the idea of a monoid action on a set, whilst in many areas of alg. top. bar constructions are a way of going from algebras to coalgebras (of similar). The two are related and perhaps I have got this wrong, but I remember feeling confused sometime ago when the resolution aspect was being called a bar construction. There is also a two sided bar construction which relates to two sided actions. (Perhaps Jim could say a word to clear up the question as it is possibly more up his street than mine.)

Posted by: Tim Porter on March 27, 2009 3:01 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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The two are related

Can you say how exactly they are related?

If you can, maybe you can even link it to the material currently at $n$ Lab: twisting cochain in the section relation to the adjunction bar-cobar.

Posted by: Urs Schreiber on March 27, 2009 3:37 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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I see what you’re saying, but the bar construction in the sense of ‘bar resolution’ is actually a special case of the two-sided bar construction, which is the terminology I got from May’s Geometry of Loop Spaces. I am honestly not sure just how standard May’s terminology is, but I’ve always assumed it was.

Just to explain to people out there: given a monoid $M$ acting on $X$ on the left and $Y$ on the right, the two-sided bar construction, which May denotes as $B(Y, M, X)$ , is the simplicial object whose face maps are of the form

$\d^n_j: Y M^{n+1} X \to Y M^n X$

$j = 0, \ldots, n$ where $d^n_0$ uses the right action on $Y$ and $d^n_n$ uses the left action on $X$ , and $d^n_j$ for $1 \leq j \leq n-1$ uses the monoid multiplication in the $j^{th}$ place. (There is an obvious bicategorical generalization (i.e., usings monads rather than monoids), and there are more abstract-nonsense-y ways of expressing it, but this is the idea. What I think you are calling the bar resolution is then the two-sided bar construction $B(M, M, X)$ .

Actually, in my notes on the bar construction, I also call this a bar resolution! Where I give a perhaps slightly idiosyncratic sense to a notion of (simplicial) ‘resolution’: a coalgebra over the decalage comonad. The bar resolution $B(M, M, X)$ , in this sense of resolution, satisfies a universal property.

Posted by: Todd Trimble on March 27, 2009 4:40 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

Thanks Todd. That confirms what I had understood. My point was exactly that `bar resolution’ is more restricted in its traditional use than `bar construction’ and that in the link with the twisting cochain and all that stuff, the extent to which that `bar construction’ is related to the `bar resolution’ is obscured. That is not to forget the cobar construction, and old jokes about co-Eilenberg and his co-coauthor co-MacLane (sorry I should have resisted that one!) (Is the dual of a co-author,….?)

Posted by: Tim Porter on March 27, 2009 7:38 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

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Urs wrote

But I am still lacking a general crisp statement that would nicely relate “monadic cohomology” with “homotopical cohomology” in an insightful way.

I wonder if there’s anything to be gained in the understanding of cohomology by thinking more about its Eckmann-Hilton duality with homotopy. Does homotopy have a ‘monadic’ face, as cohomology does?

But then maybe the duality doesn’t run deep enough. Where are all the flavours of homotopy to parallel those of cohomology? How about being able to change coefficient group for starters?

I see Hatcher considers this point:

This duality is in one respect incomplete, however, in that the cohomology statement holds for an arbitrary coefficient group, but we have not yet defined homotopy groups with coefficients. In view of the duality, one would be tempted to define $\pi_n(X; G)$ to be the set of basepoint-preserving homotopy classes of maps from the cohomology analog of a Moore space $M(G, n)$ to $X$ . The cohomology analog of $M(G; n)$ would be a space $Y$ whose only nonzero cohomology group $\tilde{H}^i(Y; \mathbb{Z})$ is $G$ for $i = n$ . Unfortunately, such a space does not exist for arbitrary $G$ , for example for $G = \mathbb{Q}$ , since we showed in Proposition 3F.12 that if the cohomology groups of a space are all countable, then they are all finitely generated. (pp. 463-4)

But there is some progress to made towards a universal coefficient theorem.

Posted by: David Corfield on April 3, 2009 9:42 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 26)

My uninformed impression was that this relation between monads and cohomologies was less intrinsic than it initially seems. More precisely, the monads (as I learned from Weibel’s book
and Barr) are playing the role of providing canonical resolutions with which one can try to calculate a cohomology theory which was already defined independent of knowing such monads are in the picture. A good example is Hochschild homology, which is defined independently of knowing the bar resolution exists, though of course the latter is a very useful way to compute the former.

I think this was one of the points of Quillen’s amazing paper “On the (co)homology of commutative rings” - there’s an intrinsic notion of homology in a category, which is the derived abelianization functor: you want to construct a derived left adjoint to the forgetful functor from {abelian group objects over some fixed A} to {objects over A}, and this (applied to A) is the Quillen homology of A (the cohomology with coefficients in an abelian group object M is given by maps from Ab(A) to M). Quillen shows that this procedure recovers many of the usual cohomology theories such as Hochschild and Andre-Quillen (cotangent complex).

The monad POV seems more extrinsic - if your initial category has some natural “forgetful functor” (eg from R-mod to k-mod for R a k-algebra) and an adjoint induction functor, you get a monad (sorry if I’m getting all my (co)s off..) which gives canonical resolutions of objects in your category. You can then hope to use this to calculate Quillen homology. But this is the usual distinction about derived functors - it’s important to know they have an intrinsic meaning independent of defining them by a specific chain complex..

A far more general POV on all this is given by Goodwillie calculus, where Quillen homology = abelianization = linearization plays a role of first derivative.. but not really understanding how this works I better make for the exit quickly..

Posted by: David Ben-Zvi on April 4, 2009 11:23 PM | Permalink | Reply to this

Read the post Convex Spaces
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Excerpt: Fritz on convex spaces as algebras for the finite Giry monad
Tracked: April 1, 2009 5:41 PM

Read the post Cohomology
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Excerpt: On the general (infinity,1)-categorical notion of cohomology.
Tracked: April 4, 2009 6:26 PM

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Excerpt: Mike Shulman reports on the 88th Peripatetic Seminar on Sheaves and Logic
Tracked: April 11, 2009 12:23 PM

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