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Stream: learning: questions

Topic: Maurer-Cartan equation

Tim Hosgood (Jul 22 2024 at 22:54):

(the conversation has moved on a bit, but i'd still like to share one more thing about Maurer–Cartan and the dg-setting)

if you want to take the idea "i have two things that are not equal but somehow equivalent, and actually the equivalence between them and the inverse of this equivalence are equivalent up to some equivalence one dimension higher, and ... and so on" and make this formal, then, if you work in the world of chain complexes, you will end up writing down exactly the Maurer–Cartan equation

John Baez (Jul 23 2024 at 08:44):

In what contexts have you met or used the Maurer-Cartan equation, @Tim Hosgood? I learned about it first when studying Lie groups: for differential forms on Lie groups: if you have a Lie group $G$ with a basis of left-invariant 1-forms $\omega^i$, then the Maurer-Cartan equation says

$d\omega^i + \frac{1}{2} c^i_{jk} \omega^j \wedge \omega^k = 0$

where $c^i_{jk}$ are the structure constants for the dual basis $e_i \in \mathfrak{g}$, meaning

$[e_j,e_k] = c_{jk}^i e_i$

John Baez (Jul 23 2024 at 08:48):

Then I learned the slicker statement of the same equation where we organize all these 1-forms $\omega^i$ on $G$ into a single $\mathfrak{g}$-valued 1-form on $G$,

$\omega = \omega^i e_i$

This obeys

$d\omega + \frac{1}{2} \omega \wedge \omega = 0$

where now we are using the wedge product of $\mathfrak{g}$-valued 1-forms, which incorporates the Lie bracket.

John Baez (Jul 23 2024 at 08:49):

All this is explained here:

• Maurer-Cartan form.

Notification Bot (Jul 23 2024 at 08:50):

4 messages were moved here from #learning: questions > Dold-Kan correspondence by John Baez.

John Baez (Jul 23 2024 at 08:54):

Much later I saw the Maurer-Cartan equation show up in Koszul duality, and I wrote about it here. I sketched how for any vector space $V$,

Making the free graded-commutative algebra on $V^*$ into a differential graded-commutative algebra is the same as making $V$ into a Lie algebra.

where $V^*$ is 'degree-shifted' dual of $V$: the linear dual of $V$ but where everything has grade 1.

The key thing here is that each basis vector for $V^\ast$ gives some element $\omega^i$ in our graded-commutative algebra. We have various choices of how to define the differential of these elements, but these correspond to different choices of Lie bracket on $V$. And the equation $d^2 = 0$ turns out to be equivalent to the the Jacobi identity for the bracket on $V$!

John Baez (Jul 23 2024 at 08:57):

And the "duality" part of Koszul duality is that also

Making the free graded Lie algebra on $V^*$ into a differential graded Lie algebra is the same as making $V$ into a commutative algebra.

John Baez (Jul 23 2024 at 09:01):

Then all of this stuff has massive generalizations.

Tim Hosgood (Jul 23 2024 at 13:00):

this is fun, because i think we can each tell stories that (at surface level (and actually quite a few levels below the surface)) look quite different! I have two stories I would tell to start: generalising vector bundles by looking at transition functions, and deforming "fake" differentials into "true" ones

Tim Hosgood (Jul 23 2024 at 13:01):

I'll find some time this afternoon to tell one of these stories at least :)

John Baez (Jul 23 2024 at 13:18):

Thanks! Somehow all these stories must eventually merge into a single story like streams flowing into a mighty river... but we'll have to see if we can make that happen here.

Tim Hosgood (Jul 23 2024 at 15:45):

yes! I'm going to start by just telling a story without paying any attention to yours, and then we can try to figure out how they could flow together

Tim Hosgood (Jul 23 2024 at 15:49):

Here and basically always I'm going to take $X$ to be a complex manifold with a cover $\mathcal{U}=\{U_\alpha\}$, but this all works in many other settings as well.

So what's a vector bundle $E$ on a space $X$? Well, my favourite way of thinking about such a thing is as a collection of transition data $g=(g_{\alpha\beta}\in\mathrm{GL}(n,\mathbb{C}))_{\alpha,\beta}$ that satisfy two conditions:

1. the invertibility condition $g_{\alpha\alpha}=\mathrm{id}$
2. the cocycle condition $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$

(note that in the cocycle condition I'm writing my matrix multiplication "backwards", i.e. thinking of composition as $f;g$ instead of $g\circ f$).

The usual intuition for this data is that we can build a space out of this data by putting a copy of $\mathbb{C}^n$ over each open subset $U_\alpha$ in our cover $\mathcal{U}$ of $X$, and then we glue them together on intersections $U_{\alpha\beta}=U_\alpha\cap U_\beta$ by using these transition matrices.

Tim Hosgood (Jul 23 2024 at 15:59):

The invertibility condition is very important, but is secretly of a different flavour than the cocycle condition, and we're going to just look at the latter for now. In fact, we're just going to write it ever so slightly differently to begin with: $-g_{\alpha\gamma}+g_{\alpha\beta}g_{\beta\gamma}=0$.

Taking a step back, we see that we're dealing with the data of matrices indexed by something of Čech degree 1, i.e. by open subsets $U_{\alpha\beta}=U_\alpha\cap U_\beta$. If we multiply together two such matrices, forgetting entirely about the two conditions for the moment, then we have another matrix but now indexed by something of Čech degree 2, i.e. by $\alpha\beta\gamma$. The justification for this is that, if we want to multiply together two such matrices then we have to be thinking about our geometric intuition, and matrix multiplication corresponds to composition of functions. This means that we can't just multiply any old $g_{\alpha\beta}$ with any old $g_{\alpha'\beta'}$, because we need the source of one to be the target of the other (we want to know what happens when we go from $U_\alpha$ to $U_\beta$ and then from $U_\alpha'$ to $U_\beta'$), so we need $\beta=\alpha'$. In summary, all this talk lets me write some neat notational shortcut: I have a multiplication map $\cdot$ defined by $(g\cdot g)_{\alpha\beta\gamma}=g_{\alpha\beta}g_{\beta\gamma}$.

Tim Hosgood (Jul 23 2024 at 16:00):

(I worry now that this wall of text with all these indices is going to give a wrong impression. This stuff is much much more "lightweight" than how I'm describing it — when you switch in the abstract maths then I could probably tell this whole story in two or three sentences.)

Tim Hosgood (Jul 23 2024 at 16:03):

But there's another thing I can do to my matrices, which is one where I'll have to appeal to "people already care about this for good reasons", namely the Čech differential. Whenever we have something indexed by open subsets in a cover, say $x_{\alpha_0\ldots\alpha_p}$, then we are interested in the construction that gives us something indexed by open subsets with one more intersection, namely $\check\delta x$, which is defined by

$$(\check\delta x)_{\alpha_0\ldots\alpha_{p+1}} = \sum_{i=0}^{p+1} (-1)^i x_{\alpha_0\ldots\widehat{\alpha_i}\ldots\alpha_{p+1}}$$

where the hat over an index means that we actually don't write that one.

Tim Hosgood (Jul 23 2024 at 16:10):

So let's look at what $\check\delta g$ is! Well, evaluating it on $\alpha\beta\gamma$ (since it's of one Čech degree higher than $g$ itself), we get

$$(\check\delta g)_{\alpha\beta\gamma} = g_{\beta\gamma} - g_{\alpha\gamma} - g_{\alpha\beta}.$$

But now we're going to appeal to some intuition again, just as before. We said that terms indexed by $\alpha\beta\gamma$ should really be talking about how to go from the fibre over $U_\alpha$ to the fibre over $U_\beta$ to the fibre over $U_\gamma$; all in all, we should be getting something that goes from $U_\alpha$ to $U_\gamma$, and only the middle term in the above description of $(\check\delta g)_{\alpha\beta\gamma}$ has the right indices to be of this form.

(Small aside that really doesn't matter: another thing that's back to front is how I say that $g_{\alpha\gamma}$ "goes from" $U_\alpha$ to $U_\gamma$; we nearly always take the opposite convention. Please ignore this, but I just know that it will likely trip me up if I ever come back and read this again.)

It turns out that this small modification (throwing away the first and the last terms) is actually also something that people have already studied: we define the deleted Čech differential $\hat\delta$ by

$$(\hat\delta x)_{\alpha_0\ldots\alpha_{p+1}} = \sum_{i=1}^{p} (-1)^i x_{\alpha_0\ldots\widehat{\alpha_i}\ldots\alpha_{p+1}}$$

because then every single term is indexed by something starting with $\alpha_0$ and ending with $\alpha_{p+1}$, meaning all this intuition about composing maps needing matching sources and targets works out just fine (I promise).

Tim Hosgood (Jul 23 2024 at 16:11):

Ok, with all that incredibly dense amount of setup, what have we gained? Just this one very cool fact: the cocycle condition is exactly the following equation:

$$(\hat\delta g + g\cdot g)_{\alpha\beta\gamma} = 0.$$

In fact, because $(\hat\delta g)_{\alpha\beta}=0$ (there just aren't any terms to sum over), and similarly for the product $g\cdot g$, this equation actually holds true in all Čech degrees, so we can just write

$$\hat\delta g + g\cdot g =0.$$

This is the Maurer–Cartan equation! In other words,

vector bundles are exactly the Maurer–Cartan elements in some (deleted) Čech bicomplex.

Tim Hosgood (Jul 23 2024 at 16:17):

Now all this faff would be pretty pointless if it didn't give us anything else, but with all the work we've done we now get a lot for free! For example,

if we also have some maps $$g_\alpha$ $of the right "shape" and ask for the combined data $g=(g_\alpha,g_{\alpha\beta})$ to satisfy the Maurer–Cartan equation in all Čech degrees, then we get exactly a chain complex of vector bundles.

To see this, note that by adding in some data of Čech degree 1, we now need to check the Maurer–Cartan equation in Čech degree 2 (since $\hat\delta$ increases the degree by one, and the degree of a product of two elements is the sum of their degrees), but, again, the $\hat\delta$ term is zero (there is nothing to sum over), and we just get that $g_\alpha g_\alpha = 0$, which is exactly saying that it defines a differential on the fibre over $U_\alpha$. Looking at what the Maurer–Cartan equation insists we satisfy in degree 2, we get that

$$g_\alpha g_{\alpha\beta} + g_{\alpha\beta}g_\beta = 0$$

which says exactly that the $g_{\alpha\beta}$ are chain morphisms, i.e. morphisms of the chain complexes.

(Here I'm cheating and using the "full" definition of multiplication: $(x\cdot y)_{\alpha_0\ldots\alpha_{p+1}}=\sum_{i=1}^p x_{\alpha_0\ldots\alpha_i}y_{\alpha_i\alpha_{p+1}}$).

Tim Hosgood (Jul 23 2024 at 16:24):

So here's a summary of the above, and the fun fact/definition that is the foundation of essentially all of my research:

A Maurer–Cartan element (in a deleted Čech bicomplex) truncated to degree 0 is exactly a graded vector space; truncated to degree 1 is exactly a complex of vector bundles (because on intersections we have isomorphisms $g_{\alpha\beta}$); and in arbitrary degree is a twisting cochain, i.e. a collection of complexes of vector spaces over each open set, "loosely glued together" by quasi-isomorphisms (not isomorphisms, so we don't get strict vector bundles), with chain homotopies between the quasi-isomorphisms, and higher chain homotopies between these, and ...

Of course, I haven't actually defined this deleted Čech bicomplex formally, but I don't think I really need to unless somebody asks me to :angel:

Tim Hosgood (Jul 23 2024 at 16:25):

So what is the Maurer–Cartan equation? In the world of bicomplexes with one "algebraic" direction and one "topological" direction, it is exactly the thing that gives us a homotopical notion of "chain complex of vector bundles".

Tim Hosgood (Jul 23 2024 at 16:28):

And to end this specific story, here's the bit that really excites (some) geometers: a twisting cochain is "exactly" an object in the (derived bounded) category of coherent sheaves. Coherent sheaves are really exciting, especially to people who care about complex (or smooth) analysis, and this derived bounded category is importantly two things: (1) omnipresent, and (2) really annoying. The category of twisting cochains, however, has a very neat presentation (it's "just" the homotopy limit of something very simple) and can actually be used for calculations in a way that coherent sheaves naturally make very difficult.

Tim Hosgood (Jul 23 2024 at 16:31):

Actually, one last thing for this story: deformation theory also has a lot to say here. It turns out that this "deleted Čech differential" I spoke about above isn't actually a differential strictly on the nose: it doesn't satisfy $\hat\delta^2=0$. But how far away is it from being a differential? Well, lets define a deformation $\hat\delta_\mathfrak{a}$ of $\hat\delta$ by some element (of some bicomplex) $\mathfrak{a}$ by

$$\hat\delta_\mathfrak{a}\colon x \longmapsto \hat\delta x + \mathfrak{a}\cdot x$$

i.e. we take our not-quite-a-differential $\hat\delta$ and just replace it by $\hat\delta+\mathfrak{a}$ (and think of "applying" $\mathfrak{a}$ as being by multiplication). When is this thing actually differential? Exactly when $\mathfrak{a}$ satisfies the Maurer–Cartan equation!

(Some spectral sequence argument can show that this deformation is exactly a "true" Čech differential; e.g. in the case of truncating everything to Čech degree 0 and 1, this deformed deleted Čech differential is exactly the usual total differential on the total complex of a bicomplex).

Tim Hosgood (Jul 23 2024 at 16:33):

I think that's enough words (if not too many) for one story, but this naturally leads in to another story about flat connections... I'll save that for another time :-)

John Baez (Jul 24 2024 at 09:14):

Tim Hosgood wrote approximately:

The invertibility condition is very important, but is secretly of a different flavour than the cocycle condition $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$, and we're going to just look at the latter for now. In fact, we're just going to write it ever so slightly differently to begin with: $-g_{\alpha\gamma}+g_{\alpha\beta}g_{\beta\gamma}=0$.

This is insane, because all of a sudden you're assuming your structure group is actually sitting in an algebra, i.e. you can add group elements as well as multiply them. Of course it is: $\mathrm{GL}(n, \mathbb{C})$ is sitting in the algebra of $n \times n$ matrices. So this is a potentially good sort of insanity. Let me see where it leads....

John Baez (Jul 24 2024 at 09:16):

Tim Hosgood said:

(I worry now that this wall of text with all these indices is going to give a wrong impression. This stuff is much much more "lightweight" than how I'm describing it — when you switch in the abstract maths then I could probably tell this whole story in two or three sentences.)

Yes, but luckily you didn't. Those sentences work best - for me, anyway - if I know ahead of time what they're elegantly summarizing And I'm not scared of a few small Greek letters.

John Baez (Jul 24 2024 at 09:24):

Tim Hosgood said:

So let's look at what $\check\delta g$ is! Well, evaluating it on $\alpha\beta\gamma$ (since it's of one Čech degree higher than $g$ itself), we get

$$(\check\delta g)_{\alpha\beta\gamma} = g_{\beta\gamma} - g_{\alpha\gamma} - g_{\alpha\beta}.$$

Again this is insane, because this is some sort of Čech differential based on adding matrix-valued functions rather than multiplying them. I.e. you're not doing Čech cohomology involving our original sheaf of (holomorphic) $\mathrm{GL}(n,\mathbb{C})$-valued functions, as we would would be if we were classifying bundles. (Since $\mathrm{GL}(n,\mathbb{C})$ is nonabelian for $n \ge 1$ that Čech cochain complex only works for very low degrees.) Now addition reigns supreme!

When I first read you mention the "(deleted) Čech complex" I thought the parenthetical meant "expletive deleted", and now I think that's actually appropriate.

Tim Hosgood (Jul 24 2024 at 12:21):

maybe things become a bit less insane if you do the more general story and replace $\mathrm{GL}$ with the endormorphism module of a graded module, but it's still a very intriguing thing that we start looking at the interplay between additive and multiplicative structures!

Tim Hosgood (Jul 24 2024 at 12:25):

basically it's as you say though: if you want to define a Čech style algebra for some graded object $V$, you can't take $\check{\mathcal{C}}^\bullet(\mathcal{U},V)$ because you'll run into some problems; in the even more specific case where $V=\mathrm{End}(V')$ for some other graded object $V'$, you'll run into this exactly problem where you can't compose things (for the multiplication structure of the algebra) unless you throw away those terms on either end of the usual Čech differential

Tim Hosgood (Jul 24 2024 at 12:27):

all of this story is told in a few papers by O'Brian, Toledo, and Tong, from the 70s and 80s, but for various reasons of mathematical history, most of their work has been forgotten, even to the extent where the nLab page for "twisting cochains" didn't even mention their four or five fundamental papers on twisting cochains!

Tim Hosgood (Jul 24 2024 at 12:28):

(this is my secret reason for wanting to write about this stuff: there are very few people working on this!)

John Baez (Jul 24 2024 at 12:38):

Tim Hosgood said:

all of this story is told in a few papers by O'Brian, Toledo, and Tong, from the 70s and 80s, but for various reasons of mathematical history, most of their work has been forgotten...

Thanks - I'd never seen this stuff.

I'm not done reading your version of the story yet: I just had to stop and say out loud how strange and interesting the basic idea is. As you say, we can do many more strange and interesting things if we replace the automorphism group of a module with its endomorphism monoid: now we can both multiply and add! :tada:

Tim Hosgood (Jul 24 2024 at 14:24):

One thing that lets you get away with the deleted Čech differential is that all the terms you throw away actually turn up in the multiplication — the whole point of the Maurer–Cartan equation (in this story) is somehow exactly that. You want the Čech complex but can't have it (because you can't get an algebra structure on it because you can't multiply elements because their Čech indices don't match up), so instead you split things into two: the terms of the Čech differential which all work (e.g. all those beginning with the same index, or beginning and ending with the same indices), and those that you just need to modify (e.g. $g_{\beta\gamma\delta\ldots}$ doesn't live over $U_\alpha$, but $g_{\alpha\beta}g_{\beta\gamma\delta\ldots}$ does!). The magic of Maurer–Cartan is that it says "hey, I'll make sure that all the terms you threw away from the Čech differential actually do turn up where you need them and still satisfy the right equations, but I'll 'pull them back' to live over the right open set for you". In other words, "I'll split up your Čech differential $\check\delta(-)$ into $\hat\delta(-)+\mathfrak{a}\cdot(-)$, and it'll be exactly what you want as long as $\mathfrak{a}$ satisfies Maurer–Cartan".

John Baez (Jul 25 2024 at 07:26):

That's an interesting hands-on attitude.

John Baez (Jul 25 2024 at 07:38):

This is quite cool:

A Maurer–Cartan element (in a deleted Čech bicomplex) truncated to degree 0 is exactly a graded vector space; truncated to degree 1 is exactly a complex of vector bundles (because on intersections we have isomorphisms $g_{\alpha\beta}$); and in arbitrary degree is a twisting cochain, i.e. a collection of complexes of vector spaces over each open set, "loosely glued together" by quasi-isomorphisms (not isomorphisms, so we don't get strict vector bundles), with chain homotopies between the quasi-isomorphisms, and higher chain homotopies between these, and ...

So let me see if I've got this right. We can think of an object in the derived category of coherent sheaves as merely a chain complex of coherent sheaves if we take the attitude that the "derived" baloney is just taking the category of chain complexes of coherent sheaves and changing the morphisms, not the objects. If we take this attitude we see that we can build an object in the derived category of coherent sheaves by gluing together chain complexes of coherent sheaves using good old-fashioned maps $g_{\alpha \beta}$ that obey the 1-cocycle condition $g_{\alpha \beta} g_{\beta \gamma} = g_{\alpha \beta}$ 'on the nose'.

But while this is true, it's very short-sighted: we're changing the morphisms in a dramatic way by inverting the quasi-isomorphisms. We're 'softening things up' and making it more homotopical. So there are much more general ways to build an object in the derived category of coherent sheaves. We can 'loosely' glue together chain complexes of coherent sheaves using maps $g_{\alpha \beta}$ that obey the 1-cocycle equation up to chain homotopy, say

$h_{\alpha \beta \gamma}: g_{\alpha \beta} g_{\beta \gamma} \to g_{\alpha \beta}$

where these homotopies obey the 2-cocycle condition up to a further chain homotopy, and so on ad infinitum (or ad nauseum, whatever comes first).

John Baez (Jul 25 2024 at 07:41):

And so, to keep from getting crushed under the weight of this extra data, we need the framework you're talking about to describe this more general way to build objects in the derived category of coherent sheaves.

Tim Hosgood (Jul 25 2024 at 09:34):

the only thing I would change in what you say is what you're gluing together here: the true statement is that (chain complexes of) coherent sheaves are (chain complexes of) locally free sheaves (vector bundles) glued together "homotopically"

Tim Hosgood (Jul 25 2024 at 09:34):

but apart from that, yes, exactly!

Tim Hosgood (Jul 25 2024 at 09:35):

you can even prove a big formal statement like "the $(\infty,1)$-category of Maurer–Cartan elements is a presentation of the derived bounded category of coherent sheaves"

Tim Hosgood (Jul 25 2024 at 09:36):

what's oddly magic is that, in the world of algebraic geometry (so whenever your complex manifold is projective, say) all of this "gluing up to homotopy" is entirely redundant — you can prove that you get exactly the same objects (complexes of coherent sheaves) if you just glue everything together strictly!

Tim Hosgood (Jul 25 2024 at 09:37):

algebraic stuff is so "rigid" that even if you try to glue it together with weak homotopical tape, then it behaves as if you'd used strict isomorphic-y superglue

John Baez (Jul 25 2024 at 10:44):

Thanks! I forgot "locally free" because somehow the word "coherent" had turned into "quasicoherent" in the sloppy workings of my brain.

John Baez (Jul 25 2024 at 10:44):

But anyway, that's very wonderful.

Tim Hosgood (Jul 25 2024 at 11:34):

i'm glad you think so! but now the real question is... what does this have to do with the story you told, or anything on the wikipedia page about Maurer–Cartan? :thinking:

John Baez (Jul 25 2024 at 11:50):

Here's how I'd try to unify the stories. I had said something homotopic to this:

If $V$ is a vector space, then making the exterior algebra $\Lambda V^*$ into a differential graded-commutative algebra is the same as making $V$ into a Lie algebra.

I got this by talking about the Maurer-Cartan equation but let's spell out how it works. If $e_i$ is a basis for $V$ let $\omega^i$ be a basis for $V^*$. We're trying to make $\Lambda V^*$ into a differential graded-commutative algebra. To specify the differential it's enough if I tell you $d\omega^i$ for each $i$. But we must have

$d \omega^i = c_{ij}^k \omega^i \wedge \omega^j$

for some constants $c_{ij}^k$. Then $d^2 = 0$ winds up implying that these constants have to be the structure constants of a Lie algebra! :tada:

That is, $d^2 = 0$ if and only if when we define

$[e_i, e_j] = c_{ij}^k e_k$

this bracket obeys the Jacobi identity!

John Baez (Jul 25 2024 at 11:52):

We can pack all these $\omega^i$ guys into a single entity called the Maurer-Cartan form - I described that last time. Also, the equation

$d \omega^i = c_{ij}^k \omega^i \wedge \omega^j$

is called the Maurer-Cartan equation. (Last time I stuck in a $\frac{1}{2}$ but it's not important.)

So all this stuff is very Maurer-Cartany.

John Baez (Jul 25 2024 at 11:54):

I think we can vaguely summarize this - and it's important let ourselves be vague when trying to make big connections - by saying "to give something a nice differential is very Maurer-Cartany".

John Baez (Jul 25 2024 at 11:59):

Well, I'm not sure the best way to go from here, but I do want to say that my motto is a baby case of something more heavy-duty and perhaps closer to what you're talking about. I said:

If $V$ is a vector space, then to make the exterior algebra $\Lambda V^*$ into a differential graded-commutative algebra is the same as to make $V$ into a Lie algebra.

But more generally:

If $V$ is a chain complex, then to make the free graded-commutative algebra on the dual cochain complex with its grading shifted by 1 into a differential graded-commutative algebra is the same as to make $V$ into a differential graded Lie algebra.

John Baez (Jul 25 2024 at 12:00):

The second motto reduces to the first one when we take a chain complex that's supported in degree 0.

John Baez (Jul 25 2024 at 12:05):

This motto is part of the thing called "Koszul duality". So one question, @Tim Hosgood, is whether those Maurer-Cartan/twisting cocycle people ever mention Koszul duality.

John Baez (Jul 25 2024 at 12:12):

This thing reminds me of some things you were saying.

Tim Hosgood (Jul 25 2024 at 14:12):

I think the link between these two stories might be in trying to understand how the nLab explains what a [[twisting cochain]] is, in terms of dg-coalgebras and dg-algebras

Tim Hosgood (Jul 25 2024 at 14:12):

that's a point of view i've never understood

John Baez (Jul 25 2024 at 14:19):

Okay, I will see if I can get anywhere understanding that. I actually do sort of understand the block of text I just quoted from the nLab, but I've never thought about "twisting cochains" except for, say, twisting a vector bundle by a cohomology class, or other lowbrow things like that.