## May 31, 2007

### On the Bar Construction

#### Posted by Urs Schreiber

– guest post by Todd Trimble

Here is a comment about the bar construction which I wanted to mention, since John told me that his course is ending soon and he might not get around to mentioning it himself.

I’d like to thank Urs for suggesting this be made a guest post, and all three hosts for the hard work they put into making the Café such a great success.

As John said, the bar construction is a wonderful gadget which can be used to compute cohomology for general algebraic theories (groups, algebras, Lie algebras, …), as well as for other things like constructing classifying bundles. Given any monad $M$ and an $M$-algebra $X$, the bar construction is an efficient machine that produces a ‘free resolution’ of $X$, to which one can then apply the machinery of derived functors to compute the cohomology of $X$.

What John gave in his course was a very nice introduction, with lots of diagrams and pictures, which gave the precise conceptual details underlying the bar construction: a simplicial object whose $n$-dimensional component is a free algebra $M M^n X$ over the monad $M$, with face operators which look like:


...      mMX
-->     mX
MmX     -->     a
... MMMX --> MMX     MX --> X
-->     -->
MMa     Ma



What I want to talk about here is the conceptual sense in which the bar construction is a resolution of $X$, i.e., the acyclicity of the bar construction as a simplicial object. I also want to talk about the sense in which the bar construction on $X$ is a universal $M$-free resolution of $X$.

In order for this note to make sense, I’ll first have to recall some of what John said. A monad $M$ is a monoid in a monoidal category, and the story begins by considering the string diagram calculus for monoids in monoidal categories. As John showed in pictures, the string diagrams, modulo the equivalence relation imposed by the monoid equations, look exactly like pictures (cospans) of order-preserving functions between finite ordinals. This was summarized in the slogan, “the category $\Delta$ is the walking monoid”, i.e., $\Delta$ equipped with its terminal object [1] is precisely the initial strict monoidal category equipped with a monoid.

Or, we could say that $\Delta^{op}$ is the “walking comonoid”. It’s nice to know by the way that $\Delta^{op}$ is equivalent to the category of finite intervals (i.e., finite totally ordered sets with a top and bottom, and maps which preserve the order and top and bottom). This is something you can see very well if you stare at John’s pictures of planar 2d cobordisms, which are planar thickenings of string diagrams for monoids, and then let your attention flicker over to their complements (of different shading) and read them upside-down as 2d cobordisms. This idea and the planar 2d cobordisms also figure prominently in Aaron Lauda’s Frobenius algebras and ambidextrous adjunctions.

With that, the bar construction is easy to construct. Let $(M, m: M M \to M, u: 1 \to M)$ be a monad on a category $E$. One has an adjunction

$(F: E \to E^M) \dashv (U: E^M \to E)$

where $E^M$ is the Eilenberg-Moore category of $M$-algebras. This in turn gives a comonad $F U$ acting on $M$-algebras, that is to say, a comonoid in a monoidal category of endofunctors. Because $\Delta^{op}$ is the walking comonoid, there is a unique monoidal functor (the bar construction)

$Bar_M: \Delta^{op} \to [E^M, E^M]$

which sends the comonoid [1] in $\Delta^{op}$ to $F U$. Applying the functor which evaluates at an $M$-algebra $X$, one gets a simplicial $M$-algebra

$B_M(X): \Delta^{op} \to E^M$

and this is the bar construction applied to $X$. This description appears in Mac Lane’s Categories for the Working Mathematician, which is where I first learned about it (see section VII.6 in the second edition).

As I say, the important point here is that this object is acyclic, in a very strong sense of the word which I need to explain. Part of the philosophy here is that the bar construction is a piece of pure equational logic, and we will correspondingly treat its acyclicity purely algebraically, not as a property involving an existential quantifier but as a structure, called a ‘resolution’ for lack of anything better. Roughly speaking, a resolution structure on a simplicial object $Y$ with augmentation,


... Y[2] --> Y[1] --> Y[0],
-->



is a contracting homotopy which realizes a homotopy equivalence between $Y$ and the constant simplicial object at $Y[0]$, and moreover a contracting homotopy with some very good properties.

Recall that a contracting homotopy $h$ on a simplicial object $Y: \Delta^{op} \to E$ is given by a collection of maps

$h_n: Y[n] \to Y[n+1]$

satisfying some equations. Observe also that the map $[n] \mapsto [n+1]$ is the object part of a comonad $[1] + (-): \Delta^{op} \to \Delta^{op}$. Here is the key notion:

• A resolution is a simplicial object $Y$ together with a right-sided coalgebra structure $h: Y \to Y \circ ([1] + (-))$ over the comonad $[1] + (-)$.

This will require some unpacking, but let’s first see how this works in the case of the bar resolution. Since $Bar_M$ preserves the monoidal and comonoid structures, we have a commutative diagram


Delta^{op} -----> [E^M, E^M]
|       Bar_M      |
|                  |
| [1]+( )          | FUo( )
V                  V
Delta^{op} -----> [E^M, E^M]
Bar_M



and we augment this diagram with another:


Delta^{op} -----> [E^M, E^M] ---> [E^M, E]
|       Bar_M       |    U o( )   |
|                   |             |
|[1]+( )      FUo( )|       UFo( )|
V                   V             V
Delta^{op} -----> [E^M, E^M] ---->[E^M, E]
Bar_M            U o( )



The bar resolution is the horizontal composite $UBar_M$. Notice there is is a canonical right $F U$-coalgebra structure on $U$, given by the coaction $u U: U \to U F U = M U$. This gives a right action of the comonad $[1]+ (-)$ on $UBar_M$, as we see from the diagram, therefore we obtain a resolution structure with components

$h_n = u M^n: M^n \to M^{n+1}.$

Now, what exactly is a resolution structure? I’d like to tease that out.

A first trivial observation is that a right-sided coalgebra for the comonad $[1]+(-)$ is the same as an ordinary (left-sided) coalgebra for the comonad on simplicial objects

$E^{[1]+(-)}: E^{\Delta^{op}} \to E^{\Delta^{op}}$

given by pulling back along $[1] + (-)$. I’ll call this pullback comonad $P$, for short.

Second, to make things easier (or perhaps more recognizable), let’s pretend for a moment that $E = Set$. The pullback comonad $P$ then has a left adjoint, given by left Kan extension along $[1]+ (-)$. What is that left Kan extension? On the representable objects, it takes the affine simplex $hom(-, [n])$ to $hom(-, [n+1])$, i.e, it takes the cone of the affine simplex. This cone construction extends to all simplicial sets (by left Kan extension). I’d like to call the Kan extension the ‘cone functor’ on simplicial sets, although we should be careful here: we are not coning a space or simplicial set to a point, because that operation does not preserve disjoint sums (therefore isn’t a left adjoint). What the left Kan extension actually does is cone a space to its set of path components, i.e., it takes the disjoint union of all the cones over all the path components. But as I say, I’m still going to call this left Kan extension the ‘cone functor’, and denote it $C$.

Now, we are in a general situation where we have a comonad $P$ with a left adjoint $C$. There is a general piece of abstract nonsense here, dating back to a famous 1965 paper by Eilenberg and Moore (Adjoint functors and triples, Ill. J. Math. 9, 381-395), which says that the comonad structure on $P$ corresponds precisely to a monad structure on $C$, in such a way that the category of $P$-coalgebras is naturally equivalent to the category of $C$-algebras. So, now we want to know: what is a $C$-algebra?

Personally, I find it more intuitive to think about it topologically. There is an analogous cone monad $C: Top \to Top$ acting on topological spaces, where $C X$ is defined to be the pushout of the diagram

$[0, 1] \times X \stackrel{inj_0}{\leftarrow} X \stackrel{q}{\to} \pi_0(X)$

where $inj_0(x) := (0, x)$ and $q$ is the canonical projection. A map $a: C X \to X$ is thus given by a pair of maps

$h: [0, 1] \times X \to X$ $i: \pi_0(X) \to X$

such that $h(0, x) = iq(x)$. If $a: C X \to X$ is a $C$-algebra structure, then $i$ picks out a basepoint in each path component, and $h$ is a homotopy which contracts each path component to its basepoint.

More precisely, the monad structure on $C$ is induced from a monoid structure on $[0, 1]$, and we will take the monoid multiplication on $[0, 1]$ to be the ‘tropical’ multiplication $(x, y) \mapsto min(x, y)$. An algebra with respect to the monad $C$ is given by a pair $(h, i)$ such that the homotopy $h$ behaves as a ‘flow’ with respect to tropical multiplication: the following equations are satisfied:

$h(s, h(t, x)) = h(min(s, t), x)$ $h(1, x) = x$ $h(0, x) = iq(x).$

[A note to experts: the topos of simplicial sets classifies the theory of intervals; in particular, geometric realization $Set^{\Delta^{op}} \to Top$ is the model which takes the generic interval to the interval [0, 1]. The generic interval carries an intrinsic monoid operation ‘min’, and geometric realization thus maps it over to the monoid $([0, 1], min)$; this is why we took the multiplication on the unit interval to be ‘tropical’.]

The cone monad on topological spaces is again left adjoint to a comonad $P$ on topological spaces, where $P X$ is the pullback of the diagram

$X^{[0, 1]} \stackrel{eval_0}{\to} X \stackrel{id}{\leftarrow} |X|$

where $|X|$ is the underlying set of $X$ equipped with the discrete topology, included in $X$ by the identity function. As before, a $P$-coalgebra structure is equivalent to a $C$-algebra structure.

• In summary: a resolution = a $P$-coalgebra = a $C$-algebra = a simplicial object $Y$ equipped with a ‘homotopy flow’ which contracts $Y$ to the discrete space of its path components $Y[0]$. In other words, a resolution is a particularly nice kind of contracting homotopy witnessing acyclicity of $Y$.

Now for the main theorem:

• The bar resolution $UBar_M$ is a universal $M$-algebra resolution for the monad $M$.

This may require some amplification. As above, let $B_M(X)$ denote the result of applying the bar resolution functor to a particular $M$-algebra $X$, i.e., let $B_M(X)$ be the composite

$\Delta^{op} \stackrel{Bar_M}{\to} [E^M, E^M] \stackrel{eval_X}{\to} E^M \stackrel{U}{\to} X.$

Let $Y$ be any $M$-algebra resolution, i.e., a simplicial $M$-algebra $\Delta^{op} \to E^M$ equipped with a $P$-coalgebra structure on the underlying simplicial object

$U Y: \Delta^{op} \to E.$

There is a forgetful functor from the category of $M$-algebra resolutions to $M$-algebras, which remembers only the augmentation part $Y[0]$.

The universal property of $B_M(X)$ is that given an $M$-algebra resolution $Y$ and an $M$-algebra map $X \to Y[0]$, there is a unique extension to a map of $M$-algebra resolutions, $B_M(X) \to Y$. This is the familiar ‘acyclic models theorem’:

• Theorem (Acyclic Models): The bar resolution $B_M(-)$ is left adjoint to the forgetful functor from $M$-algebra resolutions to $M$-algebras.

Sketch of proof: The unique simplicial extension $f: B_M(X) \to Y$ of $f_0: X \to Y[0]$ is constructed by induction on dimension. The map $f_{n+1}: M^{n+1} X \to Y$ is the unique $M$-algebra map which extends the composite

$M^n X \stackrel{f_n}{\to} Y[n] \stackrel{h_n}{\to} Y[n+1].$

It is immediate that $f$ preserves the contracting homotopies; all that remains to be checked is that $f$ is a simplicial map. This is based on a simple inductive argument; details may be found in my notes.

Posted at May 31, 2007 10:05 PM UTC

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### Re: On the Bar Construction

Just to add to Todd’s beautiful comment about bar-construction. There is another interpretaton of bar-construction as a resolution.

The following construction is in my paper “Categorical Strong Shape Theory”, Cahiers de topologie et geometrie differentielle categoriques, XXXVIII-1, 1997,3-66. If A:K → L is a simplicial functor between simplicial categories one can define a notion of coherent A-resolution (and A-coresolution) of an object x of L with respect to A (indeed, one can do it for any categories enriched over any monoidal model category). We can use these resolutions to construct strong shape (coshape) theory for A. In particular case, when L is a category of algebras of a simplicial monad T, and K is its full subcategory of free algebras (so A is a the full inclusion), the bar-construction of X gives its A-coresolution. In other word, the inclusion A has a strongly coherent left proadjoint given by bar-construction. The strong coshape theory in this case is the category of algebras of T and their strongly coherent morphisms (or weak morphisms if you prefer). I think it is somehow related to both theorems from Todd’s comment.

Another curious comment is that we can use bar-construction (both one-sided and two-sided) to construct all sort of universal categories like Δ, Ω, etc.. This is funny because Todd starts his explanation from considering Δ, but Δ itself is a bar-construction for categrical free monoid monad. For details see my paper “The Eckmann-Hilton argument and higher operads” — specifically the section about internal algebras of cartesian monads. (I hope I will post to the archive a final version of it, which will appear in Advances, very soon.)

Posted by: Michael on June 1, 2007 2:55 AM | Permalink | Reply to this

### Re: On the Bar Construction

Michael wrote:

Another curious comment is that we can use bar-construction (both one-sided and two-sided) to construct all sort of universal categories like Δ, Ω etc.. This is funny because Todd starts his explanation from considering Δ, but Δ itself is a bar-construction for categrical free monoid monad.

I’m glad you mentioned this, Michael. In part you seem to be saying

• The bar construction as applied to a cartesian monad is the nerve of a category.

This is easy to prove but actually very useful in practice, as you obviously know. For example, the free nonpermutative nonunital operad monad is cartesian, and the bar construction leads to nerves of partial orderings which give face incidence relations on classical polytopes such as the associahedra.

I’ll keep an eye out for the final version of your Eckmann-Hilton paper that you mentioned.

I’d also like to look at your strong shape theory paper soon, but my good computer is in the shop and the computer I have now can’t handle ps files. I wish I knew the meaning of ‘shape theory’, and also of the word ‘coherent’ — the results you mention do sound tantalizing!

Posted by: Todd Trimble on June 1, 2007 1:49 PM | Permalink | Reply to this

### Re: On the Bar Construction

The bar construction as applied to a cartesian monad is the nerve of a category.

Is this statement meant to hold in both directions? Is the nerve of every category the bar construction of some cartesian monad?

Posted by: urs on June 1, 2007 2:42 PM | Permalink | Reply to this

### Re: On the Bar Construction

I hadn’t intended it both ways, but it’s a fun question nevertheless!

Here’s how far I took it: a category $C$ is a monoid in Span and hence induces a monad

$(-) \circ C: Span \to Span$

which is in fact cartesian. (For readers who might not know what ‘cartesian’ means: it’s a monad which as a functor preserves pullbacks, and such that the squares which express naturality of the multiplication and of the unit of the monad are all pullback squares.)

A span $X: 1 -\mapsto C_0$ with a $C$-algebra structure is a discrete fibration over $C$ or an internal functor with domain $C$, which we may also call a $C$-action. If we take the bar construction $B_C(X)$, we get something that looks like

$...X \times_{C_0} C_1 \times_{C_0} C_1 \stackrel{\to}{\to} X \times_{C_0} C_1 \to X$

and in the case where $C$ is a groupoid, this is the nerve of the ‘action groupoid’. (I don’t know the standard term for this in the more general category case; maybe ‘translation category’ or something.)

In the case where $X$ is the span

$1 \leftarrow C_0 \stackrel{id}{\to} C_0$

we get the nerve of the category of arrows on $C$ (with morphisms = commutative triangles), augmented over $C_0$ by taking ‘domain’:

$... C_2 \stackrel{\to}{\to} C_1 \to C_0$

which is sort of ‘close but no cigar’ to the nerve of $C$ itself, which I don’t see how to get in general via bar, unfortunately.

I may return to this type of example a little later, and connect up with two-sided bar constructions, which Michael Batanin mentioned in passing.

Posted by: Todd Trimble on June 1, 2007 4:13 PM | Permalink | Reply to this

### Re: On the Bar Construction

First to reply to Todd. Shape theory is Cech homotopy’ at least in its original form. I could go on for hours about the relationships between this and the rest of maths … but won’t. If you can get hold of a copy, Jean -Marc Cordier and myself wrote a book
J.-M. Cordier and T. Porter, “Shape theory: Categorical methods of approximation”, Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1989.
Which although out of print is, I hope, shortly to be reprinted.
It tries to encode the idea of approximation of a general object (often a space) by a nice object (often a nice space such as a polyhedron or CW-complex). The categorical theory is very pretty. For coherent’ I presume Michael means homotopy coherent’ and again there is stuff by Cordier and myself on this back some 20 years ago!

Micheal’s work on Strong Shape was related to the problem of rectifying’ or rigidifying the approximations in the presence of homotopy information.

Posted by: Tim Porter on June 25, 2007 1:54 PM | Permalink | Reply to this

### Re: On the Bar Construction

Thanks, Tim. I’ll see about getting a hold of your book with Cordier; in the meantime I have your joint paper, Homotopy coherent category theory (1996), and I also have Batanin’s Categorical strong shape theory before me now; both should be very helpful. (Online references which do not require institutional subscriptions or “pay per view” are very welcome.)

By the way, I hope you do make your lecture notes mentioned
here
available; I’m pretty sure a number of people here (myself included) would be interested.

Posted by: Todd Trimble on June 26, 2007 1:10 PM | Permalink | Reply to this
Read the post Cohomology and Computation (Week 24)
Weblog: The n-Category Café
Excerpt: What makes the bar construction tick?
Tracked: June 7, 2007 5:02 PM

### Re: On the Bar Construction

Todd, this is all quite nice. Back in May, I didn’t have enough motivation to more than skim it, but this morning I wanted to find a reference for why the bar construction is acyclic. And guess where Google brought me.

I have a question, though: Is this all written down some place in a paper or a book that I could reference, or is it only on web pages (= this post + your email John posted at UCR)?

Posted by: James on November 30, 2007 6:52 AM | Permalink | Reply to this

### Re: On the Bar Construction

I think if you look at the latest version of comments on this post, you will discover that some people mean the bar conctruction to be a resolution of the ground field
others take it to correspond to the clssifying space, cf. the one-sided vs 2-sided vs no sided bar construction.

Wiki has no entry!

Posted by: jim stasheff on November 30, 2007 2:10 PM | Permalink | Reply to this

### Re: On the Bar Construction

Thanks, James. I’ve not seen this specific formulation written down elsewhere, but for the acyclicity part you might see whether either of

• M. Barr and J. Beck, Acyclic models and triples. Proceedings of the Conference on Categorical Algebra (La Jolla), New York: Springer (1966), 336-344
• M. Barr and J. Beck, Homology and standard constructions. Seminar on Triples and Categorical Homology Theory, Springer LNM 80 (1969), 245-336

meets your purposes. The formulation of acyclicity I gave above, in terms of coalgebras over the decalage comonad $P$ acting on simplicial objects, is a little stronger than the notion of acyclicity as it usually appears in the literature: there, to get the homotopy equivalence $B_M(X) \sim X$, it is sufficient to establish just the counit law

$(B_M(X) \stackrel{h}{\to} P B_M(X) \stackrel{\varepsilon}{\to} B_M(X)) = 1_{B_M(X)}$

in order to see we have a contracting homotopy $h$ to the constant simplicial object $X$. (In my post, we also have coassociativity of the coaction $h$.)

In his comment, Michael Batanin seems to be hinting that the formulation I gave is a special case of a more general result of his, but I have not found the opportunity to look into this. Perhaps if he could confirm that, then his papers would serve as suitable references.

Jim Stasheff is right that my post deals specifically with a one-sided bar construction used to resolve an algebra $X$; it can be used to produce for instance the total space of the classifying bundle for a group. There is also a nice categorical story which can be told for two-sided bar constructions $B(Y, M, X)$ (where the relevant data is a monad $M$, a left $M$-module $X$, and a right $M$-module $Y$); the story I told above is about the special case where $Y = M$). One reference I know for that (the construction, not so much the nice categorical story) is

• J.P. May, The Geometry of Iterated Loop Spaces. Springer LNM 271, 1972.
Posted by: Todd Trimble on November 30, 2007 5:32 PM | Permalink | Reply to this

### Re: On the Bar Construction

Thanks! I’ll have a look.

Posted by: James on November 30, 2007 10:53 PM | Permalink | Reply to this

### Re: On the Bar Construction

James wrote:

Is this all written down some place in a paper or a book that I could reference, or is it only on web pages (= this post + your email John posted at UCR)?

What’s ‘only’ about a web page? I reference web pages all the time in my published papers: I just write down the author, title and URL.

And these days, all my published papers are also available on the arXiv as PDF files with hyperlinks, so you can just click on the references and instantly jump to them. Anything short of this is antiquated and obsolete, since HyperTeX is incredibly easy to use for anyone who knows TeX or LaTeX. If the source is available online, why make the reader struggle to find it?

(Being an ornery cuss, I only provide links to freely available online sources, not sites where you need a paid subscription.)

If you want to see an example, try the annotated bibliography of this paper on cohomology and n-categories. You’ll see references and links to a bunch of papers on the arXiv, and also to a bunch of papers on other websites.

Posted by: John Baez on December 1, 2007 9:49 AM | Permalink | Reply to this

### Re: On the Bar Construction

What’s ‘only’ about a web page?

One thing that comes to mind is that a web page like this post hasn’t undergone the usual review process (hasn’t been vetted by anonymous referees for example), and so in the eyes of the community hasn’t yet properly established its credentials. I’m not sure to what degree such attitudes may be changing, but I think they’re certainly understandable. (This may not be at all what James was thinking; I’m just throwing it out there.)

Posts on math blogs (or for that matter on wikipedia) do seem to undergo vetting of a sort, in that there is usually a supply of people who are willing to pipe up if they see something wrong; I know I often do. So it may be possible for ideas which have been presented and worked over in a strong group blog like the $n$-Category Café to eventually earn a reasonable level of reliability, pretty much as in a seminar setting, I imagine. (Cf. Secret Blogging Seminar!) I’d think that such reliability would be cemented even further by something like an $n$-categorical wiki, if that gets off the ground.

I’d love to hear further discussion of these points.

Posted by: Todd Trimble on December 1, 2007 1:22 PM | Permalink | Reply to this

### Re: On the Bar Construction

Where is the how to’ page?

jim

Posted by: jim stasheff on December 1, 2007 1:41 PM | Permalink | Reply to this

### Re: On the Bar Construction

The only aspect of HyperTeX that I ever use is ‘hyperref’. A how-to guide for that is here. However, I only use a pathetically tiny set of the features!

If you use LaTeX on a UNIX system run by a math department, it’s likely that HyperTeX is already up and running. In that case, all you need to do is the following.

Put this line near the beginning of your LaTeX file, but after any other packages you may use:

\usepackage{hyperref}

Then, to create a hyperlink anywhere in your LaTeX file, simply type this:

\href{X}{Y}

where X is the URL you want a link to, and Y is the text you want the reader to see and click on. Then simply run LaTeX.

For example, typing this in your LaTeX file:

\href{http://math.ucr.edu/home/baez/diary/}{John Baez’s Diary}

should produce a link like this:

John Baez’s Diary

For me, this works when I create a PDF file, but not Postscript. Apparently dvips should create a Postscript file with hyperlinks if you have version 5.60a or higher.

As always, getting help from someone under age 25 is a good idea! I believe Urs was the one who showed me how to do this stuff.

Posted by: John Baez on December 3, 2007 1:20 AM | Permalink | Reply to this

### Re: On the Bar Construction

I think web pages are better than books and journals in many respects but definitely not in every respect. The one I had in mind above was longevity. For instance, I doubt that the URLs for Todd’s post here or his email message stored at UCR will work in 100 years. But I bet that if I give a standard reference to a 2007 Annals paper, say, there will be a straightforward way to look that up in 100 years. Your local library’s copy might be in storage, but it will probably be easy to look at an electronic version and almost as easy to retrieve the bound version.

On the other hand, I do believe that the Archive will still exist in some form in 100 years, so in principle I don’t have a problem with referencing it.

Posted by: James on December 2, 2007 2:06 AM | Permalink | Reply to this

### Re: On the Bar Construction

I agree that Todd and everyone else should put their papers on the arXiv, for better permanent storage. I do that with all my published papers — and someday I’ll do it with This Week’s Finds (which people often reference in its webpage form).

However, the issue of permanence isn’t as bad as you think. Lots of old webpages are available permanently via the Wayback Machine. For most people, this is much easier to access than a library in a university!

I think the bottom line is that if a scholar learns something useful from a webpage, they really do have a duty to cite it. Part of the point of references is to let people know where one got ones ideas. The alternative is called ‘plagiarism’. This is why people include references to ‘personal communications’, even though tracking down such a reference can be very tough.

Posted by: John Baez on December 2, 2007 7:29 AM | Permalink | Reply to this

### Re: On the Bar Construction

Hey John, that Wayback Machine is really great! The resolution (in time) isn’t very fine, but it’s still pretty good and a lot of fun to play with. And probably everyone will know about it in 100 years…

I would disagree with the strong form (which you probably didn’t mean) of the statement that scholars have a duty to cite where they learned useful things. For instance, with the acyclicity of the bar construction, I don’t think I have a duty to try to go back and remember which textbook, web page, friend, or whatever I learned it from. For standard results like that, I prefer to give the best reference I can find. That could be either the original source, or the one a newcomer would find most useful, or …

Posted by: James on December 2, 2007 10:11 AM | Permalink | Reply to this

### Re: On the Bar Construction

James: That could be either the original source, or the one a newcomer would find most useful, or ….

Where possible, it would be good to do both.
For me, the bar construction goes back to the Cartan seminaire 54-55 but there may be references there to earlier work.

Oh, that’s for the dg case. For ordinary algebras, see Eilenberg-Mac Lane.

Anyone have better suggestions?

Beware - sometimes the disciples are NOT as clear as the mster, viz. Dirac’s `functions’.

Posted by: jim stasheff on December 2, 2007 1:29 PM | Permalink | Reply to this

### Re: On the Bar Construction

James wrote:

I would disagree with the strong form (which you probably didn’t mean) of the statement that scholars have a duty to cite where they learned useful things.

There are certainly limits to this, especially when it comes to ‘standard results’.

If I seem overly touchy, it’s because Todd Trimble has done some very interesting research that’s only been ‘published’ on my website. So, the idea that someone could be reluctant to cite this work merely because it’s on a website makes me feel bad.

For instance, with the acyclicity of the bar construction, I don’t think I have a duty to try to go back and remember which textbook, web page, friend, or whatever I learned it from. For standard results like that, I prefer to give the best reference I can find.

There’s certainly a sense in which the acyclicity of the bar construction is a standard result… and if that sense is what you mainly care about, there must be some classic reference that’s right for your purposes. (I’d like to know what you pick!)

But in some ways Todd’s result seems new — at least to me. For example, he makes acyclicity into a structure, not a mere property. This allows him to describe the bar construction as an initial object. This characterizes it up to isomorphism, not just up to something like ‘homotopy equivalence’ (really ‘weak equivalence’). I find this quite nice, and I don’t know any other reference for this result.

On a wholly different note: I finally revised week257 to take into account your corrections. I changed the body of the text and also added a quote from you to the Addenda. I cite you merely as ‘James’. Is that optimal?

Posted by: John Baez on December 4, 2007 3:00 AM | Permalink | Reply to this

### Re: On the Bar Construction

With Zoran Škoda I am talking about the relation between algebraic/monadic definitions of descent and cohomology and “geometric” definitions (in lack of a better term).

What I believe I do understand reasonably well is generalized cohomology in the sense of “nonabelian sheaf cohomology” in terms of generalized maps as insdicated in $n$Lab:nonabelian cohomology and homotopical cohomology theory.

What I have thought about much less are what seem to be parallel developments, related to the keywords “monadic descent”, “Barr-Beck theorem” and the like, which is closely related to what Todd is talking about in the above entry.

What I am hoping for is a general picture which allows me to understand these different approaches naturally as two aspects of the same general phenomenon. I have some vague ideas about this, but nothing really satisfactory yet.

For instance, in what i am calling the “geometric” picture, Street defines a notion of cohomology for every cosimplicial $\omega$-category. I suppose this should cover large classes of examples of “monadic descent” or the like for the case that we start with a comonad on $\omega$-categories and apply a co-bar construction.

Or something like this. I wish I could formulate a more concrete question. But maybe this already rings a bell with somebody who can help out with providing a bit of a coherent picture.

Posted by: Urs Schreiber on December 29, 2008 2:51 PM | Permalink | Reply to this

### Re: On the Bar Construction

Sorry for the delay. I put down something in nLab (under the entry for stack) which attempts to get a discussion going along the lines of what you may be after here.

Posted by: Todd Trimble on January 3, 2009 12:13 AM | Permalink | Reply to this