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Stream: learning: questions

Topic: Bar construction of a monad

fosco (Jul 20 2024 at 11:19):

I need to understand a bit the old topic of co/monad co/homology for a problem I'm working on.

I am trying to get my hands on some explicit computations but I need some help with them, and with a few more conceptual questions:

• first of all I am confused by notation: a monad (from an algebra) should give a simplicial object, and dually a comonad (from a coalgebra) a cosimplicial one. Indeed, nLab agrees with me. But Barr-Beck's "Homology and standard constructions" p. 187 seems to say that a comonad gives an (augmented) simplicial object. What's up? I'm also confused by the assumption of taking an algebra: without it, $(\{\eta\},\{\mu,\eta T,T\eta\},\dots)$ still arrange in a cosimplicial object, and dually a comonad with counit and comultiplication arranges in a simplicial one. Why the dualization, why is the additional assumption more natural/less trivial?

• I started doing a computation with a very stupid monad: $A\mapsto A\oplus \mathbb Z$ on abelian groups. An hour of very fun linear algebra later I proved quite hands-on that this gives a complex with zero homology. Pretty sure that there is a conceptual reason why; again following Barr-Beck, free algebras should be acyclic, does this help? Like: what contractible space has the same homology of the one prescribed by this "additive maybe monad"?

• the previous question makes me wonder: has anyone computed the bar construction of (say) the state monad, the continuation monad, and the "combinatorial" monads that arise from other situations of life? Some ideas seems to be present in Perrone's et al. https://arxiv.org/abs/2009.07302 and I've always seen that paper as an attempt to bridge the gap between monads as in "standard constructions" and monads as in "models of computation", with the bar construction retaining some information about both notions of monad, if you know where to look at. But I can't really fill in the details without some help!

• Say a monad $T$ and a monad $T'$ have homotopy equivalent (specified in whatever sense) bar constructions; what do I know about the monads?

Todd Trimble (Jul 20 2024 at 11:39):

seems to say that a comonad gives an (augmented) simplicial object

Well, it does! A comonad on $E$ induces a monoidal functor $\Delta_a^{op} \to [E, E]$ to the endofunctor category.

In the nLab article, the input data is not just a monad, but a monad together with an algebra. The Eilenberg-Moore category $E$ where the algebra lives has a comonad on it. And then you follow the above with the evaluation map $[E, E] \to E$ at that given algebra.

fosco (Jul 20 2024 at 11:39):

aha!

fosco (Jul 20 2024 at 11:44):

I understand the comonad is the "tautological" one obtained from the free-forgetful adjunction.

But somehow I find this dualization confusing: am I correct in saying that a monad becomes a cosimplicial object in endofunctors, a comonad a simplicial one, by the universal property of $\Delta$? Is there a (computational, conceptual, else) reason why one takes a T-algebra and build a simplicial object out the comonad on T-algebras instead?

Todd Trimble (Jul 20 2024 at 11:51):

Yes, you are correct regarding the mathematical points.

As for conceptual points, the original bar construction, as far as I know, had to do with constructing classifying bundles for groups, in the category of topological spaces or similar. Those are typically gotten by applying geometric realization to simplicial objects, like the bar construction. In the end, the important data point in constructing a classifying bundle $EG \to BG$ is that $EG$ is contractible (and that $G$ act freely and transitively on it), and for that you apply the nLab bar construction to a terminal object, considered as an algebra of the monad $G \times -$. So that's one important historical reason.

John Baez (Jul 20 2024 at 12:05):

@fosco: the augmented simplex category $\Delta_a$ is the free monoidal category on a monoid, but an augmented simplicial object is a functor from $\Delta_a^{\mathrm{op}}$ to some category. This "op" creates the dualization that seems to be puzzling you. The result is that any comonoid in any monoidal category gives an augmented simplicial object!

John Baez (Jul 20 2024 at 12:07):

As a spinoff of this fact, any comonad gives an augmented simplicial object.

And then as a further spinoff, any coalgebra of any comonad gives an augmented simplicial object.

fosco (Jul 20 2024 at 12:12):

@Todd Trimble thanks, I was expecting something like this to be an explanation!
@John Baez thank you: but indeed I'm confused by the last line. A monad gives a cosimplicial object, but a monad+an algebra gives a simplicial one.

So a comonad gives a simplicial object, but a comonad+a coalgebra a cosimplicial one. No?

Todd Trimble (Jul 20 2024 at 12:20):

So a comonad gives a simplicial object, but a comonad+a coalgebra a cosimplicial one. No?

Yes, that's correct. I think John misspoke slightly. Any object in the category on which the comonad is operating gives you a simplicial object, by the composition $\Delta_a^{op} \to [E, E] \to E$ I mentioned; that object doesn't have to be a coalgebra. (It could carry a coalgebra structure, but that would be a red herring in terms of what we're discussing.)

John Baez (Jul 20 2024 at 12:25):

There are lots of different things you can do but I probably screwed up if we're trying to discuss the "usual thing people do". So let's just look at what Todd wrote.

1) He starts with a monad $T$ on a category $E$, which is a monoidal functor from the "walking monoid" $\Delta_a$ to the endofunctor category $[E,E]$. So we get a cosimplicial object in $[E,E]$.

2) The monad $T$ gives a comonad on the Eilenberg-Moore category $E^T$. This is a monoidal functor from $\Delta_a^{\textrm{op}}$ to $[E^T, E^T]$. So we get a simplicial object in $[E^T,E^T]$.

3) But now say we have an algebra of $T$. This is an object $A$ of $E^T$ so we get an 'evaluation' functor $\mathrm{ev}_A$ from $[E^T,E^T]$ to $E^T$. Using this we can map the simplicial object in part 2) to a simplicial object in $E^T$.

This last one is called the bar construction.

fosco (Jul 20 2024 at 12:27):

Clear now!

fosco (Jul 20 2024 at 12:28):

Thanks to both. What about the other questions?

John Baez (Jul 20 2024 at 12:30):

Your other questions are too hard for me.

fosco (Jul 20 2024 at 12:33):

I suspect some choices of monads like $[A,A\otimes -]$ or $[[-,A],A]$ (to avoid having to choose a coefficient functor, let's say on categories of R-modules) might be scattered in Barr-Beck paper. I will try to have a closer look, it seems a very natural question.

John Baez (Jul 20 2024 at 12:42):

By the way, I think we can could do other things with a monad. Starting from a monad on $E$ we get an augmented cosimplicial object in $[E,E]$. Then from any object in $E$ we can get an augmented cosimplicial object in $E$ using the evaluation functor $[E,E] \to E$. This is how I was getting a cosimplicial object from an algebra of a monad! Or more precisely, I was doing the dual thing: getting a simplicial object from a coalgebra of a comonad. I think this is a real thing you can do, just not "the good thing" to do.

Todd Trimble (Jul 20 2024 at 12:57):

This is how I was getting a cosimplicial object from an algebra of a monad!

Right, but my point was that in this construction, you not using the algebra structure! You're just using the underlying object that belong to the category on which the monad operates.

John Baez (Jul 20 2024 at 13:11):

Right. This wrong construction is one I often get seduced by, which is all the harder to avoid because it's a valid mathematical construction, just not a good one (for the reason you mentioned, and more). I'm glad I finally had the nerve to spell it out in public.

John Baez (Jul 20 2024 at 13:12):

Usually I just say "oh, wait, that's not getting me what I want...."

fosco (Jul 20 2024 at 17:04):

Some examples that I would like some help in computing with:

• the free monoid monad on $R$-Mod (say even R a field): it's already difficult to grasp how to write down $\mu_A : A^{**}\to A^*$ because its domain is $R + A^* + A^*\otimes A^* + A^*\otimes A^*\otimes A^* + \dots$ and on each component it acts as the Kronecker product of vectors (regarded as matrices). What's the kernel of this map? And subsequently, one has to compute the differential $\mu_{A^*} - (\mu_A)^*$...
• the tensor-hom monad $X\mapsto [A,A\otimes X]$ (again, let's say in vector spaces); the multiplication here is obtained as the linear $\lambda$-term $[A,A\otimes [A,A\otimes X]] \to [A,A\otimes X]$ sending $\Phi : A\to A\otimes [A,A\otimes X]$ to $a\mapsto ev (\Phi(a)$, which (if $\Phi(a)=\sum_{i\in I}a_i\otimes \varphi_i$ for some $a_i\in A, \varphi_i\in [A,A\otimes X]$) is $\sum_{i\in I}\varphi_i(a_i)$. What is the matrix of this linear map? And what is $\mu_{[A,A\otimes X]} - [A,A\otimes \mu_X]$?

These computations seem daunting, but also so fun, why no one wanted to write them down?!

fosco (Jul 20 2024 at 17:06):

is there a conceptual reason these objects should give contractible chain complexes?

Todd Trimble (Jul 20 2024 at 17:07):

Some people do write down such things. During one of the conversations in Tallinn, I showed Clémence a little piece of the bar construction for the free operad monad on graded sets. I think John might have written down some examples in one of his graduate courses.

Todd Trimble (Jul 20 2024 at 17:09):

Yes, there are very conceptual reasons for the contractibility. It has to do with the so-called decalage comonad on simplicial sets (or simplicial objects). I realize now that I did a bad job of getting this across at the nLab. My notes that I sent you earlier might be a little better (even though they're not complete).

fosco (Jul 20 2024 at 17:10):

ok, I will look into it!

Jean-Baptiste Vienney (Jul 20 2024 at 18:41):

fosco said:

Some examples that I would like some help in computing with:

• the free monoid monad on $R$-Mod (say even R a field): it's already difficult to grasp how to write down $\mu_A : A^{**}\to A^*$ because its domain is $R + A^* + A^*\otimes A^* + A^*\otimes A^*\otimes A^* + \dots$ and on each component it acts as the Kronecker product of vectors (regarded as matrices).

The multiplication of the free $R$-algebra monad flattens tensors of tensors into tensors:
$(x_{11} \otimes … \otimes x_{1p_{1}}) \boxtimes … \boxtimes (x_{n1} \otimes … \otimes x_{np_n}) \mapsto x_{11} \otimes … \otimes x_{np_{n}}$

Jean-Baptiste Vienney (Jul 20 2024 at 18:44):

So the kernel should be generated by the differences of every two tensors of tensors which flatten in the same way. (Or something a bit more complicated, but at least it contains these elements)

Jean-Baptiste Vienney (Jul 20 2024 at 19:32):

Yeah this a bit more complicated. For instance, elements of this form are in the kernel:
$2 x \boxtimes (y \otimes z) - (x \otimes y) \boxtimes z - x \boxtimes y \boxtimes z$

fosco (Jul 20 2024 at 19:39):

That's why I am very curious to see how one proves conceptually that this complex is contractible. Doing computations by hand seems impossible!

Todd Trimble (Jul 21 2024 at 13:50):

Well maybe I can give it a crack here.

Todd Trimble (Jul 21 2024 at 13:50):

If you are comfortable with the bar construction $B(M, A)$ where $M: E \to E$ is a monad, as a functor

$\Delta_a^{op} \to [E^M, E^M] \overset{\mathrm{ev}_A}{\to} E^M$

or in other words as an augmented simplicial $M$-algebra, now what we can do is compose further with the forgetful functor $U: E^M \to E$ to get a simplicial object in $E$. What good will this do? We know the "bad": we've forgotten the $M$-algebra structure.

Todd Trimble (Jul 21 2024 at 13:51):

The good is that, as I will show, we will pick up some extra natural maps to go along with the face and degeneracy maps that belong to the simplicial structure, and it is these extra maps that will ensure contractibility (or rather, acyclicity) of the $E$-valued simplicial object. It will give something actually contractible in the case $A = 1$, the terminal algebra.

Todd Trimble (Jul 21 2024 at 13:51):

Let me define what I mean by "acyclic". Suppose we have a simplicial set (for the moment I mean simplicial set, not augmented simplicial set) that ends in the two face maps

$\ldots Y_1 \rightrightarrows Y_0$

Todd Trimble (Jul 21 2024 at 13:52):

Define the set of connected components $\pi_0(Y)$ to be the coequalizer of these two face maps. This defines a functor

$\pi_0: [\mathsf{\Delta^{op}}, \mathsf{Set}] \to \mathsf{Set}$

and it is left adjoint to the "discrete space" functor $D: \mathsf{Set} \to \mathsf{Set}^{\Delta^{op}}$ which sends a set $S$ to the simplicial set $DS: \Delta^{op} \to \mathsf{Set}$ that is constant at $S$. (Does this remind anyone of another ongoing thread?) So we get a unit component

$Y \to D\pi_0(Y)$

which intuitively, geometrically, we imagine as the quotient map that contracts a "space" $Y$ down to its set of path components.

Todd Trimble (Jul 21 2024 at 13:53):

Roughly (I'll be more precise in a moment), we say that a simplicial set is acyclic if this unit map is a simplicial homotopy equivalence from $Y$ to the discrete space $D\pi_0(Y)$. In other words, each of the path components are contractible: $\pi_1, \pi_2, \ldots$ all vanish for every path component of $Y$, hence the word "acyclic". Only, we will export this meaning of "acyclic" from simplicial sets to simplicial $E$-objects, for any $E$.

Todd Trimble (Jul 21 2024 at 13:53):

Take the case of the bar construction $B(M, M, A)$. If we temporarily forget the augmentation, this as a simplicial object ends with

$MMA \rightrightarrows MA$

where the two maps are the multiplication $\mu_A: MMA \to MA$, and $M\alpha: MMA \to MA$ where $M$ is applied to the $M$-algebra structure map $\alpha: MA \to A$. Now, we know that the coequalizer of these two maps is just $\alpha: MA \to A$ itself; that's just the canonical presentation of an $M$-algebra as a coequalizer

$MMA \overset{\mu_A}{\underset{M\alpha}{\rightrightarrows}} MA \overset{\alpha}{\to} A \qquad (\ast)$

Thus $\pi_0(B(M, M, A)) \cong A$.

Todd Trimble (Jul 21 2024 at 13:54):

The general acyclicity condition I am asserting for the bar construction $B(M, M, A)$ says that, in a particularly nice way, the simplicial object $B(M, M, A)$ is simplicially homotopy equivalent to the discrete object on $A$. In other words, the quotient map

$B(M, M, A) \to DA$

(given by the unit component for $\pi_0 \dashv D$) is a simplicial homotopy equivalence. This map takes place at the level of simplicial $M$-algebras, but in order to manifest the contracting homotopy, we have to take leave of $M$-algebra structure.

Todd Trimble (Jul 21 2024 at 13:54):

So what is the nature of this contracting homotopy? You are guaranteed to have already seen a "baby" version of it sometime in your past. When we apply the forgetful functor $U: E^M \to E$ to the canonical coequalizer $(\ast)$ above, suddenly this coequalizer becomes a split coequalizer, or another maybe better term for it is contractible coequalizer. (It gives an absolute coequalizer.) The canonical splitting in this case is given by a pair of maps

$MMA \overset{u_{MA}}{\leftarrow} MA \overset{u_A}{\leftarrow} A,$

where $u$ is the unit of the monad $M$, so that when you put these arrows together with the arrows in the diagram $(\ast)$, you get a diagram obeying the following equations that make it a contractible coequalizer diagram:

$\alpha \circ u_A = 1_A \qquad \mu_A \circ u_{MA} = 1_{MA} \qquad M(\alpha) \circ u_{MA} = u_A \circ \alpha$

These are the types of equations one encounters when staring hard at Beck's precise monadicity theorem, which involves these $U$-contractible coequalizers among its hypotheses.

Todd Trimble (Jul 21 2024 at 13:55):

So I invite you at this point to think of the chain of unit arrows

$\ldots MMMA \leftarrow MMA \leftarrow MA \leftarrow A$,

in general $u_{M^nA}$, as giving the desired contracting homotopy. (Of course, unit components only live in the category $E$ lying below $E^M$.) Some people call these unit maps "extra degeneracies" in addition to the degeneracy maps we already had from before.

Todd Trimble (Jul 21 2024 at 13:55):

It's an interesting task to say precisely why you should think of this series of unit maps as a contracting homotopy on $UB(M, M, A)$. Evidently the unit map $u_{M^n A}$ maps the object of $(n-1)$-cells which is the $E$-object $M^n A$, to the object of $n$-cells which is $M^{n+1}A$. So evidently I have in mind a notion of contracting homotopy which for a general augmented simplicial object has for its data a bunch of maps $h_n: Y_n \to Y_{n+1}$, satisfying certain equations, which at the lowest dimensional levels are the contractible coequalizer equations. It might be suggestive to write these as $h_n: Y_{0+n} \to Y_{1+n}$.

Todd Trimble (Jul 21 2024 at 13:56):

Let me sketch out a geometric explanation of what we'll see going on more formally in a moment. For the sorts of "acyclic structures" $Y$ we're aiming to describe, think of the first map $Y_{-1} \to Y_0$ as assigning a base point $y_c \in Y_0$ to each path component $c \in Y_{-1} = \pi_0(Y)$. Then what will happen with the next map $h_0: Y_0 \to Y_1$ is that it will assign to each $0$-cell $y \in Y_0$, i.e. a point of $Y$, a 1-cell $p_y \in Y_1$ or path in $Y$, that picks out a path that goes from $y$ to the chosen basepoint of the path component that $y$ belongs to. This is a first glimmer of the contracting homotopy that contracts each path component of $Y$ down to the chosen basepoint, that are provided by the $h_n$ taken together.

In general if you think of elements of $y_n \in Y_n$ a la Yoneda as corresponding to maps $\phi_n: \Delta(-, n) \to Y$ out of the $n$-dimensional simplex, where this $n$ corresponds to an ordinal with $n+1$ elements, then applying $h_n(y_n)$ has the geometric effect of extending $\phi_n$ to a map $\psi_n: \Delta(-, 1 + n) \to Y$ which agrees with $\phi_n$ on the face given by the inclusion $0 + n \hookrightarrow 1 + n$, and has the effect of contracting $\phi_n$ down to the appropriate chosen basepoint.

Todd Trimble (Jul 21 2024 at 13:56):

So with that geometric intuition in mind for what contracting homotopies are supposed to be like, it's time for more formal details. For any category $E$, there's a certain comonad on $[\Delta_a^{op}, E]$, called the décalage comonad, which I'll denote by $P$ because I think of it as akin to a path space construction. It's very simple. First, you know that $\Delta_a^{op}$ with its ordinal sum $+$ is the "walking comonoid", just as the 1-element ordinal [1] is the monoid object of the
walking monoid $\Delta_a$. Any comonoid $C$ in a monoidal category induces a comonad $C \otimes -$, and so we have a comonad $[1] + (-): \Delta_a^{op} \to \Delta_a^{op}$. Let me denote this comonad by $D$. We have $D([m]) = D([1+m])$. We have a comonad counit $\varepsilon_m: D([1+m]) \to D([0+m])$ induced by the unique map $[1] \to [0]$ in $\Delta_a^{op}$, and we have a comonad comultiplication $\delta_m: D([1+m]) \to D([2+m])$ induced by the unique map $[1] \to [2]$ in $\Delta_a^{op}$.

Todd Trimble (Jul 21 2024 at 13:56):

The décalage comonad $P$ on $[\Delta_a^{op}, E]$ is the comonad that takes a simplicial object $Y: \Delta_a^{op} \to E$ to the composite $YD: \Delta_a^{op} \to E$. We have for example $PPY = YDD$, and the comultiplication $PY \to PPY$ is just $Y\delta: YD \to YDD$, and similarly for the counit.

Finally, with terminology as in the nLab, we define an acyclic structure on a simplicial object $Y$ to be a $P$-coalgebra structure on $Y$. A coalgebra structure is a natural transformation $h: Y \to PY$ satisfying the usual coalgebra identities. Concretely, it is given by a series of maps $h_n: Y_n \to Y_{1+n}$ starting with $n=-1$, satisfying a bunch of equations saying how they interact with faces and degeneracies, which are captured precisely by equations of naturality or coalgebra-axiom type.

Todd Trimble (Jul 21 2024 at 13:57):

I want to say a few more geometric words in a moment that explain better why $P$-coalgebraicity should be thought of as an especially nice sort of acyclic structure, that connects up with how we usually think of homotopy equivalence. But before I do, I'll state a universal property of the bar construction.

Todd Trimble (Jul 21 2024 at 13:57):

For a monad $M$ on a category $E$, let $\mathrm{AcycAlg}_M$ denote the category of simplicial $M$-algebras $Y$ that come equipped with a $P$-coalgebra structure on its underlying simplicial $E$-object $UY$. Then the path-components functor

$\pi_0: \mathrm{AcycAlg}_M \to E^M$

has a left adjoint $E^M \to \mathrm{AcycAlg}_M$ given by the bar construction $A \mapsto B(M, A)$.

Todd Trimble (Jul 21 2024 at 13:57):

That is to say: given an $M$-algebra map $\phi: A \to \pi_0(Y) = Y_{-1}$, there is a unique simplicial $M$-algebra map $\hat{\phi}: B(M, A) \to Y$ that agrees with $\phi$ at the augmentation component, and that preserves the $P$-coalgebra structures. I strongly urge you to write out what $\hat{\phi}$ has to be, based on the definitions given above. It's a follow-your-nose, bootstrapping construction that builds up the components $\hat{\phi}_n$ inductively. If you've ever gone through the usual sorts of acyclic models theorems in homological algebra, it should look very familiar. "Try it, you'll like it."

Todd Trimble (Jul 21 2024 at 13:58):

Okay, let me end by making explicit how a decalage coalgebra structure on $Y$ gives a contracting homotopy on $Y$ that goes from the identity on $Y$ to the map that constantly maps each path component to its basepoint. When I say contracting homotopy here, I literally mean a simplicial map $H: I \times Y \to Y$ where $I$ is the representable object given by the two-element ordinal, $\Delta_a(-, [2])$ where $[2] = \{0, 1\}$, such that the restriction to the "right-end" inclusion of $Y$ into $I \times Y$, given by the composite

$Y \cong \Delta_a(-, [1]) \times Y \to \Delta_a(-, [2]) \times Y \overset{H}{\to} Y,$
is the identity (the unlabeled arrow $\to$ is induced by the map $[1] \to [2]$ that names the element $1$ of $\{0, 1\}$), and whose restriction to the left-end inclusion of $Y$ into $I \times Y$ is a composite of the form

$Y \overset{\mathrm{unit}}{\to} D\pi_0(Y) \hookrightarrow Y.$

The unlabeled arrow $\hookrightarrow$ is intuitively the section of the unit map given by "basepoint inclusion", and formally comes about by the coalgebra structure on $Y$, as we will soon see.

Todd Trimble (Jul 21 2024 at 14:01):

To make the transition from $P$-coalgebras to contracting homotopies, let me recall a bit of abstract nonsense that goes back to the 1965 Eilenberg-Moore paper. (People who know me well mathematically have heard me say it many times.) It says that the left adjoint of a right adjoint comonad (like $P$) carries a monad structure that is mated to the given comonad structure, and, denoting that monad as $C: \mathcal{S} \to \mathcal{S}$, the category of $P$-coalgebras is naturally equivalent to the category of $C$-algebras, as categories over $\mathcal{S}$. In our situation, the category $\mathcal{S}$ is the category of augmented simplicial objects $[\Delta_a^{op}, E]$, and the left adjoint of $P$ is given by left Kan extension along $D: \Delta_a^{op} \to \Delta_a^{op}$. Thus the left adjoint monad $C$ is given by

$Y \mapsto \int^{[n]} \Delta_a(-, [1+n]) \cdot Y([n])$

and a $P$-coalgebra is identified with a $C$-algebra.

Todd Trimble (Jul 21 2024 at 14:01):

Now, let $C'$ be the left adjoint endofunctor that takes a simplicial object $Y$ to the pushout of the span

$D\pi_0(Y) \overset{\mathrm{unit}}{\leftarrow} Y \overset{\mathrm{left}}{\to} I \times Y$

where "left" means the left-end inclusion (all the functors appearing in $D\pi_0 \leftarrow \mathrm{Id} \to I \times -$ are left adjoints, so their pushout $C'$ is too). There's a comparison map

$C' \to C$

between left adjoints, induced by a pair of maps

$I \times \Delta_a(-, [n]) \to \Delta_a(-, [1+n]), \qquad D\pi_0(\Delta_a(-, [n]) \to \Delta_a(-, [1+n])$

natural in $[n]$, which I leave to your geometric imagination, but which I can make explicit simplicially on request. Thus, given a $P$-coalgebra structure or a $C$-algebra structure $\alpha: CY \to Y$, the composite

$C'Y \to CY \overset{\alpha}{\to} Y$

gives the desired contracting homotopy, and that's where I'll leave it for now, even though there's rather a lot more that could be added (for example, the sense in which I said the contracting homotopy is "especially nice": it has to do with the fact that we never used or took advantage of the $P$-comultiplication, or the $C$-multiplication, in describing the data of the contracting homotopy).