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Stream: theory: algebraic topology

Topic: Homotopy colimits for computing cofibrant replacements

Patrick Nicodemus (Jul 20 2022 at 21:51):

I've come across a handful of situations where one uses homotopy colimits to compute a cofibrant replacement for something and I was wondering if anybody has insight into what's going on there. I've been reading Michael Shulman's paper on homotopy colimits and enriched category theory.

More specifically, the pattern I'm talking about is : given an object $X$, you use a simplicial bar construction to construct a simplicial resolution $SX$ of $X$; the ordinary colimit is $X$ but the homotopy colimit is a cofibrant replacement for $X$.

Patrick Nicodemus (Jul 20 2022 at 21:51):

Can anybody give me some pointers to like, the idea behind how this works?

Patrick Nicodemus (Jul 20 2022 at 21:52):

My knowledge of homotopy theory is somewhat limited.

Tim Campion (Jul 20 2022 at 23:21):

Most of the time if you have a formula for a homotopy colimit, you can view it as an instance of "take a cofibrant replacement in the diagram category, then apply the colimit functor", which generally works because the colimit functor will be left Quillen. Since left Quillen functors preserve cofibrant objects, the output will be cofibrant.

John Baez (Jul 21 2022 at 03:32):

Patrick Nicodemus said:

More specifically, the pattern I'm talking about is : given an object $X$, you use a simplicial bar construction to construct a simplicial resolution $SX$ of $X$; the ordinary colimit is $X$ but the homotopy colimit is a cofibrant replacement for $X$.

This is what I was starting to explain to you, once upon a time. Did we really not get that far? I guess not. We talked about resolutions, but maybe we didn't get to how the bar construction, which is a systematic way to construct a simplicial resolution of an object in a category that has a comonad on it.

John Baez (Jul 21 2022 at 03:33):

It's funny that we never got that far.

Patrick Nicodemus (Jul 21 2022 at 11:21):

Tim Campion said:

Most of the time if you have a formula for a homotopy colimit, you can view it as an instance of "take a cofibrant replacement in the diagram category, then apply the colimit functor", which generally works because the colimit functor will be left Quillen. Since left Quillen functors preserve cofibrant objects, the output will be cofibrant.

This is interesting and i think I understand it but I'm not sure it's answering my question. If I understand your answer correctly you are explaining how we use cofibrant replacements to compute homotopy colimits, I'm trying to understand why the process works the other way. Why can I use a homotopy colimit to compute a cofibrant replacement for a thing?

Example, if I have a topological group $G$ then the Milgram bar construction defines a simplicial space where in degree $n$ we have $G^{n+1}$, the nfold Cartesian product; $d_n$ is a projection which forgets the last entry and $d_i$ is multiplication for $i<n$. The homotopy colimit of this simplicial space is $EG$ which i view as a kind of cofibrant replacement for the singleton space equipped with the trivial $G$ action - they're homotopy equivalent but the action of $G$ on the singleton space is bad, it's not a principal $G$ space. $EG$ has lifting properties that are evocative of cofibrant objects. So here my question is why does the homotopy colimit come up in the construction of $EG$? How can the cofibrancy properties I'm talking about be linked to that homotopy colimit.

Patrick Nicodemus (Jul 21 2022 at 11:22):

Presumably with the obvious minor adaptation of the Milgram construction you can give a similar cofibrant replacement for other $G$-spaces than the singleton, with similar formal properties, i.e. if $M$ is a $G$ space then define the simplicial space $X$ so that $X_n = G^{n+1}\times M$. Face maps are given by multiplication and the action.

Patrick Nicodemus (Jul 21 2022 at 11:25):

The geometric realization of $X$, I think, is a principal $G$ space which is homotopy equivalent to $M$ when the $G$ structure is forgotten and principal $G$ spaces have interesting lifting properties that make them seem like cofibrant objects.

Patrick Nicodemus (Jul 21 2022 at 11:28):

Something analogous happens with chain complexes, if $A$ is a DG algebra augmented over the ground ring, say $\mathbb{Z}$ , then define the simplicial object $X$ in Ch(Ab) with $X_n = A^{\otimes {n+1}}$. The homotopy colimit of $X$ is a complex $EA$ which carries a nice action of $A$, Cartan called it a "construction"; the quotient of $EA$ by the action of $A$ is $BA$ in the sense of Eilenberg and Mac Lane's papers on computing homology groups $H(\pi,n)$

Patrick Nicodemus (Jul 21 2022 at 11:28):

I view $EA$ as a kind of cofibrant replacement for $\mathbb{Z}$ in the category of $A$ modules.
Again it has interesting lifting properties which are characteristic of cofibrant objects, so my question is, why does the homotopy colimit come up when I'm trying to construct a cofibrant replacement for $\mathbb{Z}$ and how are the cofibrancy properties of $EA$ i'm talking about linked to the homotopy colimit

Zhen Lin Low (Jul 21 2022 at 15:28):

What do you mean by "the homotopy colimit" here? You seem to use it to refer to some specific construction that is well-defined at least up to isomorphism. (If not, then cofibrancy of "the homotopy colimit" could not be well-defined!)

Zhen Lin Low (Jul 21 2022 at 15:30):

It happens that certain specific concrete constructions function as both homotopy colimit functors and cofibrant replacement functors. I do think this is a phenomenon that deserves a proper explanation... but at the same time I want to point out that it is far from universal. I do not know of any "homotopy limit" formula that computes fibrant replacements, which suggests to me this is not abstract nonsense.

Patrick Nicodemus (Jul 21 2022 at 15:46):

Zhen Lin - in both cases i'm referring to a certain specific construction up to isomorphism, yes. For a simplicial object $X$ I mean the bar construction as $\int^n \Delta^n \otimes X_n$ to the extent where this makes sense, for example if the category is enriched and tensored over simplicial sets, as in Shulman's presentation in the paper I mentioned earlier.

I haven't thought about whether the properties I'm curious about are preserved under homotopy equivalence, but it would be interesting to figure that out.

Zhen Lin Low (Jul 21 2022 at 22:13):

OK. As Mike says in definition 6.6, the construction on its own is an "uncorrected" homotopy colimit. The qualifier is important! And the way to get the "correct" homotopy colimit is to take cofibrant replacements, which is what Tim explained. The "correct" homotopy colimit is homotopy invariant, so it is common to consider homotopy colimits as being defined only up to homotopy, rather than up to isomorphism. But cofibrancy is not a homotopy invariant – in fact, the whole point is that it is not!

Mike Shulman (Jul 22 2022 at 06:20):

I think the reason the same construction produces a homotopy colimit and a cofibrant replacement is pretty simple. To compute a homotopy colimit, you need to take a cofibrant replacement and then apply your original colimit functor. So if you take a sufficiently broad notion of "colimit" (e.g. colimits weighted by profunctors) so that the identity functor is an instance of a "colimit", then a concrete construction of the corresponding homotopy colimit will be a cofibrant replacement followed by the identity functor, i.e. just a cofibrant replacement.

Mike Shulman (Jul 22 2022 at 06:24):

Zhen Lin Low said:

As Mike says in definition 6.6, the construction on its own is an "uncorrected" homotopy colimit. The qualifier is important! And the way to get the "correct" homotopy colimit is to take cofibrant replacements, which is what Tim explained.

There's a bit more going on there than that. You can (almost) always get a homotopy colimit by taking a suitable (projective) cofibrant replacement in the functor category and then the ordinary colimit (not a bar construction of any kind, just the ordinary colimit). That's what Tim's comment sounded to me like it was referring to. But the point of the bar construction is that it gives you a cofibrant replacement in the functor category — and hence a way to compute homotopy colimits — as soon as you combine it with a pointwise cofibrant replacement in the underlying codomain category of the functors. For instance, if every object in that category is cofibrant, then the "uncorrected" homotopy colimit requires no correction at all, despite the fact that not every object will be cofibrant in the projective model structure on functors. (In fact, fairly generally the bar construction composed with a pointwise cofibrant replacement is actually a projective-cofibrant replacement.)

Zhen Lin Low (Jul 22 2022 at 10:35):

I admit I was sweeping some details under the rug when I called that "take cofibrant replacements". But I was also thinking about the old-school bar resolution by comonads and how that also functions something like a cofibrant replacement functor.