Recreational and medicinal math on NaCNO's magic scroll

Variations on an oriental folksong: The algebraist's introduction to Thom spectra

Tue, 07 Apr 2026 13:00:45 +0200

These are some notes on Thom spectra, placed within the context of the six-functor formalism of parametrised spectra. They are based on a Chinese version on 香蕉空间 that I drafted during the summer of 2025, which I translated to English and expanded quite a bit around December. There was a TeXed up Chinese version before the content expansion, which I might make public after backporting the changes (when I have time and remember).

UPDATES:

(20.apr.2026) Fixed a few typos. Minor changes in 1.2 following feedback from Bastiaan Cnossen.

Link to the notes (PDF)

The ideas are mostly based on the work of Ando–Blumberg–Gepner–Hopkins–Rezk. I should mention the work of Antolín-Camarena–Barthel as well, which hasn't come up in the notes. The notes are more centered around the use of six-functors for classical algebraic topology.

The current notes have been distributed within our working groups and to a few fellow students, but not more broadly than that, as I have hesitations about conducting much academic activity under my current legal identity, especially on the internet.

Adams tower on bounded below spectra

Tue, 17 Feb 2026 22:20:39 +0100

I've learned most techniques from Mathew–Naumann–Noel's Nilpotence and Descent paper [ND], as well as a central but mundane trick from Nikolaus–Scholze's Topological Cyclic Homology paper [TC]. Although the Adams spectral sequence is no stranger for the mathematical omnivore I have been, I've always assumed to some extent that they do what they claim to, and somewhat focused on what the machine produces, less so on what it computes.

References:

[ND]: Nilpotence and descent in equivariant stable homotopy theory by Akhil Mathew, Niko Naumann and Justin Noel.
[TC]: On topological cyclic homology by Thomas Nikolaus and Peter Scholze.

Auxiliary:

[HTT] and [HA]: by Jacob Lurie.
[LE] The localisation of spectra with respect to homology by Pete Bousfield.
[FS]: An introduction to filtered and synthetic spectra by Sven van Nigtevecht
[CC]: Conditional Convergent Spectral Sequences by John Michael Boardman

Caveat lector! For our purpose today, the Adams spectral sequence is the totalisation spectral sequence of a(n augmented) cosimplicial object associated to an \(\mathbb{E}_1\)-ring spectrum \(E\) and a spectrum \(X\). We won't discuss the spectral sequence, but rather the totalisation, which it morally computes. The totalisation spectral sequence comes from the tower of partial totalisations, also known as the coskeletal filtration, which is closely related to the titular Adams tower. The question of convergence is another topic [CC], and has to do with the right non-exactness of sequential limits (\(\operatorname{lim}^1\)). What will be addressed in this post is the vanishing of the (homotopy) limit of a certain tower, or conditional convergence in the terminology of [FS, Sec. 2.3.1].

The major examples we'd like to have in mind are \(E = \mathbb{F}_p\) and \(\mathrm{MU}\). Other common examples include Bott-periodic or connective topological \(\mathrm{K}\)-theory \(\mathrm{KU}\), \(\mathrm{ku}\), as well as Morava \(\mathrm{E}\)-theories. A well-known result (Devinatz–Hopkins cites Morava's change of rings theorem.) identifies the \(K(n)\)-local \(E_n\)-Adams spectral sequence with the homotopy fixed points spectral sequence for the Morava stabiliser group. We will not concern ourselves with these (extremely interesting) examples.

The situation we concern ourselves with in the current blog post is classical: When \(E\) is connective and \(\mathbb{S} \to E\) is surjective on \(\pi_0\). (A similar statement exists for \(\mathbb{Z} \subseteq \pi_0 E \subseteq \mathbb{Q}\).)

When \(E\) is a connective \(\mathbb{E}_1\)-ring whose unit morphism \(\mathbb{S} \to E\) is surjective on \(\pi_0\) and \(X\) is a bounded below spectrum, the \(E\)-descent object \(D_E X\) (in \(\mathrm{Sp}\)) for \(X\) (defined below) coincides with the Bousfield localisation of \(X\) at the Moore spectrum associated to \(\pi_0 E\).

For instance, if \(X\) is connective then \(D_{\mathrm{MU}} X \simeq X\) and \(D_{\mathbb{F}_p} X \simeq X_{\hat p}\) is the \(p\)-completion. This alludes to the classical case of Adams(-Novikov) spectral sequences.

The connectivity is crucial, both on \(E\) and on \(X\). At a fixed prime, the Morava \(K\)-theories are \(p\)-complete and mutually acyclic.

Before starting, we reiterate that Bousfield [LE, 6.5-7] already showed that the condition on \(\pi_0 E\) can be refined. Nevertheless, the examples we have in mind often don't require such generality.

The descent tower and the error

This section is closely following, if not taken from, [ND, Sec. 2.1].

For a spectrum \(X\), define its \(E\)-descent diagram \(D^\bullet_E X \coloneqq X \otimes E^{\otimes (1 + \bullet)}\) to be the (coaugmented) cosimplicial spectrum with face maps given by inserting units and degeneracies by multiplication of \(E\) (and constant on \(X\)). The total \(E\)-descent object of \(X\) is the totalisation \[ D_E X \coloneqq \lim_{\bullet : \Delta} D^\bullet_E X, \] which comes along with the coskeletal filtration \(D_E X \simeq \lim_{-1 \leq n \to \infty} D^{(n)}_E X\), where \[ D^{(n)}_E X \coloneqq \lim_{\bullet : \Delta_{\leq n}} X \otimes D^\bullet_E. \] We call this the descent tower. The associated spectral sequence is called the Adams spectral sequence.

We use the only reasonable convention that \(\Delta_{\leq -1}=\emptyset\). A few remarks are due (Most can be found in [ND] in a generalised setting):

There is a natural map \(X \to D_E X\) induced by the coaugmentation, which we claimed and will know to be a localisation in the classical situation. However we might like this to be true, a priori the descent construction \(D_E(-)\) has no reason to be idempotent, let alone a localisation.
If \(X\) has a right \(E\)-module structure, then the coaugmented cosimplicial object \(X \otimes D^\bullet_E\) has an extra degeneracy, and is hence a universal limit diagram. In this case, the map \(X \to D_E X\) is an isomorphism. This property propagates to the idempotent complete stable subcategory generated by \(E\)-modules.
Consequently, if \(E\) is a finite \(E_1\)-ring spectrum, so that \(E \otimes -\) commutes with limits, we have that \(X \to D_E X\) is an \(E\)-equivalence into an \(E\)-local spectrum. The descent object thus coincides with the Bousfield \(E\)-localisation functor. Typical examples include the endomorphism rings of finite spectra, and by a recent result of Burklund, also \(\mathbb{S}/p^2\) for odd primes \(p\) and \(\mathbb{S}/2^3\) provide examples of \(\mathbb{E}_1\)-algebras. (I haven't got to read the paper yet, so I can't say how canonical or unique these rings are. These are almost surely not computable anyway.)
The filtration is complete in the sense that its limit is \(D_E X\). Caution is due in translating this into computations on homotopy groups. We don't plan to cover this part in this blog post.
Although \(\Delta^{\textrm{inj}}\) is coinitial in \(\Delta\) [HTT, 6.5.3.7], the same is not true for \(\Delta_{\leq n}\) (which isn't even weakly contractible, but see [HA, 1.2.4.17]). This is one reason we demanded that \(E\) be \(\mathbb{E}_1\).

Another reason for demanding a ring structure lies with the identification of the Adams \(E_2\)-page as \(\mathrm{Ext}\)-groups of graded comodules over a Hopf algebroid. This also depends on a flatness conditions which will not be discussed.

We'll make use of the following well-known description of the Adam tower [ND, Prop. 2.14]: (The proof is [HA 1.2.4.17]

Let \(I \coloneqq \mathrm{fib}(\mathbb{S} \to E)\) be the "kernel" of the ring's unit. Set iteratively \[ T^{(-1)}_E X \coloneqq X, T^{(n+1)}_E X \coloneqq T^{(n)}_E X \otimes I \to \mathbb{S} \otimes T^{(n)}_E X. \] So that \(T^{(n)}_E X \simeq X \otimes I^{\otimes (1 + n)}\).

Then \(T^{(n)}_E X \to X \to D^{(n)}_E X\) are (a tower of) fiber sequences, where the middle term is the constant tower, and the lateral terms are the towers described above. The limit of the tower \(T_E X\) is thus the fiber of the map \(X \to D_E X\).

Arguably, both the tower \(T^\bullet_E X\) or \(D^\bullet_E X\) deserve the name the Adams tower, although the literature has assigned the name to the former. We will call \(T^{\color{gray}(n)}_E\) the error of descent (tower).

Postnikov convergence

Recall that every spectrum \(X\) is the limit of its Postnikov truncations \(\tau_{\leq m} X\), and the only spectrum which is \(\infty\)-connective is zero. The following trick is extracted from [TC, Lem. I.2.6]

We say that an exact functor \(F : \mathrm{Sp} \to \mathrm{Sp}\) is bounded below if there is a constant \(c \in \mathbb{Z}\) such that \(F\) sends connective spectra to \(c\)-connective spectra.

If \(F\) is bounded below, then \(F(X) \simeq \lim_{m \to \infty} F(\tau_{\leq m} X)\) is the limit of its value on the Postnikov tower.

Intuitively, this is a condition similar to "finite cohomological dimension" assumptions.

Proof. As \(F\) is exact, there are fiber sequences \[ F(\tau^{\gt m} X) \to F(X) \to F(\tau_{\geq m} X). \] We just have to show that the limit \(\lim_{m} F(\tau^{\gt m} X)\) is zero. The assumption implies that each \(F(\tau^{\gt m} X)\) is \((m-c)\)-connective, and a sequential limit of \(r\)-connective spectra is \((r-1)\)-connective, so the fiber is \(\infty\)-connective.

As evident from the argument, the assumption on \(F\) can be relaxed, and there is a similar generalisation, replacing \(\mathrm{Sp}\) with stable categories with \(t\)-structure. The interested reader will have already taken this as an exercise.

Evidently the identity functor is bounded below. We verify that the classical descent functors, as well as the \(p\)-completion functor, are bounded below.

If \(E\) is a connective \(\mathbb{E}_1\)-ring whose \(\pi_0\) is a quotient or localisation of \(\mathbb{Z}\), then the error of descent \(T_E\) and descent \(D_E\) are bounded below functors.

Proof For a connective ring spectrum \(E\), the fiber \(I\) of the unit morphism is \((-1)\)-connective, and \(\pi_{-1} I\) is the cokernel of \(\mathbb{Z} \to \pi_0 E\). In the stated case, \((\pi_{-1}I)^{\otimes 2} = 0\), so all the \(I^{\otimes(1+n)}\) are \((-1)\)-connective. The functor \(T_E\) decreases connectivity by at most \(2\), and is thus bounded below. It follows that the descent object \(D_E\) is also bounded below.

The \(p\)-adic completion, which is the Bousfield localisation of spectra at \(\mathbb{S}/p\), is given as a limit as follows: \[ X_{\hat p} \simeq \lim_{n \to \infty} X/p^n. \] As a result, \(p\)-completion decreases connectivity by at most \(1\).

Proof The \(p\)-completion is part of the recollement on \(\mathrm{Sp}\) induced by the idempotent algebra \(\mathbb{S}[1/p]\). More precisely, let \(C \coloneqq \mathrm{fib}(\mathbb{S} \to \mathbb{S}[1/p])\) be its complementary idempotent coalgebra.

We have (see the previous post on inverting endomorphisms) \[ \begin{gather*}\mathbb{S}[1/p] \simeq \operatorname*{colim} \left(\mathbb{S} \xrightarrow{p} \mathbb{S} \xrightarrow{p} \ldots\right)\\ C \simeq \operatorname{\Omega} \operatorname*{colim}_n \mathbb{S}/p^n \simeq \operatorname*{colim}_n (\mathbb{S}/p^n)^\vee \eqqcolon \operatorname{\Omega}\mathbb{S}/p^\infty,\end{gather*} \] and the \(p\)-completion is given by the mapping spectrum \(X_{\hat p} \simeq \mathrm{map}(\operatorname{\Omega}\mathbb{S}/p^\infty, X)\). The above formula follows.

The comparison of \(D_E\) with completion

Assume for this section that \(\pi_0 E \simeq \mathbb{Z}/n\). The classical situation is concerned with the discrete ring spectrum \(E = \mathbb{F}_p\), but everything we said above about idempotent algebras applies to \(\mathbb{S}[1/n]\). We call the \(\mathbb{S}/n\)-localisation \(n\)-(adic )completion.

Every \(\mathbb{S}/n\)-acyclic spectrum is \(E\)-acyclic. Thus every \(E\)-local spectrum, in particular every spectrum which admits an \(E\)-module structure, is \(n\)-complete.

Proof. The \(\mathbb{S}/n\)-acyclic spectra are exactly the \(\mathbb{S}[1/n]\)-modules, and \(\mathbb{S}[1/n] \otimes E = 0\).

The \(E\)-locality of \(E\)-modules follows from \[\mathrm{map}_{\mathrm{Sp}}(X, M) \simeq \mathrm{map}_{\textrm{Mod-}E}(X \otimes E, M)\]

Thus the descent diagram \(D^\bullet_E X\) consists of \(E\)-local spectra, so the limit \(D_E X\) is also \(E\)-local. We thus obtain a unique factorisation \[ X \to X_{\hat n} \to D_E X. \]

If \(E\) and \(X\) are connective, then \(X\to D_E X\) is an \(n\)-adic equivalence. In other words, \(X_{\hat n} \xrightarrow{\sim} D_E X\) is an equivalence.

Proof. As we have seen, both sides are limits of their values on truncations, and the spectra for which the conclusion holds is stable (and idempotent complete). Thus it suffices to prove the conclusion when \(X\) is discrete (i.e. an Eilenberg–Mac Lane spectrum). We can tensor the \(E\)-descent diagram with \(\mathbb{S}/n\) (a finite spectrum!), which commutes with totalisation. The goal is to show that \(X/n \to (D_E X)/n \simeq D_E(X/n)\) is an isomorphism.

We show that \(X/n\) admits an \(E\)-module structure. As \(X\) is discrete, it admits a \(\mathbb{Z}\)-module structure, and \(X/n\) admits a module structure over \(\mathbb{Z} \otimes \mathbb{S}/n \simeq \mathbb{Z}/n \simeq \tau_{\leq 0} E\). Since \(E\) was assumed to be connective, the truncation \(E \to \tau_{\leq 0} E\) is a morphism of rings, and we're done.

The comparison of \(D_E\) with localisation

Assume now that \(\pi_0 E \simeq \mathbb{Z}[T^{-1}]\), then \(E\) is an algebra over the idempotent algebra of the Moore spectrum \(\mathbb{S}[T^{-1}]\). Hence there is a factorisation \[ X \to X[T^{-1}] \to D_E X. \]

If \(E\) and \(X\) are connective, then \(X[T^{-1}] \simeq D_E X\).

Proof. As before, we reduce to the case where \(X\) is discrete. Since \(E \simeq \mathbb{S}[T^{-1}] \otimes E\), we have \(D_E X \simeq D_E(X[T^{-1}])\). Again, the \(E\)-descent diagram for \(X[T^{-1}]\) splits since it admits a module structure over \(\tau_{\leq 0} E\).

Remark. If \(\tau_{\leq 0} E \simeq \mathbb{Z}\), then \(I\) is \(1\)-connective, so already the error of descent \(T_E\) vanishes on bounded below spectra.

Bonus: a few examples of \(\operatorname*{lim^1}\) to keep in mind

I don't have a specific example of a connective spectrum whose Adams spectral sequence fails to converge for this reason, but instead I'll list a few examples of non-exactness of limits as a sketch of the issue which may arise.

Limit of surjections need not be surjection

The limit as \(n \to \infty\) of the exact sequences \[ 0 \to \mathbb{Z} \xrightarrow{p^n} \mathbb{Z} \to \mathbb{Z}/ p^n \to 0 \] (constant in the middle) is only left exact: \[ 0 \to 0 \to \mathbb{Z} \to \mathbb{Z}_{p}. \] One can see that \[ \operatorname*{lim^1} (\ldots \to \mathbb{Z} \xrightarrow{p} \mathbb{Z} \xrightarrow{p} \mathbb{Z} \to\ldots) \simeq \mathbb{Z}_{p}/\mathbb{Z} \]

Remark The limit of a diagram consisting of surjective maps of sets surjects onto the diagram.

Limit and image don't play along

In order to show that the \(E_\infty\)-page computes the graded pieces of a filtration on the colimit, one needs to look at certain intersections of images. This is an issue even when the tower conditionally converges, as in our cases above.

As \(N \to \infty\), each of the maps \[ \bigoplus_{p \geq N \text{ prime}} p\mathbb{Z} \xrightarrow{\nabla} \mathbb{Z} \] is surjective, so that the limit of image is \(\mathbb{Z}\), but the image of the limit map is of course \(0\).

Localisation and telescope: Inverting endomorphisms in (homotopy) algebra

Sat, 17 Jan 2026 13:40:38 +0100

\[ \gdef\EE{\mathbb{E}} \gdef\colim{\operatorname{colim}} \gdef\tensor{\otimes} \gdef\ZZ{\mathbb{Z}} \gdef\SSS{\mathbb{S}} \gdef\BB{\mathbf{B}} \gdef\CAlg{\operatorname{CAlg}} \gdef\Map{\operatorname{Map}} \gdef\map{\operatorname{map}} \gdef\End{\operatorname{End}} \gdef\Mod{\mathrm{Mod}} \gdef\PrL{\mathrm{Pr}^{\mathrm{L}}} \gdef\tel{\operatorname{tel}} \] This short post discusses a recent confusion about inverting things in a symmetric monoidal (\(\infty\)-)category, or the degenerate case, an "element" of an \(\EE_\infty\)-monoid.

Earlier today, I wrote quite a few paragraphs in the Algebraic Topology discord server about it, and rubberducked myself into the answer. I apologize if I wasted anyone else's time.

EDIT 2026-01-18: An error on how far we could generalise the setting in the Remark.
EDIT 2026-02-18: Reworked some storytelling.

In the introduction, I'll be vague about the precise meaning of "localisation" and "to invert". I will mention a few concrete examples, but the reader is invited to come up with their own dialect of English where sense can be made.

If you have a family \(S\) of *s*tuff (that's a technical term) that you want to invert, you can make up a notion of \(S\)-local objects, as those who "treat things in \(S\) as if they are invertible", in an appropriate sense. In nice situations, each object has a universal approximation by an \(S\)-local object, and this approximation process is called localisation. It is out there for general reasons, which means we don't know a priori what they're built out of, except that they "solve the correct problem".

The stereotypical example is to "localise at a prime \(p\)", in which case \(S\) consists of all prime numbers except \(p\). Usually I'd be talking about rings or abelian groups, and we know exactly what localisations look like here: you can just write down fractions and routinely check that it works. You've seen this during your undergrads.

It takes some work to figure out what "fractions" means in other contexts, and they're not always the correct thing to do.

What's wrong with fractions? Can't you just…

Of course you can! It appears quite often and just works in a lot of situations, which is why people talk about it a lot. It's so commonplace that I nearly stopped thinking about it, and had to think to remember those cases where it famously doesn't work.

Here are a few common localisations, see if you spot any issue.

Localisation in discrete commutative rings is given by fractions.
Localisations of associative rings are given by fraction under some form of Ore's condition. In general they are build out of words
Localising the category of chain complexes at the quasi-isomorphisms. There's a model using sequences of zig-zags, and if the stars align, a single zig-zag suffices.
Inverting elements in ring spectra \(A\), or \(v_n\)-self maps of finite spectra.
Inverting an object \(x\) in a symmetric monoidal category \(C\). This is often spoken about in motivic contexts, where one inverts the Tate motive (pointed projective line, an "twisted" version of \(S^2\)).
Group completing the core of a symmetric monoidal category to obtain its "\(K\)-theory".

In analogy of fractions is the "telescope" construction on an endomorphism \(f : A \to A\) \[ \colim (A \xrightarrow{f} A \to \ldots) \eqqcolon \tel_f(A). \] More generally, given a map \(f : 1 \to M : C\), we can form the telescope \[ \colim (A \xrightarrow{f \tensor 1} M \tensor A \xrightarrow{f \tensor 1} M^{\tensor 2} \tensor A \to \ldots) \eqqcolon \tel_f(A). \] (There's also the variant of inverting \(X \to TX\) for an endofunctor \(T\) and other ones I don't know of.)

Notoriously in the point 3. and 4., the telescope in general does not suffice to invert, even though everything does seem commutative! The issue with group completion of spaces is no stranger for those who knows higher algebraic \(K\)-theory in its classical form: The telescope isn't even guaranteed to be an \(H\)-space, as there are some issues with the fundamental group actions. One must seeks out the aid of an arcane art — an ancient blessing, since lost to time, known as the \(+\)-construction of D. G. Quillen. (Apparently the construction was introduced by Kervaire, which Quillen applied to \(\BB\mathrm{GL}\) and made famous.)

I came to realise the presence of the issue only when I started learning about things in the motivic realm: I know some \(K\)-theory and group completion. Everyone straight off made it clear that it's a big issue, use the so-and-so simplicial delooping construction and hardly talked about the telescopes ever after. In algebraic contexts, or stably, one can compute its effect on homotopy groups, and it "just works".

Recently, I went back to reflect on this topic and realised the missing of some pieces. Way too often have I seen an author just telescope and say no second word about it, and I obviously hadn't been careful enough to question my fossilised algebraic instinct, spoiled by my training in set-based mathematics.

The answer lies not far beyond reach: It's already well-established in the motivic literature. More precisely, in the literature on the construction of the category of motivic spectra, where Robalo–Voevodsky–et al.'s criterion identifies a case where telescoping does invert, namely when the cyclic permutation \( (123) : x^{\tensor 3} \to x^{\tensor 3} \) is homotopic to the identity.

I suspect that the theory has been well known in the topology circle as well, since it's where the \(+\)-construction first appeared, exactly so that one can pass from telescope to group completion. It's also where we know that topological \(K\)-theory is actually given by a telescope. I've seen these during my studies, but not in an order so that I'd recognise it on the first glance. Now it's time to make up for it.

Why does it work, why does it not? What's going on?

Fix a (presentable) symmetric monoidal category \((C, \tensor, 1)\) and an \(\EE_\infty\)-algebra \(A :\CAlg(C)\). Inverting an \(x \in \End(A)\) is to find an initial algebra \(B\) under \(A\), on which \(x\) acts invertibly.

The problem of inverting \(x : C\) is then immediately reduced to the algebra case by viewing \(C : \CAlg(\PrL)\), with the endomorphism \((- \tensor x)\). We can moreover reduce to the case \(A = 1\) by passing to \(\Mod_A(C)\).

Reduced setup \((C, \tensor, 1)\) is a presentable symmetric monoidal category, \(f\) is an endomorphism of \(1\).

Warning I'm sweeping the size issues with \(\PrL\) under the rug. One might pretend there's a cardinality bound, go up a universe, or stick their head into the sand.

The localisations: What's going on

The full subcategory \(C_f\) of \(C\) on which \(f\) acts invertible is closed under all colimits and limits, so that the inclusion has both adjoints. Let \(L_f : C \twoheadrightarrow C_f\) denote the left adjoint.

It's also not hard to see that \(L_f\) is a symmetric monoidal localisation, i.e. the \(L_f\)-equivalences are closed under tensoring with any other object. We can now define \(E_f \coloneqq 1[f^{-1}] \coloneqq L_f(1)\). Now for any \(X : C\), the map \(X \to E_f \tensor X\) is an \(L_f\)-equivalence, since \(1 \to E_f\) is. \(E_f \tensor X\) is also \(L_f\)-local, as \(E_f\) is. In other words, the algebra \(E_f\) is the idempotent algebra which universally inverts \(f\), and the localisation is smashing, i.e.: \[ L_f(X) \simeq E_f \tensor X \]

As usual we let \(T_f \coloneqq \tel_f(1)\) be the sequential colimit as defined above. We note that \(1 \xrightarrow{f} 1\) is an \(L_f\)-equivalence, hence so is any \(X \to T_f \tensor X\). We can conclude that there is a unique map \(T_f \to E_f\) under \(1\), and that that they coincide if and only if \(T_f\) is already \(L_f\)-local \[ T_f \simeq E_f \Leftrightarrow f : T_f \to T_f \text{ is an isomorphism.} \]

Remark We can generalise to the case of an morphism \(f : 1 \to I\), and consider the full subcategory on objects \(X\) where \[ f \tensor 1 : X \to I \tensor X \text{ is an isomorphism.} \] In addition, we demand that \(I\) be ~~dualisable~~ (ERRATUM: invertible!). This guarantees the following:

\((- \tensor I)\) preserves limits, so that \(C_f\) is closed under limits. The left reflection \(L_f : C \twoheadrightarrow C_f\) exists.

\(C_f\) is closed under powers, so that \(L_f\)-equivalences are closed under tensors. \[ \hom_C(Z, X) \to I \tensor \hom_C(Z, X) \simeq \hom_C(Z, I \tensor X). \] This makes \(L_f\) into a monoidal localisation.

\(T \xrightarrow{f \tensor 1} I \tensor T\) is an \(L_f\)-equivalence. Indeed, \(f \tensor 1\) being an \(L_f\)-equivalence is the same as \(I^\vee \tensor X \xrightarrow{f^\vee \tensor 1} X\) being an equivalence for all \(X : C_f\), which follows if the coevaluation \(I^\vee \tensor I \to 1\) is an isomorphism.

For the first two points to hold, we only need dualisability of \(I\). The third is where we needed invertibility, in order to make \(T_f\) even plausible as a candidate for \(E_f\).

Example In \(C = \mathrm{An}\), we consider the algebra \(A = \mathrm{core}(\mathrm{FinSet})\), the free \(\EE_\infty\)-space on a point, with the distinguished endomorphism \((- \cup \{\ast\})\). The telescope construction is \[ T_f \simeq \ZZ \times \BB\Sigma_\infty \not\simeq E_f = \Omega^\infty \SSS \simeq \ZZ \times \BB\Sigma_{\infty +}. \] We can verify that \((- \cup \{\ast\})\) doesn't act invertibly: It includes each copy of \(\Sigma_\infty\) into the the next copy as the stabiliser of a point.

When it works, it works.

Let's first summarise our setup:

\((C, \tensor, 1)\) is a symmetric monoidal category, \(I : C\) is an invertible object and \(f: 1 \to I\) is a morphism.

\(C_f\) is the bireflective subcategory on which \(f\) acts invertibly. The left localisation \(L_f\) is a smashing localisation, given by an idempotent algebra \(E_f\).

We would like to know whether \(E_f\) is equivalent to the telescope \[ T_f \coloneqq \colim ( 1 \xrightarrow{f} I \xrightarrow{f \tensor 1} I^{\tensor 2} \to \ldots) \]

The motivic literature contains the following criterion for when the telescope does give the correct answer.

Criterion (Robalo–Voevodsky and many other people) If for some \(n \geq 2\), the cyclic permutation \[ (12\ldots n) : f^{\tensor n} \Rightarrow f^{\tensor n} : 1 \to I^{\tensor n} \] is homotopic to the constant homotopy, then \(T_f \simeq E_f\).

The idea It suffices to show that \(f\) acts invertibly on \(T_f\).

There's an obvious equivalence \(T_f \xrightarrow{\sim} T_f \otimes I\), given by a cofinal inclusion of the two colimit diagrams.

The issue is that this inclusion differs from the action of \(f\) on \(T_f\). An inclusion would have identity arrows and identity homotopies everywhere. The homotopies that are involved with \(T_f \xrightarrow{f \tensor 1} I \tensor T_f\) come from the bifunctoriality of \(- \tensor -\). If one thinks about it, the homotopy is exactly the cyclic permutation \((12)\) acting on \(f^{\tensor 2}\). If we skip terms and use the cofinal subdiagram \[ A \xrightarrow{f^{\tensor n}} I^{\tensor n} \tensor A \to \ldots \] then the difference is the permutation \((12\ldots (n+1))\). If this homotopy is constant, then the construction goes through, and the action of \(f\) on \(T_f\) can be identified with a cofinal inclusion, and is hence an isomorphism.

Corollary If \(C\) is stable, then \(T_f \simeq E_f\).

Proof The action of \(\Sigma_n\) on the above map is given by a map \[ \BB\Sigma_n \to \Map_C(1, I^{\tensor n}). \] When \(C\) is stable, the right-hand side is the 0-th space of spectrum, and the map factors as \[ \SSS[\BB\Sigma_n] \to \map_C(1, I^{\tensor n}). \] The cyclic permutation then acts through its image in \[ \Sigma_n = \pi_1\BB\Sigma_n \to \pi_1\SSS[\BB\Sigma_n]. \] The latter is, by Hurewicz theorem, \(\pi_1\SSS \oplus \Sigma_n^{\textrm{ab}} \simeq \ZZ/2[\eta] \oplus \{\pm\}\). The permutation lands in the second factor as its sign, and is \(+\) whenever \(n\) is odd.

Corollary The classifying spaces \(\mathrm{Vect}_{\mathbb{K}}\) of vector bundles (\(\mathbb{K} = \mathbb{R},\mathbb{C}\)) have group completion \(\mathrm{kgl}_{\mathbb{K}}\) (under Whitney sum) given by the corresponding telescopes: \[ \mathrm{ko} \coloneqq (\mathrm{Vect}_{\mathbb{R}})^{\textrm{grp}} \simeq \ZZ \times \BB\mathrm{O}, \mathrm{ku} \coloneqq (\mathrm{Vect}_{\mathbb{C}})^{\textrm{grp}} \simeq \ZZ \times \BB\mathrm{U}. \] In fact, the topological categories \(\mathrm{Vect}_{\mathbb{K}}^{\textrm{inj}}\) of finite dimention \(\mathbb{K}\)-vector spaces and injections "Picard complete" to categories \(\mathrm{kgl}_{\mathbb{K}}^{\textrm{inj}}\) through their telescopes. The mapping spaces are \[ \Map_{\mathrm{kgl}_{\mathbb{K}}^{\textrm{inj}}}(m,n) \simeq \begin{cases}\emptyset & m > n \\ \mathrm{GL}_\mathbb{K}(\infty)/\mathrm{GL}_\mathbb{K}(-m+n) & m \leq n\end{cases} \]

Proof The action of \((123)\) on \(\mathbb{R}^3\) is homotopic to the identity. In the complex case, even \((12)\) acts as identity.

(Central) group extensions from the homotopy perspective

Fri, 26 Dec 2025 09:07:34 +0100

\[ \gdef\Bund{\mathbf{B}} \gdef\Aut{\mathrm{Aut}} \gdef\fib{\mathrm{fib}} \gdef\Ani{\mathrm{An}} \gdef\AniB{{\mathrm{An}_\ast}} \gdef\Grp{\mathrm{Grp}} \gdef\Loop{\mathrm{\Omega}} \]

The original version was posted on my Blogspot, followed subsequently by a more elaborate Chinese version, hosted on 香蕉空间 Bananaspace, where I reviewed some group theory and higher algebra for a more clear story. Feel free to make suggestions and please do point out any other error you notice!

This post tries to explain from a homotopist's perspective the following statement:

PROPOSITION. \(1 \to A \to G \to Q \to 1\) are classified by (homotopy classes of) maps \(\Bund Q \to \Bund ^2 A.\)

Under the hood of fancy language, this is really an elementary algebraic statement.

I. Central extensions of groups are classified by \(\mathrm{H}^2(Q,-)\)

It's a basic result in group theory/group cohomology that the second cohomology of \(Q\) with coefficients in a trivial module \(A\) classifies such central extensions. More generally, if \(A\) carries some \(Q\)-action, the cohomology with twisted coefficients \(\mathrm{H}^2(Q; A)\) classifies extensions where the conjugation action of \(Q\) on \(A\) coincides with the given one. The zero class in \(\mathrm{H}^2(Q; A)\) corresponds to the semi-direct product in either case.

The usual algebraic proof often consists of directly manipulating 2-cocycles to write down the multiplication rule, a boring and tedious process that is hardly illuminating. In this blogpost. I hope to shed some light onto the result by putting on the lenses of homotopy theory. We shall focus on the case of central extensions. Here we have the following Lemma 3.4.2 in May-Ponto's More Concise book. With the point-set mess cleaned up, it says:

Lemma Let \(f : X \to Y\) be a be a map of connected pointed spaces whose ﬁber \(\fib f\) is an Eilenberg-Mac Lane space \(\Bund ^n A\) for some abelian group \(A\) and \(n \geq 1\). Then the following are equivalent:

The fiber sequence of \(f\) extends one step to the right: \[ \Bund ^n A \to X \xrightarrow{f} Y \xrightarrow{k} \Bund ^{n+1} A. \]
The fundamental group \(\pi_1(Y)\) acts trivially on \(\pi_n(\fib f) = A\).

In this case, The cohomology class \(k \in \mathrm{H}^{n+1} (Y; A)\) represented by \(k : Y \to \Bund^{n+1} A\) is the only obstruction to the lifting of maps \(T \to Y\) along \(f\). These are famously known under the names of \(k\)-invariants and characteristic classes. This result is of great utility in the theory of Postnikov-Whitehead towers.

If we apply the lemma to the map \(\Bund G \to \Bund Q\) with fiber \(\Bund A\), the action of \(Q\) on \(A\) referred to above is by conjugation (see later), so it's exactly the central extensions that are classified by \(\mathrm{Map}_\Ani (\Bund Q, \Bund ^2A)\). This is a \(2\)-truncated space with \(\pi_n = \mathrm{H}^{2-n}(Q; A)\).

The proof presented in May-Ponto uses the Serre spectral sequence. In this blogpost, I'll try to explain the statement within the context of (higher) group extensions.

II. The threefold way of group extensions

Let's put on out homotopy hat and apply the machinery to (discrete) groups. We want to describe arbitrary extensions of groups \(1 \to N \to G \to Q \to 1\) .

II.1 Group extension? That's just a bundle.

Famously from classical topology, \(F\)-bundles with structure group \(G\) are classified by (unpointed) maps into \(\Bund G\), and they're the same as principal \(G\)-bundles.

Example Rank \(n\) vector bundles have structure groups \(\mathrm{GL}_n\). they are classified by \(\Bund \mathrm{GL}_n\). Due to reasons of Lie theory, we know this from classical geometry as \(\mathrm{BO}(n)\) or \(\mathrm{BU}(n)\), depending on the base field.

A direct inspection of the Puppe sequence shows that arbitrary extensions of groups \(1 \to N \to G \to Q \to 1\) are precisely the \(\Bund N\)-bundles over \(\Bund Q\). By (un)straightening they are equivalently given by functors \(\mathrm{Fun}(\Bund Q, \Ani)\) which land in the full subgroupoid generated by \(\Bund N\). In other words, the \(N\)-extensions of \(Q\) are classified by \(\Bund \Aut_\Ani (\Bund N)\).

II.2 The very important point: Semi-direct products

The careless reader (like me) might have remembered that \(Q \to \Aut_\Grp(N)\) classifies only the semi-direct products! Why does the similar construction above classify all extensions? What's different here?

It took me a while to remember that semi-direct products are split extensions. Under unstraightening, these are bundles equipped with a distinguished section.

In other words I've mixed up \(\Aut_\Ani (\Bund N)\) with \(\Aut_\AniB (\Bund N)\).

\(\Aut_\AniB (\Bund N)\) classifies pointed bundles, which are exactly the split extensions. Observe that it's equivalent to the classifying space \(\Bund \Aut_\Grp(N)\).
\(\Aut_\Ani (\Bund N)\) receives a map from the latter through forgetting the (datum witnessing the preservation of) basepoint.

These sit in a fiber sequence \[ N \to \Aut_\AniB (\Bund N) \to \Aut_\Ani(\Bund N) \to \Bund N. \] Here the last map is evaluation at the base point of \(\Bund N\). (To see this, use e.g. that \(\AniB \to \Ani\) is the universal left fibration.)

As the middle map is a group morphism, this sequence extends to the right into \[ \Aut_\Ani (\Bund N) \to \Bund N \to \Bund \Aut_\AniB (\Bund N) \to \Bund \Aut_\Ani (\Bund N), \] where the conjugation action of \(N\) on itself inducese the map \(\Bund N \to \Bund \Aut_\Grp(N) \simeq \Bund \Aut_{\AniB}(\Bund N)\). We see that \[ \pi_0\Aut_\Ani(\Bund N) = \mathrm{Out}(N), \pi_1\Aut_\Ani (\Bund N) = Z(N). \]

If we further assume that \(N = \Bund^n A, n \geq 0\) is (a delooping of) an abelian group, then the (pointed) map \(\Bund^{n+1} A \to \Bund\Aut_\AniB (\Bund^n A)\) comes equipped with a nulhomotopy, witnessing a splitting \[ \Aut_\Ani (\Bund^n A) \simeq \Aut_\AniB (\Bund^n A) \times \Bund^{n+1} A \simeq \Aut_\Grp(A) \times \Bund^{n+1}A. \] I should warn ~~you~~ myself that this splitting is only one of spaces, not of groups.

Exercise The map \(\Aut_\Ani (\Bund N) \to \Bund N\) is in general NOT a morphism of groups.

As an irrelevant side note, the extension \(1 \to G/Z(G) \to \Aut (G) \to \mathrm{Out}(G) \to 1\) doesn't always split for (non-abelian) groups.

II.3 Another attempt at central extensions

Take now the group \(A\) to be discrete abelian, then the space \(\Bund A\) is canonically a commutative group, as well as all its higher deloopings \(\Bund^n A\). Its action on itself by translation provides a group morphism \[ \Bund^n A \xrightarrow{\lambda} \Aut_\Ani (\Bund^n A). \] Moreover, the \(\Bund ^n A\)-bundle classified by the delooping \[ \Bund^{n+1} A\xrightarrow{\Bund\lambda} \Bund\Aut_\Ani (\Bund^n A) \] is exactly the path fibration \(\mathbf{E}\Bund^n A \to \Bund^{n+1} A\), a.k.a the action (\(\infty\)-)groupoid of the group \(\Bund^n A\). This is a classical construction of delooping, known as the bar construction.

We now really make use of \(A\) being DISCRETE ABELIAN. I claim that the morphism \(\Bund\lambda\) exhibits \(\Bund^{n+1} A\) as an \((n+1)\)-connective cover of \(\Bund \Aut_\Ani (\Bund^n A)\).

To see this, observe the following pullback square. \[\begin{CD} \bullet @>>> \mathrm{pt}\\ @VVV @VVV\\ \mathbf{E}\Bund^n A @>>> \Bund\Aut_\AniB(\Bund^n A)\\ @VVV @VVV\\ \Bund^{n+1} A @>>> \Bund\Aut_\Ani (\Bund^n A) \end{CD}\] The top left corner \(\bullet\) is the fiber of \(\Bund \lambda\). Seeing as \(\mathbf{E}\Bund^nA\) is contractible, it's also equivalent to \(\Loop \Bund \Aut _\AniB(\Bund^n A)= \Aut _\AniB(\Bund^n A)\). Because \(\Bund^n A\) is Eilenberg-Mac Lane, this is just the discrete group \(\Aut_\Grp(A)\).

This shows that the map \(\Bund\lambda\) is a map with discrete fiber mapping out of a simply connected space into a connected space. This means that it is a universal cover, and hence the fiber of the \(1\)-truncation map \[ \Bund\Aut_\Ani (\Bund^n A) \to \tau_{\leq1} \Bund\Aut_\Ani (\Bund^n A). \] The latter can be identified as \(\Bund\Aut _\Grp(N)\), and it classifies the action of fundamental groups on the fibers. (EXERCISE!)

From this discussion, we have obtained the fiber sequence we're looking for \[ \Aut_\Grp (A) \to \Bund^{n+1} A \to \Bund\Aut_\Ani (\Bund ^n A) \to \Bund\Aut_\Grp (A). \] and a non-computational proof of the Lemma 3.4.2 presented at the start.

As a special case in \(n = 1\): Given an extension \(1 \to A \to G \to Q \to 1\) with abelian kernel and classifying map \(g : \Bund Q \to \Bund\Aut_\Ani (\Bund A)\), a witness to the triviality of the \(Q\)-module \(A\) is exactly a lift of \(g\) to \(\Bund^2 A\).

III. Another well-known example

The theory of classifying objects work in a more general context. Most notably for (sheaf) topoi. (I'm not too confident on the details, but this seems to be what's going on in principle. PLEASE DO CORRECT ME, I VERY WELL MAY BE WRONG)

In particular, we have a central extension of group sheaves on affine schemes (with your favorite topology): \[ 1 \to \mathbb{G}_m \to \mathrm{GL}_n \to \mathrm{PGL}_n \to 1, \] These should correspond to fiber sequences \[ \Bund \mathbb{G}_m \to \Bund \mathrm{GL}_n \to \Bund\mathrm{PGL}_n \to \Bund ^2\mathbb{G}_m \] giving corresponding long exact sequences on cohomology (=global sections).

A. Further links and miscellany

i. Very much classical geometry

The space \(\Bund G\) really classifies principal \(G\)-bundles. With the datum of a \(G\)-action on \(F\), these induce \(F\)-bundles as follows: \[\begin{array}{rcl} (B \xrightarrow{f} \Bund G) &\simeq &(P = f^\ast \mathbf{E}G \to B : \text{principal bundle})\\ &\simeq &(P \times_G F \to B : \text{\(F\)-bundle with\(G\)-structure}) \end{array}\]

ii. The homotopy action of \(\pi_1(Y)\) on \(F := \operatorname{fib} (X \to Y)\) (May-Ponto 1.4-1.5)

In the sections May-Ponto 1.4-1.5, the authors defined for connected pointed spaces \(X\) and \(Y\) an action of the fundamental group \(\pi_1(Y)\) on \(\pi_0\mathrm{Map}_\AniB (X,Y)\). Upon inspection of the geometric definition, this should arise from an \(S^1\)-comodule structure on \(X\) by "extruding a neighborhood around the basepoint". This is where the conjugation action comes from. There's quite some potential confusion chasing the basepoint around.

There's another action on the fiber given heuristically by path concatenation. With the following description of the fiber \[ F = X \times_Y 1 = \big\{(x : X, \alpha : y_0 = f(x)\big\}, \] we can describe this action as \[\begin{align*} F \times \Loop Y &= X \times_Y 1 \times 1 \times_Y 1 \\ &= X \times_Y 1 \times_Y 1 \\ &\xrightarrow{y_0} X \times_Y Y \times_Y 1 \\ &= X \times_Y 1 \\ &= F,\\ \left(\left(x : X,\alpha : y_0 = f\left(x\right)\right), \beta : y_0 = y_0\right) &\mapsto\left(x : X, \alpha \circ \beta : y_0 = f\left(x\right)\right). \end{align*}\] One can also view this as given by the straightening \(Y \to \Ani, y \mapsto X \times_Y \{y\}\), plus the equivalence \(Y \simeq \Bund \Loop Y\) determined by a basepoint. At first glance, this looks much like a right translation rather than conjugation. Moreover, this map has no reason to preserve basepoints. (One should think about how these two actions relate to each other.)

Z. Changelog and TODO

TODO: Rework the appendix on \(\pi_1\)-actions.

Update 2.0 (26.dec.2025): Ported to hugo & org-mode

Update 1.3 (15.aug.2025): Reworked section III completely with a (nice and) non-sketchy proof.

Update 1.2e (14.aug.2025): Fatal error found and hotfixed.

Update 1.2 (13.aug.2025): Rewording and grammatical fixes, reorganized MathJax code.