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Stream: learning: questions

Topic: Derived bilinear and quadratic forms

Damien Robert (Mar 18 2026 at 13:21):

Hi everyone, this is my first time posting here!

I have some elementary questions about derived (non additive) functors between categories of sheaf modules, seen as functors between stable or prestable $\infty$-categories, and I hope I am in the correct channel to ask them. (Disclaimer: I have been trying to learn more about $\infty$-categories but currently I am just a beginner on this topic.)

First some context. Warning, wall of text incoming, you can skip this part and go directly in my questions below.

Damien Robert (Mar 18 2026 at 13:22):

I work in algorithmic number theory, and lately I have taken an interest in the arithmetic associated to biextensions and cubical torsor structures on abelian varieties. Biextensions were introduced by Mumford, and thoroughly developed by Grothendieck in SGA7. Later, Lawrence Breen in his book Fonctions thêta et théorème du cube introduces the notions of a symmetric biextension and of a "structure du cube sur un torseur $L$" which I'll translate as a cubical (or cubic?) torsor structure. The first Chapter of the book by Moret-Bailly Pinceaux de variétés Abéliennes gives also a nice introduction to this concept (and of hypercube torsor structures).

For now let $Sh(X,\tau)$ be an arbitrary topos, and $F,G,H$ be abelian sheaves over $X$. Then Grothendieck shows in SGA7 that the stack of biextensions of $F \times G$ by $H$ corresponds to $\tau_{\leq 0} R\mathcal{Hom}(F \otimes^L G, H[1])$ (using cohomogical grading). Then Breen shows that a symmetric biextension by $H$ over $F \times F$ corresponds to $\tau_{\leq 0} R\mathcal{Hom}(L Sym^2 F, H[1])$, and a cubical $H$-torsor structure over $F$ corresponds to $\tau_{\leq 0} R\mathcal{Hom}(L P^+_2 F, H[1])$, where $P^+_2 F$ represents the universal quadratic form $q$ on $F$, normalised by $q(0)=0$ (by quadratic form I mean an application of degree $\leq 2$, not necessarily homogeneous; since $q$ is normalised, this is equivalent to requiring $q(x+y+z)-q(x+y)-q(y+z)-q(x+z)+q(x)+q(y)+q(z)=0$ internally in the topos).

They then specialize the theory to the case of (degenerations) of abelian schemes over some base $S$, with $F, G$ the fppf sheafs associated to abelian schemes $A,B/S$, and $H$ the fppf sheaf associated to $\mathbb{G}_m$. For simplicity I'll stick to the case of abelian varieties $A,B/k$ over some field $k$ here.

Reformulating the cohomological definitions above (which I will call the "intrinsic definitions"), we obtain that:

• a biextension of $A \times B$ (by $\mathbb{G}_m$) is a derived bilinear form with values in $B\mathbb{G}_m$.
• a symmetric biextension of $A \times A$ is a derived symmetric bilinear form with values in $B\mathbb{G}_m$.
• a cubical torsor $L$ over $A$ is a derived quadratic form with values in $B\mathbb{G}_m$.

Notice that in each case the derived bilinear/quadratic structure induces a map to $B\mathbb{G}_m$, hence a $\mathbb{G}_m$-torsor (i.e. a line bundle) over $A \times B$, resp. $A$; torsor which is further endowed with a "bilinear", resp. "quadratic" structure. Taking simplicial resolutions, Grothendieck and Breen give concrete description of what this extra structure means in practice (I will call this the "external description").

For instance, a cubical structure on a $\mathbb{G}_m$-torsor $L$ should consist in a trivialisation $\sigma$ of $\theta_3(L) = m_{123}^{\ast} L \otimes m_{12}^{\ast} L^{-1} \otimes m_{23}^{\ast} L^{-1} \otimes m_{13}^{\ast} L^{-1} \otimes m_{1}^{\ast} L \otimes m_{2}^{\ast} L \otimes m_{3}^{\ast} L$ (using the notations of Moret-Bailly), where $m_{123}: A^3 \to A, (P_1, P_2, P_3) \mapsto P_1+P_2+P_3$, $m_{12}: A^3 \to A, (P_1, P_2, P_3) \mapsto P_1+P_2$, and so on. Furthermore, this trivialisation $\sigma$ should satisfy some appropriate cocycle conditions (one elegant way to state them, as explained in Moret-Bailly's book, is that up to replacing $A$ by an fppf cover $A'$, and $\mathbb{G}_m$ by some injective group $G$, we obtain an induced trivialisation $\sigma'$ associated to $\Theta_3(L')$ where $L'$ is the $G$-torsor on $A'$ associated to $L$, and this $\sigma'$ should be of the form $\Theta_3(s)$ for $s$ a trivialisation of $L'$ over $A'$, i.e. $L'$ should be endowed with a "trivial cubical structure"). Similar external descriptions hold for biextensions and symmetric biextensions.

Now the remarkable thing is that due to the rigidity of abelian varieties, given a line bundle $L$ on $A$, it is automatically endowed with a unique cubical structure. Likewise we have a bijection between morphisms $A \to B^{\vee}$ and biextensions over $A \times B$ (via pullback of the Poincare biextension over $B \times B^{\vee}$), and symmetric biextensions over $A \times A$ correspond to polarisations $\lambda: A \to A^{\vee}$.

This is pretty cool, because this means that we can think of polarisations and line bundles on an abelian variety $A$ as derived symmetric bilinear form and derived quadratic forms with values in $B\mathbb{G}_m$ respectively. And this explains many constructions we do with them: e.g., the Mumford construction that associates a polarisation $\lambda_L$ on $A$ from a line bundle $L$ mirrors the way we associate a symmetric bilinear form $b(x,y)=q(x+y)-q(x)-q(y)$ from a (normalised) quadratic form $q$; this also "explains" why having a rational line bundle $L$ is stronger than having a rational polarisation (because a quadratic form is a stronger arithmetic data than a symmetric bilinear form). More generally every elementary statement on quadratic/bilinear form seems to translate to a "categorical version" (I'll come back to this later).

For another example: a quadratic form $q$ on $X$ degenerates (i.e. becomes linear) over $X' \subset X$ when the associated bilinear form $b$ is $0$ over $X' \times X'$. Well if $L$ is a line bundle on $A$, the associated biextension is trivial over $K(L) \times K(L)$ where $K(L)$ is the kernel of the polarisation $\lambda_L$, so the cubical structure on $L$ degenerates to a squared structure on $L \mid K(L)$, i.e. we have a "derived linear form" with values in $B\mathbb{G}_m$, i.e. an element of $Ext^1(K(L), \mathbb{G}_m)$, i.e. a group extension $G(L)$ of $K(L)$ by $\mathbb{G}_m$. We recover Mumford's theta group $G(L)$. In fact, the biextension associated to $\lambda_L$ is trivial over $K(L) \times A$, this gives the usual action of $G(L)$ on the sections of $L$. We also recover the standard Weil pairing as a degeneration of the biextension structure.

Yet another (related) example: a quadratic form $q$ on $X$ descends through a morphism $\phi: X \to Y$ with kernel $X'$ when $q(X')=0$, but also the associated bilinear form $b$ is trivial over $X' \times X$. There is a corresponding statement for descending a cubical structure over $A$ through a morphism $\phi: A \to B$ with kernel $A'$: there should be a trivialisation of the cubical structure over $A'$ and of the biextension over $A' \times A$, trivialisations compatible with each other over $A' \times A'$. Applying this to a complex abelian variety $A=\mathbb{C}^g/\Lambda$, and because cube structures over $\mathbb{C}^g$ are necessarily trivial, we get that cube structures over $A$ can be described as suitable descent of the trivial cube structure over $\mathbb{C}^g$ through $\Lambda$. As explained by Breen in his book this gives an alternative algebraic theory of theta functions.

A last comment on this: we can think of a cube structure on a line bundle $L$ over $A$ as some sort of "quadratic arithmetic" on $L$ which lifts the group arithmetic (i.e. a "linear arithmetic") on $A$. It turns out that instantiating this arithmetic explicitly gives surprisingly efficient formulas. This is what I have been working on: even in the very special case of an elliptic curve $E/\mathbb{F}_q$ over a finite field, it turns out that using this "cubical arithmetic" (on the torsor associated to $(0_E)$) to compute e.g. pairings beat in certain cases the state of the art.

Ok, that's it for the context, apologies that was longer than I expected. Now for my questions.

Damien Robert (Mar 18 2026 at 13:22):

Question 1

As we saw above, the stack of "derived symmetric bilinear form" is given by $\tau_{\leq 0} R\mathcal{Hom}(L Sym^2 A, \mathbb{G}_m[1])$, and a "derived quadratic form" is given by $\tau_{\leq 0} R\mathcal{Hom}(L P^+_2 A, \mathbb{G}_m[1])$ (using cohomological grading).

In both cases we need to derive a quadratic functor $G$ (in particular not additive), like $G=Sym^2$, on abelian sheaves in a topos. For this, Breen takes simplicial resolutions. I expect that the modern point of view of derived categories as stable $\infty$-categories clarifies the construction of $LG$.

More precisely (this is essentially a copy/paste of this question I asked on mathoverflow):

First we consider the case of a topos over a point, and of an additive functor. So let $G:Ab→Ab$ be a right exact functor on the abelian category of abelian groups.

Since $Ab$ has enough projectives, we can build the left derived functor $LG:D_−(Ab)→D_Ab)$(Lurie, Higher Algebra, Example 1.3.3.4.), which we can extend to $LG:D(Ab)→D(Ab)$ using the fact that $D(Ab)$ (using the construction of Higher Algebra, Definition 1.3.5.8. for Grothendieck abelian categories) is left complete.

More generally, I also want to consider functors $G$ which are not necessarily additive, but at least preserve 1-sifted colimits. In which case, I think that we can also define $LG:D(Ab)→D(Ab)$ as the stabilisation of $Ani(G):D(Ab)_{≥0}→D(Ab)_{≥0}$ (using homological grading).

Now let $(X,τ)$ be a 1-site, then we can consider the 1-topos $Sh(X,Ab)$. Then, if I understand correctly "Lurie, Spectral Algebraic Geometry, Corollary 2.1.2.3", we have that $D(Sh(X,Ab))$ is the hypercompletion of the topos of $h\mathbb{Z}$-sheaf of spectras $Sh(X,D(Ab))$.

On the one hand we can extend $G$ to a functor $\mathcal{G}$ on $Sh(X,Ab)$ by sheafification. On the other hand we can also extend $LG$ to a functor $\mathcal{LG}$ on $D(Sh(X,Ab))$ by (hyper)sheafification.

Now I would expect $\mathcal{LG}$ to be a left derived functor $L\mathcal{G}$ of $\mathcal{G}$. Is it true?

N.B.: for my applications, I am interested in $\tau_{\ge 0} \mathcal{Hom}(\mathcal{LG}, B \mathbb{G}_m)$ (in homological grading this time). Since I am truncating to the connective part, I guess that hypercompleting does not really change the result, and I could take the sheafification of $LG$ rather than the hypersheafification. Is that correct? (I am not really confortable with hypercompletion subtleties).

Damien Robert (Mar 18 2026 at 13:22):

Question 2

So we have seen that we have an intrinsic definition of biextensions, symmetric biextensions and cubical torsor structures as derived bilinear form, derived symmetric bilinear forms and derived quadratic forms with values in $B\mathbb{G}_m$.

But translating this definition into an external statement is quite painful, see e.g. $8 of Breen's book. Now in a standard 1-topos, I am fairly confortable translating/compiling an internal statement in the internal logic into an external statement.

Here we are in the derived world, so in an $\infty$-topos, but from what I understand HoTT gives us precisely an internal logic of this $\infty$-topos, and a way to "compile" internal statements into external statements. (Disclaimer: I know even less about HoTT than about $\infty$-categories.)

So the question is whether compiling the internal statement "we have a quadratic form with values in $B\mathbb{G}_m$" gives back the external characterisation of cubical torsor structures, without having to jump through the hoop of computing explicit simplicial resolutions as Breen does in his book?

To be fair, since I am looking at 1-truncations of Hom spaces/anima, HoTT is probably overkill and I probably just need to look at 1-stack semantics, rather than a fully fledged $\infty$-topos semantic (let alone the univalence principle).

Still, from the few thing I learned about HoTT, is that from afar it seems that it would work. The very rough idea I get from HoTT is that the equality type should be a fully fledged type/Anima, not just as a Proposition. And so an equality should be 'witnessed' by some sort of extra data, maybe subject to some sort of coherence condition, and recursively so.

If I apply this guiding principle to what should be a "derived quadratic form with values in $B\mathbb{G}_m$, I get that if $q$ corresponds to the torsor $L$, the statement $q(x+y+z)-q(x+y)-q(y+z)-q(x+z)+q(x)+q(y)+q(z)=0$ corresponds to a choice trivialisation $\sigma$ of $\theta_3(L) = m_{123}^{\ast} L \otimes m_{12}^{\ast} L^{-1} \otimes m_{23}^{\ast} L^{-1} \otimes m_{13}^{\ast} L^{-1} \otimes m_{1}^{\ast} L \otimes m_{2}^{\ast} L \otimes m_{3}^{\ast} L$. But now this choice $\sigma$ is subject to some coherence conditions.
Except that I have moved one categorical level up, from the delooping $B \mathbb{G}_m$ to $\mathbb{G}_m$; and if $\sigma$ does give a cubical structure, I can tweak it by an actual quadratic form $\sigma'$ with values in $\mathbb{G}_m$ to get a potentially non isomorphic cube structure $\sigma \sigma'$. (And I can stop the recursion here, because the $\pi_i$ are trivial for $i>1$.) Which is precisely what happens in practice from the explicit description of the cubical torsor structure. (One needs to be a bit careful about moving one level up from $B \mathbb{G}_m$ to $\mathbb{G}_m$: for instance if we have $\sigma$ inducing a trivial symmetric biextension, then we have to tweak $\sigma$ by $\sigma'$ an alternate bilinear form rather than a symmetric bilinear form to get another symmetric biextension.)

Damien Robert (Mar 18 2026 at 15:50):

Damien Robert said:

This is pretty cool, because this means that we can think of polarisations and line bundles on an abelian variety $A$ as derived symmetric bilinear form and derived quadratic forms with values in $B\mathbb{G}_m$ respectively.

Of course, retrospectively, I should have understood this a lot sooner. Already to define the dual abelian variety $A^{\vee}$, we cannot use $\mathcal{Hom}(A, \mathbb{G}_m)$ because it is trivial, so we instead go through $\tau_{\leq 0} R\mathcal{Hom}(A, \mathbb{G}_m[1])$, whose' $\pi_1$' is trivial, and whose '$\pi_0$' is $\mathcal{Ext}^1(A, \mathbb{G}_m)$; which we can also reinterpret as $Hom(A, B\mathbb{G}_m)$ where here in the Hom we want morphisms that respects the group structure, i.e. "morphisms of Picard stacks". In other words the dual abelian variety is described by "1-truncated derived linear forms" already.

(One might naively expect that $R \mathcal{Hom}(A, \mathbb{G}_m)$ would be concentrated in degree $[0,1]$, but although $\mathcal{Ext}^2(A, \mathbb{G}_m)=0$, Breen has examples of abelian varieties in characteristic 2 where $\mathcal{Ext}^3(A, \mathbb{G}_m) \ne 0$, so that's why we need to truncate to have something sane.)

And as a bilinear form $b: X \times Y \to Z$ may be thought as a linear map $\phi: X \to Y^{\vee}$ where $Y^{\vee}=Hom(Y, Z)$ is the "dual" of $Y$ (and, in the case $Y=X$ with $X \simeq {X^{\vee}}^{\vee}$ is "bidual", then $b$ is symmetric precisely when $\phi$ is autodual), it indeed is the case that a biextension, i.e. a derived bilinear form on $A \times B$, is the same as a linear map $A \to B^{\vee}$, where $B^{\vee}$ is the "derived dual" of $B$ (and in the case $A=B$, the biextension is symmetric precisely when $\phi$ is symmetric, i.e. equal to its dual). So it makes sense that polarisations are a way to encode (1-truncated) derived bilinear forms on abelian varieties.

Kevin Carlson (Mar 18 2026 at 15:58):

Hi Damien, welcome! This is really an awfully, awfully long, very technical post. I think you’d have a lot better luck getting a conversation started if you could pick somewhere to start where an expert would have at least a decent chance of being able to say something quickly.

Damien Robert (Mar 18 2026 at 16:09):

Hi Kevin, thanks! Yes I wanted to add some context to my questions, but maybe that was a bad idea... I am mostly interested in Question 1 which should be self contained: basically to what extent can we expect sheafification to commute with derivation. Or more precisely, given a functor $G: Ab \to Ab$ (say which preserves 1-sifted colimits), if we sheafify and then derive it, is it the same thing as sheafifying the derivation of $G$ (up to hypercompletion subtleties)?

Kevin Carlson (Mar 18 2026 at 16:16):

Thanks for clarifying that. As far as I can tell you have a sequence of two questions, even in that: first, about deriving sifted colimit-preserving functors, and then about how this hypothetical operation interacts with sheafification. No?

Damien Robert (Mar 18 2026 at 16:38):

Indeed, but I am fairly sure that for $G$ a 1-sifted colimit preserving functor, then under the identification of $Ani(Ab)$ with $D_{\geq 0}(Ab)$, $Ani(G)$ coincides with $LG$. Here $Ani(G)$ is the unique functor $Ani(Ab) \to Ani(Ab)$ that commutes with $\infty$-sifted limit and whose value on projective f.t. $\mathbb{Z}$-modules (or even just free f.t. $\mathbb{Z}$-modules) $M$ is given by $G(M)$.

On the other hand there is indeed also an implicit question about whether stabilisation also commutes with sheafification, because I need to have $B \mathbb{G}_m$, the delooping of $\mathbb{G}_m$, in my sheaf topos, but that's less important.

So forgetting about stabilisation, I guess my main question is, given $(X,\tau)$ a $1$-site and denoting $\mathcal{G}: Sh(X, Ab) \to Sh(X, Ab)$ the sheafification of $G$, then is the hypersheafification of $Ani(G)$ a derived functor of $\mathcal{G}$ under the identification of $D_{\geq 0}(Sh_{\tau}(X,Ab))$ as the hypercompletion of $Sh_{\tau}(X, D_{\geq 0} (Ab))$?

John Baez (Mar 18 2026 at 17:16):

I'm really interested in polarizations of abelian varieties, and curious about quantifying the way they contain less information than line bundles, but I'd have to overcome a laziness barrier to follow how it relates to all this more general abstract stuff. :cry: In short, I'm the guy who was interested in your intended application more than the techniques.

Damien Robert (Mar 18 2026 at 19:03):

John Baez said:

I'm really interested in polarizations of abelian varieties, and curious about quantifying the way they contain less information than line bundles

So that's a very good question, and IMHO a good rule of thumb to have an intuition about the difference between line bundles and polarisations is that it is exactly (a derived version of) the difference between a (normalised) quadratic form and a symmetric bilinear form.

To a (normalised) quadratic form $q: X \to Z$ (where $X, Z$ are say some abelian groups) we can associate a symmetric bilinear form $b: X \times X \to Z$ via $b(x,y)=q(x+y)-q(x)-q(y)$. Conversely to a bilinear form $b$, we can associate a quadratic form $Q(x)=b(x,x)$. Crucially, these are not inverse operations!

A few remarks on this:

• Two (normalised) quadratic forms $q, q'$ induce the same $b$ precisely when they differ by a linear form, i.e. by an element of the "dual" $X^{\vee}=Hom(X,Z)$
• When $q$ is homogeneous of degree 2 (or equivalently it is symmetric, i.e. $[-1]^{\ast} q=q$, i.e. $q(-x)=q(x)$), then $Q=2q$. So as usual with quadratic forms we have problems with $2$, notably if $2$ is not invertible, we cannot recover $Q$ from $q$.
• If we encode $b$ as the associated "polarisation" $\phi: X \to X^{\vee}$, then $Q$ is the pullback by $Id \times \phi$ of the universal quadratic form on $X \times X^{\vee}$ given by $(x,\psi) \mapsto \psi(x)$. Notice also that $\phi$ is given by $\phi: x \mapsto t_x^{\ast} q - q$ (up to renormalisation to have a linear form).

All of this generalizes to abelian varieties, replacing $X$ by $A$ and $Z$ by $\mathbb{G}_m$, or rather by $B\mathbb{G}_m$:

• to a line bundle $L$ over $A$ we can associate a biextension over $A\times A$, encoded via the polarisation $\phi_L: A \to A^{\vee}, P \mapsto t_P^{\ast} L \otimes L^{-1}$.
• Two line bundles $L, L'$ induce the same $\phi$ precisely when they differ by an element of $Pic^0(A)=Hom(A, B\mathbb{G}_m)$, the dual $A^{\vee}$ of $A$.
• The Poincaré line bundle $P$ on $A \times A^{\vee}$ corresponds to the universal quadratic form. Pulling it back to $A$ via $Id \times \phi$ gives us a line bundle which is isomorphic to $L^2$ if $L$ is symmetric.

In particular, given a rational polarisation $\phi: A \to A^{\vee}$ over $k$, $\phi$ is induced by a line bundle only over an étale extension of $k$ in general (notice that although the multiplication by $2$ is not surjective on $\mathbb{G}_m$ it is étale locally surjective, at least in characteristic different from $2$). The set of such line bundles form a torsor $A'$ under $A^{\vee}$, and we can find a rational line bundle on $A$ inducing $\phi$ iff $A'$ is trivial, i.e. has a point over $k$. If $k$ is a finite field, this is always the case by Lang's theorem, so there is no obstruction! Likewise, if we want a rational symmetric line bundle $L$ representing $\phi$, the obstruction is measured by an étale $A[2]$-torsor.

(to be continued later)

Damien Robert (Mar 18 2026 at 21:26):

As an aside, I would like to point out that the above statements on line bundles on abelian varieties are treated in pretty much every book about them, but they do require some work, unless the trivial corresponding statements about standard quadratic and biliear forms. Related to my Question 2, is the question of how much the general theory of (symmetric) biextensions and cubical structures could simply be seen as the specialisation of a general theory of quadratic and bilinear forms in an $\infty$-topos; which would give the statements above "for free". Of course we would still need to prove the specific feature of abelian varieties, which is that all line bundles over them have a unique cubical structure. By the way, a key insight of Grothendieck-Breen-Moret Bailly is that it is the later notion rather than the former which is the "correct" one when looking at deformations of abelian schemes.

Damien Robert (Mar 18 2026 at 21:26):

Anyway, going back to quadratic forms, if we have a $X$-torsor $X'$ (i.e. a set $X'$ with a free and transitive action from $X$), then we can still define a notion of quadratic form $q: X' \to Z$ on $X'$. Namely, $q(w+x+y+z)-q(w+x+y)-q(w+y+z)-q(w+z+x)+\\q(w+x)+q(w+y)+q(w+z)-q(w) =0\quad \forall w \in X', x,y,z \in X$ (in fact it suffices to check this for one base point $w \in X'$). To this $q$ over $X'$, we can still associate a bilinear form $b: X \times X \to Z$ via $b(x,y)=q(w+x+y)+q(w)-q(w+x)-q(w+y)$.

Damien Robert (Mar 18 2026 at 21:26):

The same construction works for abelian varieties. If we have a $A$-torsor $A'$ and a rational line bundle $L$ on $A'$, this $L$ still gives a rational polarisation $\phi: A \to A^{\vee}$ on $A$.

The most important example is the case of Jacobians. If $C/k$ is a (proper smooth geometrically connected) curve, the theta divisor $\Theta$ (the locus of effective divisors of degree $g-1$) is rational and lives in $Pic^{g-1}(C)$, which is a torsor under the Jacobian $J(C)=Pic^0(C)$. So the Jacobian always has a rational principal polarisation $\lambda$, but it may not have a rational line bundle associated to $\lambda$ if $Pic^{g-1}(C)$ has no rational points.

Damien Robert (Mar 18 2026 at 21:26):

There is a famous article by Poonen and Stoll about this situation. When $A/k$ is a principally polarised abelian variety over a number field $k$, the Cassel-Tate pairing $ш(A) \times ш(A) \to \mathbb{Q}/\mathbb{Z}$ is non degenerate and antisymmetric (assuming the Tate-Shafarevich group is finite). If furthermore $\lambda$ is represented by a rational line bundle, then the Cassel-Tate pairing is even alternate, so the Tate-Shafarevich group has cardinal a square. But Poonen and Stoll construct explicit examples of curves where the Tate-Shafarevich group of the Jacobian is twice a square; hence their Jacobian cannot admit a rational line bundle representing the canonical principal polarisation!

John Baez (Mar 18 2026 at 22:01):

Damien Robert said:

John Baez said:

I'm really interested in polarizations of abelian varieties, and curious about quantifying the way they contain less information than line bundles

So that's a very good question, and IMHO a good rule of thumb to have an intuition about the difference between line bundles and polarisations is that it is exactly (a derived version of) the difference between a (normalised) quadratic form and a symmetric bilinear form.

To a (normalised) quadratic form $q: X \to Z$ (where $X, Z$ are say some abelian groups) we can associate a symmetric bilinear form $b: X \times X \to Z$ via $b(x,y)=q(x+y)-q(x)-q(y)$. Conversely to a bilinear form $b$, we can associate a quadratic form $Q(x)=b(x,x)$. Crucially, these are not inverse operations!

I've mainly thought about this in two contexts, which you might be interested in. Maybe you know all about this, and I'm not really an expert on it, but anyway:

1. The classification of finite simple groups includes those of 'Lie type' like the special orthogonal groups SO(n,F) and symplectic groups Sp(n,F) for finite fields F, but this becomes tricky when F has characteristic two, because the difference between symmetric and antisymmetric bilinear forms evaporates, but also the difference between bilinear forms and quadratic forms becomes very significant. (There's also the added complication that the classification of nondegenerate quadratic forms over finite fields is unfamiliar to physicists such as myself who are used to working over $\mathbb{R}$ or $\mathbb{C}$.)

2. Mac Lane classified homotopy 2-types of double loop spaces in terms of 'quadratic functions' from an abelian group $G$ to an abelian group $A$. Joyal and Street realized that Mac Lane was secretly classifying braided 2-groups $X$: braided monoidal groupoids where every object has an inverse under tensor product. $G$ is the group of isomorphism classes of objects of $X$ (aka $\pi_0$) while $A$ is the group of automorphisms of the unit object (aka $\pi_1)$). The quadratic function comes from the 'self-braiding' of objects $x$, namely $B_{x,x} : x \otimes x \to x \otimes x$, together with a standard trick for turning an endomorphism of $x \otimes x$ into an endomorphism of the unit object.

I feel the second one might even be relevant to what you're saying, though you're probably dealing with (derived versions of) symmetric 2-groups rather than braided ones.

Damien Robert (Mar 19 2026 at 22:08):

John Baez said:

I've mainly thought about this in two contexts, which you might be interested in. Maybe you know all about this,

On the contrary, I am quite out of my depth on all this higher category / derived stuff.

1. The classification of finite simple groups includes those of 'Lie type' like the special orthogonal groups SO(n,F) and symplectic groups Sp(n,F) for finite fields F, but this becomes tricky when F has characteristic two, because the difference between symmetric and antisymmetric bilinear forms evaporates, but also the difference between bilinear forms and quadratic forms becomes very significant. (There's also the added complication that the classification of nondegenerate quadratic forms over finite fields is unfamiliar to physicists such as myself who are used to working over $\mathbb{R}$ or $\mathbb{C}$.)

I tend to avoid characteristic 2 in my work because I use symmetric theta structures on ppavs a lot, and things get a bit weird when the $2$-torsion is not étale.
So apart from the Arf invariant, which is of course very important to classify even and odd theta characteristics, I don't know much about quadratic forms in char 2.

You might be interested in the following paper by Breen: Biextensions alternées. He defines a notion of antisymmetric and alternate biextension (in an analog way of his earlier definition of a symmetric biextension in his book); show that this is the 'correct' derived generalisation of a antisymmetric, resp. alternate bilinear form; and that being alternate is a stronger property than being antisymmetric, as expected.

Damien Robert (Mar 19 2026 at 22:09):

(For some reason most of the litterature on biextensions/cube structure seem to be in french)

Damien Robert (Mar 19 2026 at 22:34):

John Baez said:

2. Mac Lane classified homotopy 2-types of double loop spaces in terms of 'quadratic functions' from an abelian group $G$ to an abelian group $A$. Joyal and Street realized that Mac Lane was secretly classifying braided 2-groups $X$: braided monoidal groupoids where every object has an inverse under tensor product. $G$ is the group of isomorphism classes of objects of $X$ (aka $\pi_0$) while $A$ is the group of automorphisms of the unit object (aka $\pi_1)$). The quadratic function comes from the 'self-braiding' of objects $x$, namely $B_{x,x} : x \otimes x \to x \otimes x$, together with a standard trick for turning an endomorphism of $x \otimes x$ into an endomorphism of the unit object.

This seems very interesting, do you have a reference for this?

More generally, it has been quite a surprise to me to learn that the polarisations/line bundles on abelian varieties I had been working with were actually (1-truncated) derived bilinear forms and quadratic forms in disguise, and I'd be very interested in other examples of this kind in nature. Notably:

1) Going up in degree/dimension: of course biextension can be generalised to n-multi-extension (the derived version of multilinear forms), and cube structures to $n+1$-hypercube structure (the derived version of a form of degree $\leq n$; there is an unfortunate shift on the $n$ here), and an $n+1$-hypercube structure on $A$ gives a symmetric $n$-multiextension on $A^n$ in exactly the same way a form of degree $\leq n$$ gives a $n$-multilinear form.

However, triextensions (and more) of an abelian variety by $G_m$ are trivial, and so are the $n+1$-hypercube structures. It seems that the interesting arithmetic on abelian varieties is inherently quadratic.

Apparently, if $\pi: X \to S$ is a proper flat morphism of schemes of relative dimension $n$, then the determinant functor $R det \pi_{\ast}$ has a $(n+2)$-hypercube structure.

2) Going up in categorical level. I would also be curious to see some "natural examples" of a derived quadratic or bilinear form with value in $B^2 \mathbb{G}_m$, i.e. in gerbes rather than in torsors.

John Baez (Mar 20 2026 at 02:16):

Damien Robert said:

John Baez said:

2. Mac Lane classified homotopy 2-types of double loop spaces in terms of 'quadratic functions' from an abelian group $G$ to an abelian group $A$. Joyal and Street realized that Mac Lane was secretly classifying braided 2-groups $X$: braided monoidal groupoids where every object has an inverse under tensor product.

This seems very interesting, do you have a reference for this?

Sorry, I forgot to include a link: this is available in the unpublished version of their paper "Braided monoidal categories", which contains more than the published version! It's in section 7, "Cohomology of abelian groups", though you should also look at section 6, "Cohomology of groups".