Duality of Monads in Geometric Homotopy Theory


These are notes accompanying a talk that I once gave:

Abstract It is fairly well known that the category-theoretic concept of adjoint pairs captures a good deal of the informal philosophical sentiment of duality (Lambek 82). But ancienct metaphysics also tends to associate a primordial dynamical progression to duality. Indeed, adjoint pairs may progress, first of all to adjoint triples, which in turn constitute adjoint pairs of (co-)mondads. That these should serve a broader organizing role in geometry and physics was suggested starting around (Lawvere 91). Indeed, a fair bit of modern differential geometry and of modern physics springs out of a natural progression of adjoint (co-)monads in homotopy theory (Schreiber 11). In the talk I illustrate this by way of the formal derivations from dualities of the Stokes theorem (Bunke-Nikolaus-Völkl 13) and of the Noether theorem (Khavkine-Schreiber 15).

Telated talks include


Introduction – From philosophy to mathematics

According to the Oxford Dictionary, the informal meaning of duality is:

an instance of opposition or contrast between two concepts or two aspects

A useful mathematical formalization of this dialectic meaning of duality is provided by the concept of adjunction in category theory (or rather in 2-category theory).

LR L \dashv R

Joachim Lambek recalls (Lambek 82):

I spent my sabbatical 1965-6 in Zurich, where I had many conversations with the young American mathematician Bill Lawvere. We kicked around the idea that an interesting illustration of dialectic contradictions could be found in the “adjoint functors” of modern mathematics, which had recently been popularized by Peter Freyd in his book on Abelian categories.

Notice that various famous mathematical dualities are special cases or restrictions of adjunctions:

  1. A dual object V *V^\ast in a monoidal category (𝒞,)(\mathcal{C},\otimes) is the adjoint of VV in the delooping 2-category of 𝒞\mathcal{C}.

  2. Those equivalences of categories that are widely known as dualities are typically the restriction of adjunctions to their fixed points. This includes

William Lawvere goes on to refine this formalization. In (Lawvere 91) a duality of opposites is specifically taken to be an adjunction between idempotent (co-)monads, an adjoint modality

. \bigcirc \dashv \Box \,.

Philosophy knows the idea that duality of concepts is, or causes, a metaphysical dynamical process, providing emergence out of a primordial opposition. In western philosophy this perspective culminates in Hegel 1813.

3 3 2 2 1 1 \array{ \vdots && \vdots \\ \bigcirc_3 &\dashv& \Box_3 \\ \\ \bigcirc_2 &\dashv& \Box_2 \\ \\ \bigcirc_1 &\dashv& \Box_1 }

This vaguely resonates with Dana Scott‘s advice (Scott 70):

Here is what I consider one of the biggest mistakes in all of modal logic: concentration on a system with just one modal operator.

William Lawvere suggests in (Lawvere 92) that mathematics would be served by seriously considering the idea of progressions of adjoint modalities (the “objective logic”):

It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.



Geometric homotopy types

After so much general abstraction, let’s consider a concrete particular.



SuperFormalSmoothMfd SuperFormalSmoothMfd

for the category of super formal smooth manifolds, hence of products of smooth manifolds with formal duals of super local Artin algebras over \mathbb{R}.

Regard this category as a site via its canonical Grothendieck topology given by the open covers of the underlying ordinary manifolds. Write

HSh (SuperFormalSmoothMfd) \mathbf{H} \coloneqq Sh_\infty(SuperFormalSmoothMfd)

for the ∞-topos of (∞,1)-sheaves (“∞-stacks”) on this site. (The topological localization coincides here with its hypercompletion.)

We call this the \infty-topos of super formal smooth ∞-groupoids.


(Schreiber 13,15 with Dave Carchedi)

On H\mathbf{H} there is a progression of idempotent ∞-(co-)monads of the following form

Id Id R loc 0|1 \ loc 𝔻 0|1 & loc 𝔻 ʃ 0loc ¬¬ / * \array{ && & \mathbf{Id} &\dashv& \mathbf{Id} \\ && & \vee && \vee \\ && & \mathbf{\rightrightarrows} &\dashv& \mathbf{\rightsquigarrow} \\ && & \bot &\stackrel{}{}& \bot \\ && & \rightsquigarrow &\dashv& \mathbf{\R} & \simeq loc_{\mathbb{R}^{0|1}} \\ && & \vee &\backslash& \vee \\ \stackrel{}{} & & & \mathbf{\Re} &\stackrel{}{\dashv}& \mathbf{\Im} & \\ && & \bot &\stackrel{}{}& \bot \\ \stackrel{}{} && loc_{\mathbb{D}} \simeq_{\mathbb{R}^{0\vert 1}} & \mathbf{\Im} &\stackrel{}{\dashv}& \mathbf{\&} & && \\ && & \vee && \vee \\ \stackrel{}{} && loc_{\mathbb{R}} \simeq_{\mathbb{D}} & \mathbf{ʃ} &\stackrel{}{\dashv}& \mathbf{\flat} & & \mathbf{}& \\ && & \bot &\stackrel{}{}& \bot \\ \stackrel{}{} & & & \mathbf{\flat} &\stackrel{}{\dashv}& \mathbf{\sharp} & \simeq_0 loc_{\not\not} & & \\ && & \vee &/& \vee \\ && & \mathbf{\emptyset} &\dashv& \mathbf{\ast} && }


  • \emptyset and *\ast denote the (co-)monads constant on the initial object and terminal object, respectively;

  • =ΔΓ\flat = \Delta\circ \Gamma is the composite of taking global sections followed by forming constant ∞-stacks;

  • \bigcirc \dashv \Box means that \bigcirc is left adjoint to \Box (adjoint modality);

  • \array{\bigcirc \\ \vee \\ \Box} means that if XX\Box X \simeq X then also XX\bigcirc X \simeq X;

  • the “//” and “\\backslash” indicate where this also holds diagornally, i.e. \sharp \emptyset \simeq \emptyset and XX\stackrel{\rightsquigarrow}{\Im X} \simeq \Im X;

  • loc Aloc_A for AHA \in \mathbf{H} means localization at the terminal morphism A*A \to \ast;

  • B,C,loc A\bigcirc \simeq_{B,C,\cdots} loc_A means that \bigcirc is equivalent to such a localization on those objects that are already local with respect to A,B,A, B, \cdots;

  • 0loc ¬¬\sharp \simeq_{0} loc_{\neg \neg} finally means that \sharp is equivalent on 0-types to sheafification with respect to the double negation topology.

The \infty-(co-)monads here operate as follows

Id Id even / e bosonic/fermionic supergeometry bosonic R loc 0|1 rheonomic infinitesimal/reduced ¯ / (T J ) frames/jets differentialgeometry infinitesimalshape loc 𝔻 0|1 & étalé shape loc 𝔻 ʃ / ¯ flat/rational differentialcohomology discrete 0loc ¬¬ continuous basetopos nothing * () possibility/necessity, \array{ && & Id &\dashv& Id \\ && & \vee && \vee \\ \stackrel{even}{} && & \mathbf{\rightrightarrows} &\dashv& \mathbf{\rightsquigarrow} & / & \mathbf{\e} & \stackrel{bosonic/fermionic}{} \\ && & \bot &\stackrel{super \atop geometry}{}& \bot \\ \stackrel{bosonic}{} && & \rightsquigarrow &\dashv& \mathbf{\R} & \simeq loc_{\mathbb{R}^{0|1}} && \stackrel{rheonomic}{} \\ && & \vee &\stackrel{}{}& \vee \\ \stackrel{infinitesimal/reduced}{} &\mathbf{\overline{\Re}} & / & \mathbf{\Re} &\stackrel{}{\dashv}& \mathbf{\Im} & & (\mathbf{T}^\infty \dashv \mathbf{J}^\infty) & \stackrel{frames/jets}{} \\ && & \bot &\stackrel{differential \atop geometry}{}& \bot \\ \stackrel{infinitesimal \atop shape}{} && loc_{\mathbb{D}} \simeq_{\mathbb{R}^{0\vert 1}} & \mathbf{\Im} &\stackrel{}{\dashv}& \mathbf{\&} & && \text{étalé} \\ && & \vee && \vee \\ \stackrel{shape}{} && loc_{\mathbb{R}} \simeq_{\mathbb{D}}& \mathbf{ʃ} &\stackrel{}{\dashv}& \mathbf{\flat} & / & \mathbf{\overline{\flat}}& \stackrel{ flat/rational }{} \\ && & \bot &\stackrel{differential \atop cohomology}{}& \bot \\ \stackrel{discrete}{} & & & \mathbf{\flat} &\stackrel{}{\dashv}& \mathbf{\sharp} & \simeq_0 loc_{\not\not} & & \stackrel{continuous\; }{} \\ && & \vee &\stackrel{base \atop topos}{}& \vee \\ \stackrel{nothing}{} && & \mathbf{\emptyset} &\dashv& \mathbf{\ast} && (\mathbf{\lozenge} \dashv \mathbf{\Box}) & \stackrel{possibility/necessity}{} } \,,


  • /¯\bigcirc / \overline{\bigcirc} denotes a comonad and the homotopy cofiber of its counit η :Id\eta_\bigcirc \colon \bigcirc \longrightarrow Id;

  • (LR)\bigcirc \;\; (L \dashv R) denotes a comonad and the base change adjunction along its counit: (η ) *(η ) !(η ) *(η ) *(\eta_\bigcirc)^\ast(\eta_\bigcirc)_! \dashv (\eta_\bigcirc)^\ast(\eta_\bigcirc)_\ast.

For more details see at

as well as


The existence of a nondegenerate system of adjoint modalities as in theorem is a strong condition on an ∞-topos. I have called it solid differential cohesion in (Schreiber 13), following Lawvere’s terminology of cohesion for 1-toposes.

I am only aware of slight variants of super formal smooth ∞-groupoids that admit solid differential cohesion in a non-degenerate way. For instance also the super and formal version of complex analytic ∞-groupoids does.

Hence, if indeed a system of adjoint modalities as above more or less characterizes higher differential supergeometry (smooth or complex analytic), then it is of interest to turn this around and ask if standard constructions in higher differential supergeometry may be axiomatized exclusively in terms of universal constructions involving just these adjoint modalities.

In other words:


How much of higher differential geometry and of physics

follows formally

from abstract progressive duality of opposites?


(One says “this follows formally” to mean that a proof doesn’t require to get our hands dirty. Often this is meant derogatively, as if dirty work is more valuable. Here we are after the opposite: axioms for geometric theory that allow to prove the key facts “formally”.)


The claim of (Schreiber 13, 15) is that the answer is: “A fair bit does.”



Abstract differential geometry

As illustration, we survey two classical theorems that may be stated and proven from just the progression of dualities of theorem .

Stokes’s theorem

This is due to (Bunke-Nikolaus-Völkl 13).


For H\mathbf{H} a cohesive (∞,1)-topos with shape modality ʃʃ and flat modality \flat, then for every stable homotopy type AStab(H)A \in Stab(\mathbf{H}) the canonical hexagon diagram

ʃ¯A d ¯A θ A ʃ¯A A ʃ¯A ch A A ʃA, \array{ && \overline{ʃ} A && \stackrel{\mathbf{d}}{\longrightarrow} && \overline{\flat} A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \overline{ʃ} \flat A && && A && && ʃ \overline{\flat} A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_A}} \\ && \flat A && \longrightarrow && ʃ A } \,,

formed from the ʃʃ-unit and \flat-counit – the “differential cohomology hexagon” – is homotopy exact in that

  1. the two squares are homotopy pullback squares (“fracture squares”);

  2. the two diagonals are the homotopy fiber sequences of the Maurer-Cartan form θ A\theta_A and its dual;

  3. the bottom morphism is the canonical points-to-pieces transform;

  4. the top and bottom outer sequences are long homotopy fiber sequences.


Here ʃAʃ A is a spectrum. By the Brown representability theorem this represents a (stable) cohomology theory. In (Simons-Sullivan 07, Simons-Sullivan 08) it was suggested that (stable) differential cohomology should be characterized by exact hexagons as in theorem . From this perspective the objects in the hexagon are moduli stacks for the following structures:

connectionformsontrivialbundles deRhamdifferential curvatureforms curvature deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundle topol.class Cherncharacter geometricbundleswithflatconnection comparison/regulatormap shapeofbundle \array{ && {{connection\;forms}\atop{on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature}\atop{forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat}\atop{differential\;forms}} && && {{geometric\;bundles}\atop{with\;connection}} && && {{rationalized}\atop{bundle}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison/regulator\;map}{\longrightarrow} && {{shape}\atop{of\;bundle}} }


A =B n(/Γ) conn DK[𝒪d dRΩ 1d dRd dRΩ n] \begin{aligned} A & = \mathbf{B}^{n}(\mathbb{R}/\Gamma)_{conn} \\ & \coloneqq DK[\; \mathbb{Z}\hookrightarrow \mathcal{O} \stackrel{d_{dR}}{\to} \Omega^1 \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n \;] \end{aligned}

the Deligne complex (regarded as a smooth ∞-groupoid under the Dold-Kan correspondence) whose sheaf cohomology is ordinary differential cohomology H^ n+1(,)\hat H^{n+1}(-,\mathbb{Z}), then the image of the hexagon in theorem under π 0H(X,)\pi_0 \mathbf{H}(X,-) is

Ω n(X)/im(d dR) d dR Ω cl n+1(X) a H n(X,) H^ n+1(X,) H n+1(X,) H n(X,U(1)) H n+1(X,). \array{ && \Omega^{n}(X)/im(d_{dR}) && \stackrel{d_{dR}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && \hat H^{n+1}(X,\mathbb{Z}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && H^{n}(X,U(1)) && \underset{}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) } \,.

Consider a cohesive (∞,1)-topos H\mathbf{H} with shape modality of the form ʃloc ʃ \simeq loc_{\mathbb{R}} where \mathbb{R} admits two distinct points 0,1:*0,1 \colon \ast \to \mathbb{R}. Then there is a canonical morphism

0 1:[,¯A]ʃ¯A \int_0^1 \;\colon\; [\mathbb{R}, \overline{\flat} A] \longrightarrow \overline{ʃ} A

which is induced via the homotopy cofiber property of ¯A\overline{\flat} A from the counit naturality square of the flat modality on [(**(0,1)),][(\ast \coprod \ast \stackrel{(0, 1)}{\to} \mathbb{R} ), -], using that this square exhibits a null homotopy due to the \mathbb{R}-homotopy invariance of A\flat A.


In the context of the previous example then the abstractly defined map 0 1\int_0^1 of prop. is the standard operation of integration of differential forms.


In the general abstract situation of prop. , Stokes' theorem holds:

0 1d()| 1()| 0 \int_0^1 \circ \mathbf{d} \;\;\;\simeq \;\;\; (-)|_1 - (-)|_0

as an equivalence on ʃ¯Stab(H)\overline{ʃ} Stab(\mathbf{H}).

Noether’s theorem

The classical first variational Noether theorem says (see Vinogradov 84, theorem 11.2 for this sharp version):



  1. a field bundle EΣE \to \Sigma over a spacetime/worldvolume Σ\Sigma of dimension p+1p+1, with jet bundle J EJ^\infty E;

  2. a system of local Lagrangians L iΩ H p+1(J E i)\mathbf{L}_i \in \Omega^{p+1}_H(J^\infty E_i) for a sufficiently regular non-gauge field theory (BBH 00, 5.1), defined on an open cover {E i}\{E_i\} of EE and differing by trivial Lagrangians on double overlaps of charts;

then the Dickey bracket Lie algebra 𝔠𝔲𝔯(L)\mathfrak{cur}(\mathbf{L}) of conserved currents is a central Lie algebra extension of the Lie algebra 𝔰𝔶𝔪(L)\mathfrak{sym}(\mathbf{L}) of infinitesimal symmetries by the de Rham cohomology H dR p(E)H^{p}_{dR}(E), i.e. there is a short exact sequence of Lie algebras

0H dR p(E)𝔠𝔲𝔯(L)𝔰𝔶𝔪(L)0. 0 \to H^p_{dR}(E) \longrightarrow \mathfrak{cur}(\mathbf{L}) \longrightarrow \mathfrak{sym}(\mathbf{L}) \to 0 \,.

We discuss now how this may be recovered essentially formally in the presence of the system of adjunctions of theorem .

  1. For parameterized WZW models

  2. For general Lagrangian field theories.

For parameterized WZW models

The following is due to (Sati-Schreiber 15) and owes much to discussion with Domenico Fiorenza and with Igor Khavkine.

Consider first the special case that

  1. EΣ×XE \simeq \Sigma \times X – in this case the field theory is called a sigma-model with target space XX;

  2. the L i\mathbf{L}_i are pullbacks along J (Σ×X)XJ^\infty(\Sigma \times X) \to X of local potential forms of a closed (p+1)(p+1)-form ω\omega on XX – in this case one says that the L i\mathbf{L}_i are higher WZW terms.

Then the locally defined L i\mathbf{L}_i are to be promoted to a cocycle in Deligne cohomology, hence to a map in H\mathbf{H} of the form

X L B p+1(/) conn \array{ X \\ \downarrow^{\mathbf{L}} \\ \mathbf{B}^{p+1}(\mathbb{R}/\mathbb{Z})_{conn} }

(where the codomain is the Deligne complex from example ).

By inspection of the local data one finds that from this perspective a symmetry of L\mathbf{L} is a diagram in H\mathbf{H} of the form

X X L L B p+1(/) conn \array{ X && \stackrel{\simeq}{\longrightarrow} && X \\ & _{\mathllap{\mathbf{L}}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}(\mathbb{R}/\mathbb{Z})_{conn} }

such that there exists a homotopy filling the diagram as shown, while the corresponding conserved Noether current is the datum of that homotopy.


(Fiorenza-Rogers-Schreiber 13)

In any cohesive (∞,1)-topos the above situation leads to a homotopy fiber sequence of ∞-groups that is schematically of the form

{ X L L B p+1(/) conn}{X X L L B p+1(/) conn}{X X} \left\{ \array{ && X \\ & _{\mathllap{\mathbf{L}}}{\swarrow \atop \searrow} &\Leftarrow& {\searrow \atop \swarrow}_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}(\mathbb{R}/\mathbb{Z})_{conn} } \right\} \longrightarrow \left\{ \array{ X && \stackrel{\simeq}{\longrightarrow} && X \\ & _{\mathllap{\mathbf{L}}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}(\mathbb{R}/\mathbb{Z})_{conn} } \right\} \longrightarrow \left\{ \array{ X && \stackrel{\simeq}{\longrightarrow} && X } \right\}

This ∞-group extension yields, infinitesimally, an extension of L-∞ algebras. The 0-homology truncation of that to an extension of Lie algebras is the extension of the classical Noether theorem, .

Hence the abstract theorem is the Noether theorem for point symmetries of higher WZW models, refined from infinitesimal to finite symmetries and reflecting also all higher conserved currents.

In particular, applied to the higher WZW models that are Green-Schwarz sigma models for super p-branes propagating on super-spacetimes, then this is yields the BPS charge extensions of superisometries of superspacetime that are known as the M-theory super Lie algebra, etc.


For general Lagrangian field theories

The following is due to (Khavkine-Schreiber 15).

Consider the restriction of the jet comonad J Σ J^\infty_\Sigma from theorem to the full subcategory Diff /ΣH /Σ=SuperFormalSmoothGrpd /ΣDiff_{/\Sigma} \hookrightarrow \mathbf{H}_{/\Sigma} = SuperFormalSmooth\infty Grpd_{/\Sigma} of diffeological bundles over Σ\Sigma.


(Marvan 86)

The Eilenberg-Moore category of the jet comonad J Σ J^\infty_\Sigma is Vinogradov’s category of partial differential equations with variables in Σ\Sigma:

J Σ AlgPDE Σ. J^\infty_\Sigma Alg \simeq PDE_\Sigma \,.

In particulat the coKleisli category of J J^\infty is the category of non-linear differential operators between sections of bundles over Σ\Sigma.

Notice that by comonadic descent we have

J Σ Alg(H)H /Σ. J^\infty_\Sigma Alg(\mathbf{H}) \simeq \mathbf{H}_{/\Im \Sigma} \,.

(Khavkine-Schreiber 15)

Replacing the de Rham complex by the Euler-Lagrange complex in example yields a coefficient object

B H p+1(/) connH /Σ \mathbf{B}^{p+1}_{H} (\mathbb{R}/\mathbb{Z})_{conn} \in \mathbf{H}_{/\Im \Sigma}

such that

  1. morphisms L:EB H p+1(/) conn\mathbf{L} \colon E \longrightarrow \mathbf{B}^{p+1}_{H} (\mathbb{R}/\mathbb{Z})_{conn} are properly globalized local Lagrangians for Lagrangian field theories;

  2. their curvature E¯B H p+1(/) connE \longrightarrow \overline{\flat}\mathbf{B}^{p+1}_{H} (\mathbb{R}/\mathbb{Z})_{conn} is the corresponding Euler-Lagrange equations of motion;

  3. the corresponding \infty-group extension

    { E L L B H p+1(/) conn}{E E L L B H p+1(/) conn}{E E} \left\{ \array{ && E \\ & _{\mathllap{\mathbf{L}}}{\swarrow \atop \searrow} &\Leftarrow& {\searrow \atop \swarrow}_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}_H(\mathbb{R}/\mathbb{Z})_{conn} } \right\} \longrightarrow \left\{ \array{ E && \stackrel{\simeq}{\longrightarrow} && E \\ & _{\mathllap{\mathbf{L}}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^{p+1}_H(\mathbb{R}/\mathbb{Z})_{conn} } \right\} \longrightarrow \left\{ \array{ E && \stackrel{\simeq}{\longrightarrow} && E } \right\}

    is the globalized stacky lift of the classical Noether theorem, .

Notice that since now EE is allowed to be a stack, it may contain gauge symmetries. For more on this see also the exposition at Higher field bundles for gauge fields.


Last revised on September 3, 2015 at 17:45:51. See the history of this page for a list of all contributions to it.