These are notes accompanying a talk that I once gave:
Duality of Monads in Geometric Homotopy Theory,
talk at at the workshop:
Duality in contemporary mathematics ? philosophical aspects
3.-5. September 2015
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
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).
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:
A dual object $V^\ast$ in a monoidal category $(\mathcal{C},\otimes)$ is the adjoint of $V$ in the delooping 2-category of $\mathcal{C}$.
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
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.
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.
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After so much general abstraction, let’s consider a concrete particular.
Write
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
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 $\mathbf{H}$ there is a progression of idempotent ∞-(co-)monads of the following form
where
$\emptyset$ and $\ast$ denote the (co-)monads constant on the initial object and terminal object, respectively;
$\flat = \Delta\circ \Gamma$ 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 $\Box X \simeq X$ then also $\bigcirc X \simeq X$;
the “$/$” and “$\backslash$” indicate where this also holds diagornally, i.e. $\sharp \emptyset \simeq \emptyset$ and $\stackrel{\rightsquigarrow}{\Im X} \simeq \Im X$;
$loc_A$ for $A \in \mathbf{H}$ means localization at the terminal morphism $A \to \ast$;
$\bigcirc \simeq_{B,C,\cdots} loc_A$ means that $\bigcirc$ is equivalent to such a localization on those objects that are already local with respect to $A, B, \cdots$;
$\sharp \simeq_{0} loc_{\neg \neg}$ 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
where
$\bigcirc / \overline{\bigcirc}$ denotes a comonad and the homotopy cofiber of its counit $\eta_\bigcirc \colon \bigcirc \longrightarrow Id$;
$\bigcirc \;\; (L \dashv R)$ denotes a comonad and the base change adjunction along its counit: $(\eta_\bigcirc)^\ast(\eta_\bigcirc)_! \dashv (\eta_\bigcirc)^\ast(\eta_\bigcirc)_\ast$.
For more details see at
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
reduction modality$\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality
shape modality$\dashv$ flat modality $\dashv$ sharp modality
as well as
possibility$\dashv$ necessity
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:
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How much of higher differential geometry and of physics
follows formally
from abstract progressive duality of opposites?
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(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”.)
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The claim of (Schreiber 13, 15) is that the answer is: “A fair bit does.”
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As illustration, we survey two classical theorems that may be stated and proven from just the progression of dualities of theorem .
This is due to (Bunke-Nikolaus-Völkl 13).
For $\mathbf{H}$ a cohesive (∞,1)-topos with shape modality $ʃ$ and flat modality $\flat$, then for every stable homotopy type $A \in Stab(\mathbf{H})$ the canonical hexagon diagram
formed from the $ʃ$-unit and $\flat$-counit – the “differential cohomology hexagon” – is homotopy exact in that
the two squares are homotopy pullback squares (“fracture squares”);
the two diagonals are the homotopy fiber sequences of the Maurer-Cartan form $\theta_A$ and its dual;
the bottom morphism is the canonical points-to-pieces transform;
the top and bottom outer sequences are long homotopy fiber sequences.
Here $ʃ A$ 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:
For
the Deligne complex (regarded as a smooth ∞-groupoid under the Dold-Kan correspondence) whose sheaf cohomology is ordinary differential cohomology $\hat H^{n+1}(-,\mathbb{Z})$, then the image of the hexagon in theorem under $\pi_0 \mathbf{H}(X,-)$ is
Consider a cohesive (∞,1)-topos $\mathbf{H}$ with shape modality of the form $ʃ \simeq loc_{\mathbb{R}}$ where $\mathbb{R}$ admits two distinct points $0,1 \colon \ast \to \mathbb{R}$. Then there is a canonical morphism
which is induced via the homotopy cofiber property of $\overline{\flat} A$ from the counit naturality square of the flat modality on $[(\ast \coprod \ast \stackrel{(0, 1)}{\to} \mathbb{R} ), -]$, using that this square exhibits a null homotopy due to the $\mathbb{R}$-homotopy invariance of $\flat A$.
In the context of the previous example then the abstractly defined map $\int_0^1$ of prop. is the standard operation of integration of differential forms.
In the general abstract situation of prop. , Stokes' theorem holds:
as an equivalence on $\overline{ʃ} Stab(\mathbf{H})$.
The classical first variational Noether theorem says (see Vinogradov 84, theorem 11.2 for this sharp version):
Given
a field bundle $E \to \Sigma$ over a spacetime/worldvolume $\Sigma$ of dimension $p+1$, with jet bundle $J^\infty E$;
a system of local Lagrangians $\mathbf{L}_i \in \Omega^{p+1}_H(J^\infty E_i)$ for a sufficiently regular non-gauge field theory (BBH 00, 5.1), defined on an open cover $\{E_i\}$ of $E$ and differing by trivial Lagrangians on double overlaps of charts;
then the Dickey bracket Lie algebra $\mathfrak{cur}(\mathbf{L})$ of conserved currents is a central Lie algebra extension of the Lie algebra $\mathfrak{sym}(\mathbf{L})$ of infinitesimal symmetries by the de Rham cohomology $H^{p}_{dR}(E)$, i.e. there is a short exact sequence of Lie algebras
We discuss now how this may be recovered essentially formally in the presence of the system of adjunctions of theorem .
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
$E \simeq \Sigma \times X$ – in this case the field theory is called a sigma-model with target space $X$;
the $\mathbf{L}_i$ are pullbacks along $J^\infty(\Sigma \times X) \to X$ of local potential forms of a closed $(p+1)$-form $\omega$ on $X$ – in this case one says that the $\mathbf{L}_i$ are higher WZW terms.
Then the locally defined $\mathbf{L}_i$ are to be promoted to a cocycle in Deligne cohomology, hence to a map in $\mathbf{H}$ of the form
(where the codomain is the Deligne complex from example ).
By inspection of the local data one finds that from this perspective a symmetry of $\mathbf{L}$ is a diagram in $\mathbf{H}$ of the form
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
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.
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For general Lagrangian field theories
The following is due to (Khavkine-Schreiber 15).
Consider the restriction of the jet comonad $J^\infty_\Sigma$ from theorem to the full subcategory $Diff_{/\Sigma} \hookrightarrow \mathbf{H}_{/\Sigma} = SuperFormalSmooth\infty Grpd_{/\Sigma}$ of diffeological bundles over $\Sigma$.
The Eilenberg-Moore category of the jet comonad $J^\infty_\Sigma$ is Vinogradov’s category of partial differential equations with variables in $\Sigma$:
In particulat the coKleisli category of $J^\infty$ is the category of non-linear differential operators between sections of bundles over $\Sigma$.
Notice that by comonadic descent we have
Replacing the de Rham complex by the Euler-Lagrange complex in example yields a coefficient object
such that
morphisms $\mathbf{L} \colon E \longrightarrow \mathbf{B}^{p+1}_{H} (\mathbb{R}/\mathbb{Z})_{conn}$ are properly globalized local Lagrangians for Lagrangian field theories;
their curvature $E \longrightarrow \overline{\flat}\mathbf{B}^{p+1}_{H} (\mathbb{R}/\mathbb{Z})_{conn}$ is the corresponding Euler-Lagrange equations of motion;
the corresponding $\infty$-group extension
is the globalized stacky lift of the classical Noether theorem, .
Notice that since now $E$ 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.
Dana Scott, Advice on Modal Logic, in Karel Lambert (ed.) Philosophical problems in Logic – Some recent developments, Reidel 1970
Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111?21. Boston: D. Reidel Publishing Co. (1982)
Alexandre Vinogradov, The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, Journal of Mathematical Analysis and Applications 100, Issue 1, 30 April 1984, Pages 41-129 (publisher)
Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)
William Lawvere, Some Thoughts on the Future of Category Theory in A. Carboni, M. Pedicchio, G. Rosolini, Category Theory , Proceedings of the International Conference held in Como, Lecture Notes in Mathematics 1488, Springer (1991)
Glenn Barnich, Friedemann Brandt, Marc Henneaux, Local BRST cohomology in gauge theories, Physics Reports 338 (2000), 439-569 (hep-th/0002245)
James Simons, Dennis Sullivan, Axiomatic Characterization of Ordinary Differential Cohomology, Journal of Topology 1.1 (2008): 45-56. (arXiv:math/0701077)
James Simons, Dennis Sullivan, Structured vector bundles define differential K-theory (arXiv:0810.4935)
Urs Schreiber, differential cohomology in a cohesive topos (pdf)
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)
Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)
Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (pdf)
Igor Khavkine, Urs Schreiber, Lie n-algebras of higher Noether currents, in preparation
Last revised on September 3, 2015 at 21:45:51. See the history of this page for a list of all contributions to it.