In (Lawvere 67, Lawvere 86, Lawvere 97) there was proposed a notion of toposes of laws of motion meant to formalize classical continuum mechanics in synthetic differential geometry/in topos theory. This page here gives an introductory survey of the refinements possible when lifting this from topos theory to higher topos theory and of the applications of the resulting formalism to quantum field theory.
This text originates in a talk at the Eighth Scottish Category Theory Seminar. Accordingly, these notes amplify aspects of category theory and topos theory and generally stick to a Lawverian perspective.
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Backround
Definition
Presentation over a site
Models
Urban legend has it that there was a time when only three people understood Einstein‘s theory of classical gravity – “general relativity”.
Whether true or not, one of the three was David Hilbert. He made sure every beginning student today can understand general relativity, he did so by giving it a clear and precise (= rigorous) formalization in mathematics:
classical Einstein gravity is simply the study of the critical points of the integral of the scalar curvature density functional on the moduli space of pseudo-Riemannian metrics on spacetime.
By trusting that a fundamental theory of physics should have a fundamental formulation in mathematics, Hilbert was able to essentially scoop Einstein (see here for the history), that’s why this functional is now called the Einstein-Hilbert action functional.
Hilbert had promoted this general idea before as part of the famous eponymous Hilbert's problems in mathematics, from 1900. Here Hilbert's 6th problem asks mathematicians generally to find axioms for theories in physics.
Since then a list of such axiomatizations has been found, for instance
Two aspects of this list are noteworthy: on the one hand, it contains crown jewels of mathematics, on the other the items appear unrelated and piecemeal.
As a student, William Lawvere was exposed to the proposal to axiomatize thermodynamics as what was called “rational thermodynamics”. He realized that a fundamental foundation of such continuum physics first of all requires a good foundation of differential geometry itself. Looking over his life publication record (see here) one sees that he pursued the following grand plan.
Plan.
lay the foundations of mathematics in topos theory (“ETCS”)
lay the foundations of geometry in topos theory (synthetic differential geometry, cohesion)
lay the foundations of classical continuum physics in synthetic differential geometry (toposes of laws of motion).
Lawvere became famous for his groundbreaking contributions to the first two items (categorical logic, elementary topos theory, algebraic theories, SDG). For some reason the motivation of all this by the third item is not as widely recognized, even thought Lawvere continuously emphasized this third point, see the list of quotations here.
Grandiose as this plan is, we have to note that in the above form it falls short in each item, by modern standards:
modern mathematics is naturally founded not in topos theory/type theory, but in higher topos theory/homotopy type theory.
modern geometry is not just about “variable sets” (sheaves) but is higher geometry about “variable homotopy types”, “geometric homotopy types”, “higher stacks”;
modern physics goes beyond classical continuum physics; at high energy (small distance) classical physics is refined by quantum physics, specifically by quantum field theory.
Therefore what is needed is a foundation of high energy physics in higher differential geometry formulated in higher topos theory.
In the following we illustrate three aspects of such a refined theory, following (dcct, sythQFT). We close by indicating how this theory serves to solve subtle open problems in modern quantum field theory and string theory.
One basic point emphasized in (Lawvere 67) is that central to the formulation of physics is the existence of mapping spaces which satisfy the exponential law.
(notice: mapping space … space of trajectories … path integral)
Indeed, generically a physical system is specified by
a space $\Sigma$ called spacetime or worldvolume;
a space $X$ called target space or field bundle or moduli space of fields.
such that a trajectory or history of field configurations is a map
For instance for $\Sigma = \mathbb{R}$ the abstract worldline and $X$ spacetime, then $\Sigma \to X$ may be taken to be the trajectory of a particle in spacetime.
(Notice that after second quantization the roles change. First the domain $\Sigma$ is worldvolume and the codomain $X$ is spacetime (“sigma-model)”, then after second quantization spacetime $X$ becomes the domain (hence becomes $\Sigma$ in the above).)
Hence the mapping space $[\Sigma,X]$ is the space of all trajectories (the path space, famous as the domain of the infamous path integral).
Lawvere observed that this not only needs to exist as a decent “space”, it also needs to satisfy the axiom of an cartesian internal hom. Because if we consider a split
into time and space, then we want that spacetime field configurations
are equivalently trajectories of fields on space
and also equivalently a collection of field trajectories for each point of space
This led Lawvere to recognize that physics (prequantum physics, to be precise) is to be formulated in a cartesian closed category, such as a topos.
The category $SmthMfd$ of smooth manifolds is too small to accomplish this. But the category of sheaves $Sh(SmthMdf)$ on the site of smooth manifolds is the canonical improvement. Objects in here include smooth manifolds, also diffeological spaces and general smooth spaces.
Better still, there is the category of sheaves $Sh(FSmthMfd)$ on the site of formal smooth manifolds – known as the Cahiers topos . This also contains infinitesimal objects and indeed interprets the axioms of synthetic differential geometry.
But actually in modern physics one needs a bit more than this. Physics is fundamentally governed by gauge equivalence, which means that there is no sense in asking if field configurations are equal, we must ask if they are equivalent.
A fundamental example of this is Einstein‘s notion of general covariance. This says that if
is a region in spacetime, and $\phi \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma$ is a diffeomorphism acting on spacetime, then the “translated region”
is “the same”, for all physical purposes. Stated this way in ordinary topos theory this is confusing, and historically it was confusing: this confusion is essentially what is known as the “hole paradox”.
This apparent paradox is resolved in higher topos theory. Here a space $S$ has internal symmetries, it is a groupoid, a homotopy type. This means that it is not sensible to ask if two maps into it are equal, but between any two maps there is a space of equivalences between them.
For general covariance this means by the above that spacetime is not actually the spacetime manifold $\Sigma$, but is the action groupoid/quotient stack
of $\Sigma$ by its diffeomorphism group (regarded as a diffeological group). By the very definition of “stack”, this $\Sigma//Diff(\Sigma)$ is the thing which is such that maps $U \longrightarrow \Sigma//Diff(\Sigma)$ into it behave just as they should as demanded by general covariance.
This is the formalization of “general covariance” for regions inside spacetime. The other thing now is general covariance for fields on spacetime. It turns out that the formalism automatically handles these now:
For notice that the above means now that we also need to consider the mapping spaces refined to higher topos theory. A fundamental fact is that for $G \in Grp(\mathbf{H})$ group object then the higher slice topos over its delooping is equivalently the collection of G-actions
Under this equivalence a higher action is identified with the universal associated bundle which it induces
Lawvere also introduced categorical logic and understood dependent product $\prod$ and dependent sum $\sum$ as base change. In higher topos theory this becomes representation theory as follows:
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type on $\mathbf{B}G$ | $G$-∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | restricted representation |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
Using this we have the mapping space of covariant fields formed in the context of $Diff(\Sigma)$-covariance/equivariance
Theorem. Down in the absolute context, under dependent sum, this is
Here $[\Sigma, X]$ is the space of fields on spacetime as one would find it in ordinary topos theory, and $[\Sigma,X]//Diff(\Sigma)$ is its homotopy quotient by the action of the diffeomorphism group by pullback of fields. This is precisely the group of gauge equivalences on fields in general relativity.
So we obtain a formalization of the famous insight of Einstein, derived by lifting Lawvere‘s argument about mapping spaces in physics from topos theory to higher topos theory.
In a slogan we may conclude, in view of the above table, that in the formalization of physics in higher topos theory we have:
in mapping spaces.
In (Lawvere 97) it was observed that equations of motion in physics can (almost, see below) be formalized in synthetic differential geometry as follows.
Let $\mathbf{H}$ be an ambient synthetic differential topos (such as the Cahiers topos of smooth spaces and formal smooth manifolds).
The canonical line object $\mathbb{A}^1 = \mathbb{R}$ of this models the continuum line, the abstract worldline. Let
be the inclusion of the first order infinitesimal neighbourhood of the origin of $\mathbb{R}$ – in the internal logic this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the spectrum of the ring of dual numbers over $\mathbb{R}$.
Then consider $X \in \mathbf{H}$ any object which we are going to think of as a configuration space of a physical system. For instance if the system is a particle propagating on a spacetime, then $X$ is that spacetime. Or $X$ may be the phase space of the system.
Accordingly the mapping space $[\mathbb{R}, X] \in \mathbf{H}$ is the smooth path space of $X$. This is the space of potential trajectories of the physical system.
If $X$ is thought of as phase space, then every point in there determines a unique trajectory starting at that point. This means that time evolution is then an action of $\mathbb{R}$ on $X$. As $X$ here might be any space, we have the collection
of all $\mathbb{R}$-actions on objects in $\mathbf{H}$. This is again a topos, and hence this is a first version of what one might call a topos of laws of motion.
On the other hand, if we think of $X$ as configuration space, then it is (in the simplest but common case of physical systems) a tangent vector in $X$ that determines a trajectory, hence a point in $[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth path space to the tangent bundle, which sends each path to its tangent vector/derivative at $0 \in \mathbb{R}$. A section of this map is hence an assignment that sends each tangent vector to a trajectory which starts out with this tangent. Specifying such a section is hence part of what it means to have equations of motion in physics. Accordingly in Toposes of laws of motion Lawvere called the collection of such data a Galilean topos of laws of motion.
Of course this is not quite yet what is actually used and needed in physics. On p. 9 of (Lawvere 97) this problem is briefly mentioned:
But what about actual dynamical systems in the spirit of Galileo, for example, second-order ODE’s? (Of course, the symplectic or Hamiltonian systems that are also much studied do address this question of states of Becoming versus locations of Being, but in a special way which it may not be possible to construe as a topos;
We observe now (following Classical field theory via Cohesive homotopy types) that it does exist as a “higher topos”.
First notice that in physics a phase space is not any space $X$, but is a space $X$ equipped with a closed differential 2-form, a “presymplectic form”. Speaking in terms of mapping spaces as above let $\mathbf{\Omega}^2_{cl}$ be the moduli space of closed 2-forms, then this means that a phase space is really a map
In fact this is still not quite the accurate statement. Rather a phase space is a “prequantization” of such data. This means the following.
The circle group $S^1$ naturally acts on the space of differential 1-forms $\mathbf{\Omega}^1$ by
where $\mathbf{d}$ is the de Rham differential. The resulting quotient stack we write
The de Rham differential $\mathbf{d} \;\colon\; \mathbf{\Omega}^1 \longrightarrow \mathbf{\Omega}^2_{cl}$ descends to this quotient to yield a map
A prequantization of a presymplectic form is a lift $\nabla$ through this map
Now suppose $\omega$ is actually a symplectic form. Then:
Theorem. Concrete actions of $\mathbb{R}$ on $(X, \nabla) \in \mathbf{H}_{/\mathbf{B}S^1_{conn}}$ are equivalent to “Hamiltonians” $H \in [X,\mathbb{R}]$, where under the equivalence an element $t \in \mathbb{R}$ is sent to the slice automorphism
where $\exp(t \{H,-\})$ denotes the flow of Hamilton's equations of motion induced by $H$ and where $S_t = \int_0^t L \, d t$ is the Hamilton-Jacobi action given by the integral of the Lagrangian $L$ (the Legendre transform of $H$).
This statement subsumes the core ingredients of classical mechanics. See at prequantized Lagrangian correspondence for details.
In conclusion we find that $\mathbb{R}$-actions in the higher slice topos $\mathbf{H}_{/\mathbf{B}S^1_{conn}}$ over the moduli stack of circle group-principal connections are equivalent to actual laws of motion in classical mechanics
More precisely, this applies to laws of motion in mechanics. One obtains more generally the Hamilton-de Donder-Weyl equations of motion? of $n$-dimensional local classical field theory by replacing $\mathbf{B} S^1_{conn}$ here with $\mathbf{B}^n S^1_{conn}$ (Schreiber 13, Schreiber 13b).
In physics and especially in continuum mechanics and thermodynamics, a physical quantity associated with a physical system extended in space is called
intensive if it is a function on (the physical system extended in) space;
extensive if it is a density or linear distribution on (the physical system extended in) space.
For instance for a solid body its temperature is intensive, but its mass is extensive: there is a temperature assigned to every point of the body (in the idealization of classical continuum mechanics anyway) but a mass is assigned only to every little “extended” piece of the body, not to a single point.
This terminology in physics apparently originates with Richard Tolman in 1917.
In (Lawvere 86) it is amplified that this duality is generally a fundamental one also in mathematics: given a topos $\mathbf{H}$ with a commutative ring object $R \in CRing(\mathbf{H})$, then
the space of intensive quantities on an object $X \in \mathbf{H}$ is the mapping space $[X,R]_{\mathbf{H}} \in CRing(\mathbf{H})$ formed in $\mathbf{H}$;
the space of extensive quantities on $X$ is the $R$-linear dual, namely the mapping space $[X,R]^\ast \coloneqq [[X,R], R]_{R Mod}$ formed in $R$-modules in $\mathbf{H}$.
the integration map is the canonical evaluation pairing
Viewed this way, this naturally generalizes to the case where $\mathbf{H}$ is in fact an (∞,1)-topos and $R \in CRing(\mathbf{H})$ an E-∞ ring. In this case $[X,R]$ is called the $R$-cohomology spectrum of $X$ and $[X,R]^\ast$ is the corresponding generalized homology spectrum. In this form intensive and extensive properties appear in physics in the context of motivic quantization of local prequantum field theory.
More generally, for $\chi$ an $R$-(∞,1)-line bundle over $X$ then the corresponding extensive object is the $\chi$-twisted Thom spectrum $R_{\bullet + \chi}(X)$ and the intensive object is the $\chi$-twisted cohomology spectrum $R^{\bullet + \chi}(X) = [R_{\bullet+ \chi}(X),R]_{R Mod}$. See at motivic quantization for how this appears in physics.
In particular ordinary quantum mechanics is recovered by settin $R =$ KU, the complex K-theory spectrum (Nuiten 13).
The monoidal (infinity,1)-category $KU Mod$ is the refined ambient home for $Hilb = \mathbb{C} Mod$ (used for finite quantum mechanics in terms of dagger-compact categories).
On first sight, formalization of physics in (higher) topos theory might seem like a fruitless exercise. But on the contrary, it is hardly possible to understand the deep structure of quantum field theory without such (geometric) homotopy theory.
We close here by briefly indicating one example problem of recent interest, concerned with the fine-structure of quantum anomaly cancellation in 2d QFT.
One reason why the need for geometric homotopy theory in QFT is not mentioned in the bulk of the QFT literature is that traditionally the bulk of the discussion of quantum field theory is in perturbation theory (perturbative both in Planck's constant and in terms of the coupling constant). This perspective tends to hide the rich nature of what QFT fundamentally is, as non-perturbative quantum field theory.
Phenomena that arise from the global structure of a moduli of field configurations in physics are alien to perturbation theory, and hence are anomalies. Such an anomalous action functional is something that ought to be a function $[\Sigma,X] \longrightarrow S^1$ on configuration space, but possibly comes out just as a section of a bundle over configuration space (examples include gravitational anomalies, the conformal anomaly, the Freed-Witten-Kapustin anomaly, the Green-Schwarz anomaly, the Diaconescu-Moore-Witten anomaly.)
The anomaly bundles on $[\Sigma,X]$ typically arise as the transgression of higher bundles on the moduli space of fields $X$ itself (see at twisted smooth cohomology in string theory for more on this). So these are phenomena which are intrinsically phenomena in geometric homotopy theory/(infinity,1)-topos theory.
We consider now specifically a general aspect of what is called the Freed-Witten-Kapustin anomaly. It is usually read out as follows:
Just as above we saw that the basic example of a quantum field theory on $\Sigma = \mathbb{R}$ descibes the dynamics of a particle, so the basic example of a quantum field theory on a 2-dimensional $\Sigma$ describes the dynamics of a string. This naturally feels forces excerted in particular by two background gauge fields called the B-field and the RR-field.
The global nature of these fields is more subtle than for, say, the electromagnetic field, since they are higher gauge fields. To a first approximation one finds that the RR-field is a cocycle in twisted K-theory, where the twist is the B-field which in turn is a cocycle in ordinary cohomology.
But this is not the full story, in the full story these fields are cocycles in differential cohomology. The RR-field is a cocycle in twisted differential K-theory twisted by the B-field which is a cocycle in ordinary differential cohomology.
In (DFM 09) is indicated the rich subtleties in the quantum anomaly consistency conditions on these background fields, assuming that twisted differential K-theory exists with some properties, but without having constructed it.
The infamous “landscape of string theory vacua” is essentially the moduli space of certain 2d field theories satisfying consistency conditions like this.
The central problem in showing the existence of a differential cohomology theory is to show that this cohomology theory sits inside a double square diagram called the “differential cohomology diagram”.
Now a miracle happens. After developing synthetic differential geometry, Lawvere explored a more fundamental axiomatization of differential geometry, which he called cohesion (Lawvere 94, Lawvere 07 ) (Earlier: “being and becoming” (Lawvere 91)). Slightly paraphrased, cohesion means that the ambient type theory is equipped with an adjoint triple of (co-)modalities
called: shape modality $\dashv$ flat modality $\dashv$ sharp modality.
This has an immediate extension to homotopy type theory (cohesive homotopy type theory). But there it has more dramatic consequences. In (Bunke-Nikolaus-Völkl 13) it was observed that on stable homotopy types $A$ cohesion implies that the canonical diagram formed from modality units and counits
is guaranteed to consist of homotopy pullback squares, by the nature of adjoint triples of modalities (see at tangent cohesion for more on this).
In (Bunke-Nikolaus-Völkl 13) it is shown that this is universally the “differential cohomology diagram” which hence exhibits every stable homotopy type $A$ in cohesive homotopy type theory as a differential cohomology theory, hence as the moduli stack for abelian higher gauge fields in quantum field theory. Hence cohesive homotopy type theory is a universal ambient context for differential cohomology and hence for higher gauge fields appearing in quantum field theory – whence the title “differential cohomology in a cohesive topos”.
Using this and the twisted cohomology available in tangent cohesion (Bunke-Nikolaus) show the existence of twisted differential K-theory, the way it needs to exist for 2d QFT to be consistent.
William Lawvere, Categorical dynamics, 1967 Chicago lectures (pdf)
William Lawvere, Introduction to Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986
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)
William LawvereCohesive Toposes and Cantor's "lauter Einsen" Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. (pdf)
William Lawvere, Axiomatic cohesion, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)
Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)
Urs Schreiber, Classical field theory via Cohesive homotopy types (arXiv:1311.1172)
Urs Schreiber, Quantization via Linear homotopy types (arXiv:1402.7041)
Joost Nuiten, Cohomological quantization of local boundary prequantum field theory, 2013
Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)
Ulrich Bunke, Thomas Nikolaus, Twisted differential cohomology, in preparation
Last revised on April 22, 2020 at 22:39:19. See the history of this page for a list of all contributions to it.