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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Higher toposes of laws of motion} \begin{quote}% In (\hyperlink{Lawvere67}{Lawvere 67}, \hyperlink{Lawvere86}{Lawvere 86}, \hyperlink{Lawvere79}{Lawvere 97}) there was proposed a notion of \emph{[[toposes of laws of motion]]} meant to formalize [[classical mechanics|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 [[(infinity,1)-topos theory|higher topos theory]] and of the applications of the resulting formalism to [[quantum field theory]]. This text originates in a \href{http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos#TalkAt8thScottishCategorySeminar}{talk} at the \emph{\href{http://homepages.inf.ed.ac.uk/als/SCT/sct131129.html}{Eighth Scottish Category Theory Seminar}}. Accordingly, these notes amplify aspects of [[category theory]] and [[topos theory]] and generally stick to a [[William Lawvere|Lawverian]] perspective. \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - contents]] \hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes} [[!include cohesive infinity-toposes - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{introduction_and_motivation}{Introduction and motivation}\dotfill \pageref*{introduction_and_motivation} \linebreak \noindent\hyperlink{HigherMappingSpaces}{\textbf{I)} Mapping spaces in gauge theory and General covariance}\dotfill \pageref*{HigherMappingSpaces} \linebreak \noindent\hyperlink{ii_toposes_of_laws_of_motion_and_hamiltonjacobilagrange_mechanics}{\textbf{II)} Toposes of laws of motion and Hamilton-Jacobi-Lagrange mechanics}\dotfill \pageref*{ii_toposes_of_laws_of_motion_and_hamiltonjacobilagrange_mechanics} \linebreak \noindent\hyperlink{iii_extensiveintensive_duality_and_cohomological_quantization}{\textbf{III)} Extensive/intensive duality and Cohomological quantization}\dotfill \pageref*{iii_extensiveintensive_duality_and_cohomological_quantization} \linebreak \noindent\hyperlink{application_cohesion_and_superstring_anomaly_cancellation}{Application: Cohesion and Superstring anomaly cancellation}\dotfill \pageref*{application_cohesion_and_superstring_anomaly_cancellation} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{introduction_and_motivation}{}\subsection*{{Introduction and motivation}}\label{introduction_and_motivation} Urban legend \href{http://en.wikipedia.org/wiki/History_of_general_relativity#Sir_Arthur_Eddington}{has it} that there was a time when only three people understood [[Einstein]]`s [[theory (physics)|theory]] of [[classical field theory|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]] [[action functional|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 \href{Einstein-Hilbert+action#History}{here} for the history), that's why this functional is now called the \emph{[[Einstein-Hilbert action functional]]}. Hilbert had promoted this general idea before as part of the famous eponymous \emph{[[Hilbert's problems]] in mathematics, from 1900. Here [[Hilbert's 6th problem]] asks [[mathematics|mathematicians]] generally to find [[axioms]] for [[theory (physics)|theories]] in [[physics]].} Since then a list of such axiomatizations has been found, for instance \begin{tabular}{l|l} [[physics]]&[[mathematics]]\\ \hline [[mechanics]]&[[symplectic geometry]]\\ [[gravity]]&[[Riemannian geometry]]\\ [[gauge theory]]&[[Chern-Weil theory]]\\ [[quantum mechanics]]&[[operator algebra]]\\ [[TFT&topological]] [[local quantum field theory]]\\ $\vdots$&$\vdots$\\ \end{tabular} 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 [[William Lawvere|here]]) one sees that he pursued the following grand plan. \textbf{Plan.} \begin{enumerate}% \item lay the [[foundations]] of [[mathematics]] in [[topos theory]] (``[[ETCS]]'') \item lay the foundations of [[geometry]] in [[topos theory]] ([[synthetic differential geometry]], [[cohesion]]) \item lay the foundations of [[classical mechanics|classical]] [[continuum physics]] in [[synthetic differential geometry]] ([[toposes of laws of motion]]). \end{enumerate} Lawvere became famous for his groundbreaking contributions to the first two items ([[categorical logic]], [[topos theory|elementary topos theory]], [[algebraic theories]], [[synthetic differential geometry|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 \href{William%20Lawvere#MotivationFromFoundationsOfPhysics}{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: \begin{enumerate}% \item modern [[mathematics]] is naturally founded not in [[topos theory]]/[[type theory]], but in [[(infinity,1)-topos theory|higher topos theory]]/[[homotopy type theory]]. \item modern [[geometry]] is not just about ``variable sets'' ([[sheaves]]) but is [[higher geometry]] about ``variable [[homotopy types]]'', ``[[geometric homotopy types]]'', ``[[infinity-stack|higher stacks]]''; \item modern [[physics]] goes beyond [[classical mechanics|classical]] [[continuum physics]]; at high [[energy]] (small distance) classical physics is refined by [[quantum physics]], specifically by [[quantum field theory]]. \end{enumerate} 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 (\hyperlink{dcct}{dcct}, \hyperlink{syntheticQFT}{sythQFT}). We close by indicating how this theory serves to solve subtle open problems in modern [[quantum field theory]] and [[string theory]]. \hypertarget{HigherMappingSpaces}{}\subsection*{{\textbf{I)} Mapping spaces in gauge theory and General covariance}}\label{HigherMappingSpaces} One basic point emphasized in (\hyperlink{Lawvere67}{Lawvere 67}) is that central to the formulation of [[physics]] is the existence of \emph{[[mapping spaces]]} which satisfy the [[exponential law]]. (notice: [[mapping space]] \ldots{} space of [[trajectories]] \ldots{} [[path integral]]) Indeed, generically a [[physical system]] is specified by \begin{itemize}% \item a [[space]] $\Sigma$ called \emph{[[spacetime]]} or \emph{[[worldvolume]]}; \item a [[space]] $X$ called \emph{[[target space]]} or \emph{[[field bundle]]} or \emph{[[moduli space]] of [[field (physics)|fields]]}. \end{itemize} such that a \emph{[[trajectory]]} or \emph{history} of [[physical field|field configurations]] is a [[map]] \begin{displaymath} \Sigma \longrightarrow X \,. \end{displaymath} 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 \emph{[[path space]]}, famous as the [[domain]] of the infamous \emph{[[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 closed category|cartesian]] [[internal hom]]. Because if we consider a split \begin{displaymath} \Sigma \coloneqq \mathbb{R} \times \Sigma_{d-1} \end{displaymath} into [[time]] and [[space]], then we want that spacetime field configurations \begin{displaymath} \mathbb{R} \times \Sigma_{d-1} \longrightarrow X \end{displaymath} are equivalently trajectories of fields on space \begin{displaymath} \mathbb{R} \longrightarrow [\Sigma_{d-1}, X] \end{displaymath} and also equivalently a collection of field trajectories for each point of space \begin{displaymath} \Sigma_{d-1} \longrightarrow [\mathbb{R}, X] \,. \end{displaymath} This led Lawvere to recognize that physics ([[prequantum field theory|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 \emph{[[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 (physics)|field]] configurations are equal, we must ask if they are [[equivalence|equivalent]]. A fundamental example of this is [[Einstein]]`s notion of \emph{[[general covariance]]}. This says that if \begin{displaymath} s \;\colon\; U \hookrightarrow \Sigma \end{displaymath} is a region in [[spacetime]], and $\phi \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma$ is a [[diffeomorphism]] acting on spacetime, then the ``translated region'' \begin{displaymath} \phi^\ast s \;\colon\; U \hookrightarrow \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma \end{displaymath} 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 \emph{[[groupoid]]}, a \emph{[[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]] \begin{displaymath} \Sigma//Diff(\Sigma) \end{displaymath} 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 \emph{inside} spacetime. The other thing now is general covariance for [[field (physics)|fields]] \emph{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 in an (infinity,1)-category|group object]] then the [[slice (infinity,1)-topos|higher slice topos]] over its [[delooping]] is equivalently the collection of [[infinity-action|G-actions]] \begin{displaymath} \mathbf{H}_{/\mathbf{B}G} \simeq G Act(\mathbf{H}) \,. \end{displaymath} Under this equivalence a [[infinity-action|higher action]] is identified with the universal [[associated infinity-bundle|associated bundle]] which it induces \begin{displaymath} \left( \itexarray{ \Sigma &\longrightarrow& \Sigma//Diff(\Sigma) \\ && \downarrow \\ && \mathbf{B}Diff(\Sigma) } \right) \;\; \in \;\; \mathbf{H}_{/\mathbf{B}Diff(\Sigma)} \simeq Diff(\Sigma)Act(\mathbf{H}) \,. \end{displaymath} Lawvere also introduced [[categorical logic]] and understood [[dependent product]] $\prod$ and [[dependent sum]] $\sum$ as [[base change]]. In [[(infinity,1)-topos theory|higher topos theory]] this becomes [[representation theory]] as follows: [[!include homotopy type representation theory -- table]] Using this we have the mapping space of [[general covariance|covariant]] [[field (physics)|fields]] formed in the [[context]] of $Diff(\Sigma)$-covariance/[[equivariance]] \begin{displaymath} \mathbf{B}Diff(\Sigma) \;\; \vdash \;\; \left[ \Sigma//Diff(\Sigma) \,,\; X \right] \end{displaymath} \textbf{Theorem.} Down in the absolute [[context]], under [[dependent sum]], this is \begin{displaymath} \vdash \;\; \underset{\mathbf{B}Diff(\Sigma)}{\sum} \left[ \Sigma//Diff(\Sigma) \,,\; X \right] \;\; \simeq \;\; [\Sigma, X]//Diff(\Sigma) \,. \end{displaymath} 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 [[(infinity,1)-topos theory|higher topos theory]] we have: \begin{itemize}% \item \emph{[[general covariance]] is homotopy [[coinvariants]]} \end{itemize} in [[mapping spaces]]. \hypertarget{ii_toposes_of_laws_of_motion_and_hamiltonjacobilagrange_mechanics}{}\subsection*{{\textbf{II)} Toposes of laws of motion and Hamilton-Jacobi-Lagrange mechanics}}\label{ii_toposes_of_laws_of_motion_and_hamiltonjacobilagrange_mechanics} In (\hyperlink{Lawvere97}{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 [[smooth topos|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 \begin{displaymath} D \hookrightarrow \mathbb{R} \end{displaymath} be the inclusion of the first order [[infinitesimally thickened point|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 a commutative ring|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 space|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 \begin{displaymath} \mathbb{R}Act(\mathbf{H}) \in Topos \end{displaymath} 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 \emph{[[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 \emph{[[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 (\hyperlink{Lawvere97}{Lawvere 97}) this problem is briefly mentioned: \begin{quote}% But what about actual dynamical systems in the spirit of [[Galileo]], for example, second-order ODE's? (Of course, the [[symplectic geometry|symplectic]] or [[Hamiltonian mechanics|Hamiltonian systems]] that are also much studied do address this question of states of [[becoming|Becoming]] versus locations of [[being|Being]], but in a special way which it may not be possible to construe as a topos; \end{quote} We observe now (following \emph{[[schreiber:Classical field theory via Cohesive homotopy types]]}) that it does exist as a ``[[higher topos theory|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 \begin{displaymath} \itexarray{ X \\ & \searrow \\ && \mathbf{\Omega}^2_{cl} } \end{displaymath} 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 [[action|acts]] on the space of [[differential 1-forms]] $\mathbf{\Omega}^1$ by \begin{displaymath} \mathbf{\Omega^1} \times S^1 \longrightarrow \mathbf{\Omega}^1 \end{displaymath} \begin{displaymath} (A, g) \mapsto A + \mathbf{d}log(g) \,, \end{displaymath} where $\mathbf{d}$ is the [[de Rham differential]]. The resulting [[quotient stack]] we write \begin{displaymath} \mathbf{B}S^1_{conn} = \mathbf{\Omega}^1//S^1 \,. \end{displaymath} The [[de Rham differential]] $\mathbf{d} \;\colon\; \mathbf{\Omega}^1 \longrightarrow \mathbf{\Omega}^2_{cl}$ descends to this quotient to yield a map \begin{displaymath} F_{(-)} \;\colon\; \mathbf{B}S^1_{conn} \longrightarrow \mathbf{\Omega}^2_{cl} \,. \end{displaymath} A [[prequantization]] of a [[presymplectic form]] is a lift $\nabla$ through this map \begin{displaymath} \itexarray{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ & {}_{\mathllap{\omega}}\searrow & \downarrow^{\mathrlap{F_{(-)}}} \\ && \mathbf{\Omega}^2 } \,. \end{displaymath} Now suppose $\omega$ is actually a [[symplectic form]]. Then: \textbf{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 \begin{displaymath} \itexarray{ X && \stackrel{\exp(t \{H,-\})}{\longrightarrow} && X \\ & {}_{\mathllap{\theta}} \searrow & \swArrow_{\exp( i S_t )} & \swarrow_{\mathrlap{\theta}} \\ && \mathbf{B}U(1)_{conn} } \,, \end{displaymath} 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 [[integration|integral]] of the [[Lagrangian]] $L$ (the [[Legendre transform]] of $H$). This statement subsumes the core ingredients of [[classical mechanics]]. See at \emph{[[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]] \begin{displaymath} LawsOfMotion(\mathbf{H}) \simeq \mathbb{R}Act\left( \mathbf{H}_{/\mathbf{B}S^1_{conn}} \right) \,. \end{displaymath} 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 field theory|local]] [[classical field theory]] by replacing $\mathbf{B} S^1_{conn}$ here with $\mathbf{B}^n S^1_{conn}$ (\hyperlink{dcct}{Schreiber 13}, \hyperlink{Schreiber}{Schreiber 13b}). \hypertarget{iii_extensiveintensive_duality_and_cohomological_quantization}{}\subsection*{{\textbf{III)} Extensive/intensive duality and Cohomological quantization}}\label{iii_extensiveintensive_duality_and_cohomological_quantization} In [[physics]] and especially in [[continuum mechanics]] and [[thermodynamics]], a physical quantity associated with a [[physical system]] extended in [[space]] is called \begin{itemize}% \item \emph{[[intensive]]} if it is a [[function]] on (the physical system extended in) space; \item \emph{[[extensive]]} if it is a [[density]] or [[linear distribution]] on (the physical system extended in) space. \end{itemize} 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 mechanics|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 (\hyperlink{Lawvere86}{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|commutative]] [[ring object]] $R \in CRing(\mathbf{H})$, then \begin{itemize}% \item the space of \emph{intensive quantities} on an [[object]] $X \in \mathbf{H}$ is the [[mapping space]] $[X,R]_{\mathbf{H}} \in CRing(\mathbf{H})$ formed in $\mathbf{H}$; \item the space of \emph{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 $\mathbf{H}$. \item the [[integration]] map is the canonical [[evaluation]] pairing \begin{displaymath} \int_X \;\colon\; [X,R] \times [X,R]^\ast \longrightarrow R \,. \end{displaymath} \end{itemize} 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 \emph{[[motivic quantization]]} for how this appears in [[physics]]. In particular ordinary [[quantum mechanics]] is recovered by settin $R =$ [[KU]], the [[complex K-theory]] [[spectrum]] (\hyperlink{Nuiten13}{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]]). \hypertarget{application_cohesion_and_superstring_anomaly_cancellation}{}\subsection*{{Application: Cohesion and Superstring anomaly cancellation}}\label{application_cohesion_and_superstring_anomaly_cancellation} On first sight, formalization of [[physics]] in ([[higher topos theory|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|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 (physics)|field]] configurations in [[physics]] are alien to [[perturbation theory]], and hence are \emph{[[quantum anomaly|anomalies]]}. Such an \href{quantum+anomaly#AnomalousActionFunctional}{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 anomaly|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 [[principal infinity-bundles|higher bundles]] on the [[moduli space]] of [[field (physics)|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 \emph{[[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 \emph{[[B-field]]} and the \emph{[[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 [[twisted cohomology|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 cohomology|twisted]] [[differential K-theory]] twisted by the [[B-field]] which is a [[cocycle]] in [[ordinary differential cohomology]]. In (\hyperlink{DFM09}{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 \emph{[[cohesion]]} (\hyperlink{Lawvere94}{Lawvere 94}, \hyperlink{Lawvere07}{Lawvere 07} ) (Earlier: ``being and becoming'' (\hyperlink{Lawvere91}{Lawvere 91})). Slightly paraphrased, cohesion means that the ambient [[type theory]] is equipped with an [[adjoint triple]] of (co-)[[modalities]] \begin{displaymath} \int \;\dashv\; \flat \;\dashv\; \sharp \end{displaymath} 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 (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13}) it was observed that on [[stable homotopy types]] $A$ cohesion implies that the canonical diagram formed from modality units and counits \begin{displaymath} \itexarray{ && \int_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat \int_{dR} \Omega A && && A && && \int \flat_{dR}\Sigma A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{\int \theta_A}} \\ && \flat A && \longrightarrow && \int A } \,, \end{displaymath} is guaranteed to consist of [[homotopy pullback]] squares, by the nature of [[adjoint triples]] of [[modalities]] (see at \emph{[[tangent cohesion]]} for more on this). In (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13}) it is shown that this is universally the ``differential cohomology diagram'' which hence exhibits \emph{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 ``[[schreiber:differential cohomology in a cohesive topos]]''. Using this and the [[twisted cohomology]] available in [[tangent cohesion]] (\hyperlink{BunkeNikolaus}{Bunke-Nikolaus}) show the existence of [[twisted differential K-theory]], the way it needs to exist for 2d QFT to be consistent. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[William Lawvere]], \emph{[[Categorical dynamics]]}, 1967 Chicago lectures (\href{http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf}{pdf}) \item [[William Lawvere]], Introduction to \emph{[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986} \item [[William Lawvere]], \emph{[[Some Thoughts on the Future of Category Theory]]} in A. Carboni, M. Pedicchio, G. Rosolini, \emph{Category Theory} , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991) \item [[William Lawvere]] \emph{[[Cohesive Toposes and Cantor's ``lauter Einsen'']]} Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]]) \item [[William Lawvere]], \emph{[[Toposes of laws of motion]]}, 1997 \item [[William Lawvere]], \emph{Axiomatic cohesion}, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf}) \item [[Urs Schreiber]] \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930}) \item [[Urs Schreiber]], \emph{[[schreiber:Classical field theory via Cohesive homotopy types]]} (\href{http://arxiv.org/abs/1311.1172}{arXiv:1311.1172}) \item [[Urs Schreiber]], \emph{[[schreiber:Quantization via Linear homotopy types]]} (\href{http://arxiv.org/abs/1402.7041}{arXiv:1402.7041}) \item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local boundary prequantum field theory]]}, 2013 \item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides}) \item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188}) \item [[Ulrich Bunke]], [[Thomas Nikolaus]], \emph{Twisted differential cohomology}, in preparation \end{itemize} \end{document}