Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeDec 22nd 2016
   • (edited Dec 23rd 2016)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexI recently accepted to take part to a series of seminars on topos theory. My idea is to offer a glance to higher toposes using cohesive-infty-stuff in Physics, profiting from the huge amount of material present here on the $n$Lab. My current idea is to give at least a vague intuition about why nice things happen if we choose a higher setting to do Physics. This is of course extremely well explained in several places here, but I feel I'm lacking a birdseye view of the topic. This is obviously my fault! I don't know Physics :-( But I'm trying my best. I kindly ask for your help. My initial attempt was reading the page [[higher category theory and physics]]; here's a list of the things I'm not able to understand. > "Dirac had not only identified the [[electromagnetic field]] as a [[line bundle]] with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological [[Chern class]] with the (physically hypothetical but formally inevitable) [[magnetic charge]] located in [[spacetime]]. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the [[field strength]] of the [[electromagnetic field]] is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle." This is rather mysterious. I don’t understand what’s going on. Here’s my attempt to an interpretation: when we write Maxwell equations we have to forget the contribution given by a magnetic charge to obtain a properly-defined 2-form that fits into the “gauges-areconnections” POV. But this seems to conflict with the above sentence that Maxwell outlined that the electromagnetic field is modeled by this 2-form, and Dirac observed that this 2-form is the curvature of a U(1)- principal bundle over the spacetime! What’s going on? > "In attempts to better understand the structure of these two theories and their interrelation, theoretical physicists were led to consider variations and generalizations of them that are known as [[supergravity]] and [[string theory]]." This connection is interesting. Can string theory and sugra be seen as some sort of homotopization of classical theories in a sense similar to the one we have in matehmatics? From my layman POV supergeometry basically consists of “algebraic geometry on supervarieties” as defined in Deligne’s notes in the book about strings and fields for mathematicians. > "In these theories the notion of [[gauge field]] turns out to generalize: instead of just [[Lie algebra]]s, [[Lie group]]s and connections with values in these, one finds structures called [[Lie 2-algebra]]s, [[Lie 2-group]]s and the gauge fields themselves behave like generalized connections with values in these." I wonder if John Baez’s work on 2-versions of several mathematical structures fits into this framework: I expect that if one tries to interpret those results in an (∞, 1)-framework something nice happens. But again, nobody ever mentioned me these topics (and when I had the opportunity to read Baez work I found it absoutely valuable and still modern). Please, elighten me! I want to have clear especially why the natural move is to categorify groups and Lie algebras. . . I also have several questions about the [[cohesive topos]] page: here's the list 1. What is the meaning of the intuitions behind shape - flat - sharp modality? Why should $\int\colon \mathcal{E} \to \mathcal{E}$ be thought as a "geometric realization" of an object X? I am inclined to regard $\int X$ as some sort of cofibrant approximation of X, but why should the discrete E-object on the set of pieces of X play this role? 2. How do you prove the last statement in Definition 2.13 of cohesive topos page? I.e.: if $\Omega$ is contractible, then $\Omega^X$ is contractible too. Usually, functor that behave well wrt exponentials are right adjoints, not left like $f_! = \Pi_0$... 3. Is there any interest in studying the subcategory of those $X \in\mathcal{E}$ such that $\Gamma(X)\simeq *$? Does the inclusion of " $\mathcal{K} = \ker\Gamma$ " admit adjoints on some side? 4. I'm familiar with the definition of "discrete truth" for a proposition, I've seen it studying [[reflective factorization systems]]! Every proposition $S\to A$ seems to factor through its "evaluation on discretes", and I'd guess there is some interest in (dis)proving that discretely-true proposition form an orthogonal class in a factorization system in a (cohesive) topos. A comment on the kind of help I need: it would be incredibly helpful to have an expanded version of [this document](https://ncatlab.org/schreiber/files/SchreiberHalifax2013.pdf): that's precisely the kind of motivation I was looking for, and the kind of connection I'm trying to unravel.

   I recently accepted to take part to a series of seminars on topos theory. My idea is to offer a glance to higher toposes using cohesive-infty-stuff in Physics, profiting from the huge amount of material present here on the $n$Lab.

   My current idea is to give at least a vague intuition about why nice things happen if we choose a higher setting to do Physics. This is of course extremely well explained in several places here, but I feel I’m lacking a birdseye view of the topic. This is obviously my fault! I don’t know Physics :-(

   But I’m trying my best. I kindly ask for your help.

   My initial attempt was reading the page higher category theory and physics; here’s a list of the things I’m not able to understand.

   “Dirac had not only identified the electromagnetic field as a line bundle with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological Chern class with the (physically hypothetical but formally inevitable) magnetic charge located in spacetime. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the field strength of the electromagnetic field is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle.”

   This is rather mysterious. I don’t understand what’s going on. Here’s my attempt to an interpretation: when we write Maxwell equations we have to forget the contribution given by a magnetic charge to obtain a properly-defined 2-form that fits into the “gauges-areconnections” POV. But this seems to conflict with the above sentence that Maxwell outlined that the electromagnetic field is modeled by this 2-form, and Dirac observed that this 2-form is the curvature of a U(1)- principal bundle over the spacetime! What’s going on?

   “In attempts to better understand the structure of these two theories and their interrelation, theoretical physicists were led to consider variations and generalizations of them that are known as supergravity and string theory.”

   This connection is interesting. Can string theory and sugra be seen as some sort of homotopization of classical theories in a sense similar to the one we have in matehmatics? From my layman POV supergeometry basically consists of “algebraic geometry on supervarieties” as defined in Deligne’s notes in the book about strings and fields for mathematicians.

   “In these theories the notion of gauge field turns out to generalize: instead of just Lie algebras, Lie groups and connections with values in these, one finds structures called Lie 2-algebras, Lie 2-groups and the gauge fields themselves behave like generalized connections with values in these.”

   I wonder if John Baez’s work on 2-versions of several mathematical structures fits into this framework: I expect that if one tries to interpret those results in an (∞, 1)-framework something nice happens. But again, nobody ever mentioned me these topics (and when I had the opportunity to read Baez work I found it absoutely valuable and still modern). Please, elighten me! I want to have clear especially why the natural move is to categorify groups and Lie algebras…

   I also have several questions about the cohesive topos page: here’s the list

   1. What is the meaning of the intuitions behind shape - flat - sharp modality? Why should $\int\colon \mathcal{E} \to \mathcal{E}$ be thought as a “geometric realization” of an object X? I am inclined to regard $\int X$ as some sort of cofibrant approximation of X, but why should the discrete E-object on the set of pieces of X play this role?
   2. How do you prove the last statement in Definition 2.13 of cohesive topos page? I.e.: if $\Omega$ is contractible, then $\Omega^X$ is contractible too. Usually, functor that behave well wrt exponentials are right adjoints, not left like $f_! = \Pi_0$…
   3. Is there any interest in studying the subcategory of those $X \in\mathcal{E}$ such that $\Gamma(X)\simeq *$? Does the inclusion of ” $\mathcal{K} = \ker\Gamma$ ” admit adjoints on some side?
   4. I’m familiar with the definition of “discrete truth” for a proposition, I’ve seen it studying reflective factorization systems! Every proposition $S\to A$ seems to factor through its “evaluation on discretes”, and I’d guess there is some interest in (dis)proving that discretely-true proposition form an orthogonal class in a factorization system in a (cohesive) topos.

   A comment on the kind of help I need: it would be incredibly helpful to have an expanded version of this document: that’s precisely the kind of motivation I was looking for, and the kind of connection I’m trying to unravel.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2016
   • (edited Dec 23rd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I recently accepted to take part to a series of seminars on topos theory. My idea is to offer a glance to higher toposes using cohesive-infty-stuff in Physics, profiting from the huge amount of material present here on the $n$Lab. Nice! > > "Dirac had not only identified the [[electromagnetic field]] as a [[line bundle]] with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological [[Chern class]] with the (physically hypothetical but formally inevitable) [[magnetic charge]] located in [[spacetime]]. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the [[field strength]] of the [[electromagnetic field]] is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle." > This is rather mysterious. $[$...$]$ What’s going on? In the presence of magentic charge $J_3$ the Maxwell equations say that $$ d F_2 = J_3 \,. $$ For non-vanishing $J_3$ this is incompatible with regarding $F_2$ as the curvature of a $U(1)$-principal bundle, since curvature 2-forms are necessarily closed. Dirac removed the support of $J_3$ from the spacetime, so that on the remaining complement we have $d F_2 = 0$ and hence may (and have to) regard $F_2$ as the curvature 2-form of a connection. If one does want to include the supprt of $J_3$, then one needs to pass to a more sophisticated formulation. Now $J_3$ is interpreted in compactly supported differential cohomology in degree 3, and $F$ is interpreted as a trivialization of the image of $J_3$ in not-necessarily compactly supported differential cohomology. > This connection is interesting. Can string theory and sugra be seen as some sort of homotopization of classical theories in a sense similar to the one we have in matehmatics? From my layman POV supergeometry basically consists of “algebraic geometry on supervarieties” as defined in Deligne’s notes in the book about strings and fields for mathematicians. The reason that supergravity implies homotopification is that it so happens that supersymmetric completions of the Einstein-Hilbert action in dimensions greater than 4 (or 5, depending on whether one dualizes a bit or not) necessarily involve higher gauge fields. The usual colloquial way to explain this is that the degrees of freedom in spinors grow quickly with dimension, and counting arguments show that only higher differential form fields are the appropriate bosonic fields to pair to these such as to get a supersymmetric matching. But the real explanation is a bit more technical. In any case, the end result is: in dimension $\geq 5$ then local spacetime supersymmetry necessarily involves higher gauge fields. > Please, elighten me! I want to have clear especially why the natural move is to categorify groups and Lie algebras. The quick easy observation that is at the origin of much of the history of the development is simply this: Just like a charged particle couples to a 1-form (the electromagnetic vector potential, whose integral over the worldline of the particle gives the action term whose variation is the Lorentz force that the particle experiences) so a string couples to a 2-form. Just as gauge transormations of 1-forms $A$ are given by 0-forms $\kappa$ via $A' = A + d \kappa$, so gauge transformations of 2-forms $B$ are given by 1-forms $a$ with $B' = B + d a$. But this means that these gauge transformations of 2-forms have higher order gauge-of-gauge transformations between them, labeled by 0-forms $\rho$ such that $a' = a + d \rho$. In conclusion, where gauge fields to which particles couple form a groupoid, the higher gauge fields to which the string couples form a 2-groupoid. Infintiesimally the former gives Lie algebroids and Lie algebras, while the latter gives Lie 2-algebroids and Lie 2-algebras.

   I recently accepted to take part to a series of seminars on topos theory. My idea is to offer a glance to higher toposes using cohesive-infty-stuff in Physics, profiting from the huge amount of material present here on the $n$Lab.

   Nice!

   “Dirac had not only identified the electromagnetic field as a line bundle with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological Chern class with the (physically hypothetical but formally inevitable) magnetic charge located in spacetime. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the field strength of the electromagnetic field is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle.”

   This is rather mysterious. $[$…$]$ What’s going on?

   In the presence of magentic charge $J_3$ the Maxwell equations say that

   $$d F_2 = J_3 \,.$$

   For non-vanishing $J_3$ this is incompatible with regarding $F_2$ as the curvature of a $U(1)$-principal bundle, since curvature 2-forms are necessarily closed. Dirac removed the support of $J_3$ from the spacetime, so that on the remaining complement we have $d F_2 = 0$ and hence may (and have to) regard $F_2$ as the curvature 2-form of a connection. If one does want to include the supprt of $J_3$, then one needs to pass to a more sophisticated formulation. Now $J_3$ is interpreted in compactly supported differential cohomology in degree 3, and $F$ is interpreted as a trivialization of the image of $J_3$ in not-necessarily compactly supported differential cohomology.

   This connection is interesting. Can string theory and sugra be seen as some sort of homotopization of classical theories in a sense similar to the one we have in matehmatics? From my layman POV supergeometry basically consists of “algebraic geometry on supervarieties” as defined in Deligne’s notes in the book about strings and fields for mathematicians.

   The reason that supergravity implies homotopification is that it so happens that supersymmetric completions of the Einstein-Hilbert action in dimensions greater than 4 (or 5, depending on whether one dualizes a bit or not) necessarily involve higher gauge fields. The usual colloquial way to explain this is that the degrees of freedom in spinors grow quickly with dimension, and counting arguments show that only higher differential form fields are the appropriate bosonic fields to pair to these such as to get a supersymmetric matching. But the real explanation is a bit more technical. In any case, the end result is: in dimension $\geq 5$ then local spacetime supersymmetry necessarily involves higher gauge fields.

   Please, elighten me! I want to have clear especially why the natural move is to categorify groups and Lie algebras.

   The quick easy observation that is at the origin of much of the history of the development is simply this: Just like a charged particle couples to a 1-form (the electromagnetic vector potential, whose integral over the worldline of the particle gives the action term whose variation is the Lorentz force that the particle experiences) so a string couples to a 2-form. Just as gauge transormations of 1-forms $A$ are given by 0-forms $\kappa$ via $A' = A + d \kappa$, so gauge transformations of 2-forms $B$ are given by 1-forms $a$ with $B' = B + d a$. But this means that these gauge transformations of 2-forms have higher order gauge-of-gauge transformations between them, labeled by 0-forms $\rho$ such that $a' = a + d \rho$. In conclusion, where gauge fields to which particles couple form a groupoid, the higher gauge fields to which the string couples form a 2-groupoid. Infintiesimally the former gives Lie algebroids and Lie algebras, while the latter gives Lie 2-algebroids and Lie 2-algebras.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2016
   • (edited Dec 23rd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRegarding the questions 1) and 3) in #1, these I have replied to before via email. Please give me feedback on what these replies did to you, so that I know what else I could say. Question 2) you should forward to the thread here on the $n$Forum titled "cohesive topos", so that people see it. Probably in the whealth of material in this thread here most readers will miss it. Regarding 4): Could you spell out the definition of "discrete truth" that you are after? It sounds plausible that this is related to "flat-modal types" in cohesive type theory, but to be sure I'd need to know what you actually have in mind.

   Regarding the questions 1) and 3) in #1, these I have replied to before via email. Please give me feedback on what these replies did to you, so that I know what else I could say.

   Question 2) you should forward to the thread here on the $n$Forum titled “cohesive topos”, so that people see it. Probably in the whealth of material in this thread here most readers will miss it.

   Regarding 4): Could you spell out the definition of “discrete truth” that you are after? It sounds plausible that this is related to “flat-modal types” in cohesive type theory, but to be sure I’d need to know what you actually have in mind.

4. • CommentRowNumber4.
   • CommentAuthorFosco
   • CommentTimeDec 23rd 2016
   • (edited Dec 23rd 2016)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexFor 1) and 3); I have to think about it some more; for the moment I simply reported the questions as-they-were. :-) digestion is a long process. Especially during Christmas holidays. For 2) I'll do it, thanks Regarding 4): here's what I understand: a propositon of type $A$ is a monomorphism $\varphi : S\to A$. Now we say that $\varphi$ is "discretely true" if in $$\begin{array}{ccc} \varphi^*(S) &\to& \flat S \\ \downarrow & \pb &\downarrow \\ A &\to& \flat A \end{array} $$ the arrow $\varphi^*(S) \to A$ is an isomorphism. I've met this construction elsewhere: whenever you have an idempotent monad $\flat$, then you can obtain a (pre)factorization system factoring $\varphi \colon A\to B$ as $A\to \varphi^*(A)\to B$ using the same pullback above. I was wondering if this construction is a special case of what I have in mind, and what are the consequences/interpretations of this POV (is there a factorization system having discretely true propositions as right class? What is the left class? What are the co/fibrants of this factorization system, i.e. objects $\varnothing \xrightarrow{l}X$ and $Y \xrightarrow{r}*$?). I'll add other questions later. Be prepared ;-)

   For 1) and 3); I have to think about it some more; for the moment I simply reported the questions as-they-were. :-) digestion is a long process. Especially during Christmas holidays.

   For 2) I’ll do it, thanks

   Regarding 4): here’s what I understand: a propositon of type $A$ is a monomorphism $\varphi : S\to A$. Now we say that $\varphi$ is “discretely true” if in

   $$\begin{array}{ccc} \varphi^*(S) &\to& \flat S \\ \downarrow & \pb &\downarrow \\ A &\to& \flat A \end{array}$$

   the arrow $\varphi^*(S) \to A$ is an isomorphism. I’ve met this construction elsewhere: whenever you have an idempotent monad $\flat$, then you can obtain a (pre)factorization system factoring $\varphi \colon A\to B$ as $A\to \varphi^*(A)\to B$ using the same pullback above. I was wondering if this construction is a special case of what I have in mind, and what are the consequences/interpretations of this POV (is there a factorization system having discretely true propositions as right class? What is the left class? What are the co/fibrants of this factorization system, i.e. objects $\varnothing \xrightarrow{l}X$ and $Y \xrightarrow{r}*$?).

   I’ll add other questions later. Be prepared ;-)

5. • CommentRowNumber5.
   • CommentAuthorFosco
   • CommentTimeDec 23rd 2016
   • (edited Dec 23rd 2016)
   • PermaLink
   Author: Fosco
   Format: MarkdownItex1. Is it true that if $X$ is infinitesimal then $\Pi_0(X)\cong *$? (is it reasonable to expect it from a "infinitesimally extended nbd of a point). 2. Is there a "deep meaning" in the [big diagram](https://ncatlab.org/nlab/show/tangent+(infinity,1)-category) $$\array{ Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} \\ \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }$$ My intuition is that there is a "fiber sequence" $$\begin{array}{ccc} FiberOfCohesion &\to&TangentCohesion\\ \downarrow &&\downarrow \\ * &\to&Cohesion \end{array}$$ that purportedly underlines a "micro-macrocosm" principle?

   1. Is it true that if $X$ is infinitesimal then $\Pi_0(X)\cong *$? (is it reasonable to expect it from a “infinitesimally extended nbd of a point).
   2. Is there a “deep meaning” in the big diagram

   $$\array{ Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} \\ \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }$$

   My intuition is that there is a “fiber sequence”

   $$\begin{array}{ccc} FiberOfCohesion &\to&TangentCohesion\\ \downarrow &&\downarrow \\ * &\to&Cohesion \end{array}$$

   that purportedly underlines a “micro-macrocosm” principle?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItex> a propositon of type $A$ is a monomorphism $\varphi : S\to A$. Now we say that $\varphi$ is "discretely true" if in $$\begin{array}{ccc} \varphi^*(S) &\to& \flat S \\ \downarrow & \pb &\downarrow \\ A &\to& \flat A \end{array} $$ the arrow $\varphi^*(S) \to A$ is an isomorphism. I've met this construction elsewhere: whenever you have an idempotent monad $\flat$, then you can obtain a (pre)factorization system factoring $\varphi \colon A\to B$ as $A\to \varphi^*(A)\to B$ using the same pullback above. Yes, but beware that all this is for monads, while $\flat$ happens to be a comonad. What you describe (modulo the monad/comonad switch) is the fact that a factorization system where the reflector preserves fiber products over local objects and on a suitably nice category induces factorizations systems on all slices, by the pullback construction that you indicate. In cohesion, for instance 1. applied to the shape modality this yields Galois theory, see section 5.2.7 in dcct ([pdf](https://ncatlab.org/schreiber/files/dcct161227.pdf)). 1. applied to the infinitesimal shape modality this yields the theory of jet bundles, see section 5.3.7. In both cases I am not sure if the restriction of the system to propositions, hence to (-1)-truncated objects is all that interesting. The interest here arises really from the modality acting on the full homotopy-type theory.

   a propositon of type $A$ is a monomorphism $\varphi : S\to A$. Now we say that $\varphi$ is “discretely true” if in

   $$\begin{array}{ccc} \varphi^*(S) &\to& \flat S \\ \downarrow & \pb &\downarrow \\ A &\to& \flat A \end{array}$$

   the arrow $\varphi^*(S) \to A$ is an isomorphism. I’ve met this construction elsewhere: whenever you have an idempotent monad $\flat$, then you can obtain a (pre)factorization system factoring $\varphi \colon A\to B$ as $A\to \varphi^*(A)\to B$ using the same pullback above.

   Yes, but beware that all this is for monads, while $\flat$ happens to be a comonad.

   What you describe (modulo the monad/comonad switch) is the fact that a factorization system where the reflector preserves fiber products over local objects and on a suitably nice category induces factorizations systems on all slices, by the pullback construction that you indicate.

   In cohesion, for instance

   1. applied to the shape modality this yields Galois theory, see section 5.2.7 in dcct (pdf).

   2. applied to the infinitesimal shape modality this yields the theory of jet bundles, see section 5.3.7.

   In both cases I am not sure if the restriction of the system to propositions, hence to (-1)-truncated objects is all that interesting. The interest here arises really from the modality acting on the full homotopy-type theory.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Is it true that if $X$ is infinitesimal then $\Pi_0(X)\cong *$? (is it reasonable to expect it from a "infinitesimally extended nbd of a point). No, $\Pi(X) \simeq \ast$ just means that $X$ is contractible. And $\Pi_0(X) \simeq \ast$ even just means that $X$ is connected. Certainly there are many connected and also contractible geometric spaces which are not infinitesimally thickened points, e.g. $\mathbb{R}^n$. > Is there a "deep meaning" in the [big diagram](https://ncatlab.org/nlab/show/tangent+(infinity,1)-category) $[$...$]$ My intuition is that there is a "fiber sequence" $$\begin{array}{ccc} FiberOfCohesion &\to&TangentCohesion\\ \downarrow &&\downarrow \\ * &\to&Cohesion \end{array}$$ That seems right. Maybe one should says that $Stabl(\mathbf{H})$ is the "fiber of the tangent cohesion", namely of $T \mathbf{H} \stackrel{base}{\to} \mathbf{H}$. > that purportedly underlines a "micro-macrocosm" principle? Here I am not sure what you have in mind. I am guessing that you are referring to what is stated at _[[microcosm principle]]_? But how do you see this applying to the above situation?

   Is it true that if $X$ is infinitesimal then $\Pi_0(X)\cong *$? (is it reasonable to expect it from a “infinitesimally extended nbd of a point).

   No, $\Pi(X) \simeq \ast$ just means that $X$ is contractible. And $\Pi_0(X) \simeq \ast$ even just means that $X$ is connected. Certainly there are many connected and also contractible geometric spaces which are not infinitesimally thickened points, e.g. $\mathbb{R}^n$.

   Is there a “deep meaning” in the big diagram $[$…$]$ My intuition is that there is a “fiber sequence”

   $$\begin{array}{ccc} FiberOfCohesion &\to&TangentCohesion\\ \downarrow &&\downarrow \\ * &\to&Cohesion \end{array}$$

   That seems right. Maybe one should says that $Stabl(\mathbf{H})$ is the “fiber of the tangent cohesion”, namely of $T \mathbf{H} \stackrel{base}{\to} \mathbf{H}$.

   that purportedly underlines a “micro-macrocosm” principle?

   Here I am not sure what you have in mind. I am guessing that you are referring to what is stated at microcosm principle? But how do you see this applying to the above situation?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 4th 2017
   • (edited Jan 4th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIf you are after more motivation, there is now an article in the _PhysicsForums Insights_ series titled: * _[Why Higher Category Theory in Physics?](https://www.physicsforums.com/insights/higher-category-theory-physics/)_ This is my personal story. It is really a reply to one question in an interview that Greg Bernhard is doing with me.

   If you are after more motivation, there is now an article in the PhysicsForums Insights series titled:

   • Why Higher Category Theory in Physics?

   This is my personal story. It is really a reply to one question in an interview that Greg Bernhard is doing with me.