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brief category:people-entry for hyperlinking references at topological subspace
brief category:people entry for hyperlinking references at gravity, general relativity and other entries
With Igor Khavkine we finally have a polished version of what is now “Part I” of a theory of variational calculus in a differentially cohesive -topos. It’s now called:
Synthetic geometry of differential equations
We keep our latest version of the file here.
Comments are most welcome.
Abstract:
We give an abstract (synthetic) formulation of the formal theory of partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields.
A remarkable observation of Marvan 86 is that the jet bundle construction in ordinary differential geometry has the structure of a comonad, whose (Eilenberg-Moore) category of coalgebras is equivalent to Vinogradov’s category of PDEs. We give a synthetic generalization of the jet bundle construction and exhibit it as the base change comonad along the unit of the “infinitesimal shape” functor, the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry. This comonad structure coincides with Marvan’s on ordinary manifolds. This suggests to consider PDE theory in the more general context of any topos equipped with an “infinitesimal shape” monad (a “differentially cohesive” topos).
We give a new natural definition of a category of formally integrable PDEs at this level of generality and prove that it is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad. When restricted to ordinary manifolds, Marvan’s result shows that our definition of the category of PDEs coincides with Vinogradov’s, meaning that it is a sensible generalization in the synthetic context.
Finally we observe that whenever the unit of the “infinitesimal shape” ℑ\Im operation is epimorphic, which it is in examples of interest, the category of formally integrable PDEs with independent variables ranging in Σ is also equivalent simply to the slice category over ℑΣ. This yields in particular a convenient site presentation of the categories of PDEs in general contexts.
brief category:people entry for hyperlinking references at topological algebra and Gelfand duality
Created double-negation shift, with a proof that it is equivalent to double-negated excluded middle.
brief category:people entry for hyperlinking references at asymptotic safety and inhomogeneous cosmology
I’ve started sufficiently cohesive topos. Here are a couple of remarks and questions:
The corresponding terminology in def. 2.13 at cohesive topos strikes me as odd: is connectedness not contractability.
It isn’t quite clear to me yet at which level of generality to optimally state the definition of ’sufficient cohesion’. It seems that what one wants to get here are the minimal assumptions ensuring that the connectedness of is equivalent to its contractibility and this presumably requires only preservation of finite products by and not the Nullstellensatz (nor even the existence of !?).
Since the entry so far lives on the (0,1)Lab maybe somebody here has an idea what to say for the (,1)-case e.g. assuming connectedness of the (higher) object classifier !?
brief category:people entry for hyperlinking references at inhomogeneous cosmology
his website produces Seite nicht erreichbar
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I am splitting off from the geometry of physics cluster a chapter geometry of physics – homotopy types.
For the moment I have there mostly section outline as well as some material copied over from my homological algebra lecture notes. My aim is now to put in a gentle discussion of Dold-Kan that leads an audience familiar with chain complexes from homological algebra to simplicial homotopy theory.
I’ll be touching a bunch of related entries in the process.
It’s still not quite right, is it? (here) After
Moreover, up to equivalence, every Grothendieck topos arises this way:
isn’t there the clause of accessible embedding missing? I.e. instead of
the equivalence classes of left exact reflective subcategories of the category of presheaves
it should have
the equivalence classes of left exact reflective and accessivley embedded subcategories of the category of presheaves
Or else, by the prop that follows, it should say
the equivalence classes of left exact reflective and locally presentable subcategories of the category of presheaves
No?
(This is just a question. I didn’t make an edit. Yet.)
I have started a (stubby) entry on multiagent systems, to link into certain of the modal logic entries.
I started putting down some thoughts at theory (physics). Not meant to be comprehensive or anything, but just a quick note. I am not claiming that the state the entry is in is the state it should remain in at all. But maybe it’s a start that helps to develop something.
brief category:people-entry for the purpose of hyperlinking references at gauge coupling unification and naturalness
created some minimum at gauge coupling unification
Added a bit more to proximity space.
am creating a minimum entry here, for the moment just for completeness, to go along with Pr(∞,1)Cat and Ho(CombModCat)
brief category:people-entry for the purpose of hyperlinking references at 2-localization of a 2-category and at Ho(CombModCat).
brief category:people-entry for the purpose of hyperlinking references at gauge coupling unification and GUT
Added to global element, which seems not to have had a Latest Changes-thread so far (hence this newly created one), a remark on a formalization of “name of a morphism” which I just stumbled upon and find a noteworthy thing.
Perhaps this should go somewhere on the nLab, but to me global element seemed the most fitting place.
I had always thought something like, “Well, if one really has to be careful and formal about the distinction between names or morphisms and morphisms per se, then the protocategories and protomorphisms in the sense of Freyd and Scedrov give one way to do so, and there is an introduction to this in the Elephant.” I was surprised to find someone connecting this to internal homs, hence made this note in global element.
If there are some “situating comments” on this that can conveniently be made, I would be happy to read them here.
There is now a diagonal functor entry.
For ease of referencing from other entries, I am splitting this off from combinatorial model category. Currently this is a blind copy-and-paste. Looking back at this old material, I see it may need some attention. For instance the model structure on sSet-enriched presheaves needs mentioning, I suppose.
I moved various subsections (Monoidal structure, closed structure, Adjunctions) on properties of sSet from the entry simplicial set to sSet.
brief category:people entry for hyperlinking references at AdS-CFT in condensed matter physics
Added some more intuition for duploids now that I understand them and cbpv better. Duploids only axiomatize effectful morphisms, whereas an adjunction (CBPV) axiomatizes pure morphisms (as homomorphisms) and effectful morphisms (as heteromorphisms). Then thunkable and linear are the maximal way to recover pure morphisms from effectful morphisms. I.e., we should think of duploids as presenting a kind of “Morita equivalence” of adjunctions where we only care about the equivalence of the notion of heteromorphism.
the direct formal-geometric analogue of smooth sets. This used to just redirect to geometry of physics – manifolds and orbifolds, but it deserves its own entry, on par with related variants that long had their own. To be expanded.
the direct supergeometric analogue of smooth sets. This is discussed in some detail in geometry of physics – supergeometry, but it deserves its own entry, on par with related variants that long had their own. To be expanded..