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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus 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 finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory 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 planar 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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- Discussion Type
- discussion topicvariational caclulus - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 6th 2011

I have started a new subject complex variational calculus - contents and have included it as a floating TOC into relevant entries

- Discussion Type
- discussion topiccharge
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 6th 2011

stub for charge

- Discussion Type
- discussion topicdifferential monad
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active May 6th 2011

I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

- Discussion Type
- discussion topic[[cartesian category]]
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by SridharRamesh
- Last Active May 6th 2011

I’ve disambiguated links to cartesian category. I suggest that we avoid this term.

- Discussion Type
- discussion topiccoherent (oo,1)-operad
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active May 5th 2011

stub for coherent (infinity,1)-operad

- Discussion Type
- discussion topicnew entries
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active May 4th 2011

Created a stubby entry for Gereon Quick, and added some more into (or fixed typos on) Daniel Isaksen, Cech homotopy,profinite homotopy theory and pro-homotopy theory.

- Discussion Type
- discussion topicJon Pridham
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active May 4th 2011

I created an entry for Jonathan Pridham.

- Discussion Type
- discussion topicO-monoidal (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 3rd 2011

have added to monoidal (infinity,1)-category the definition of $\mathcal{O}$-monoidal $(\infty,1)$-category, for $\mathcal{O}$ an $\infty$-operad

(though maybe this definition either deserves its own entry or ought to be included instead at symmetric monoidal (infnity,1)-category)

- Discussion Type
- discussion topiccategory of factorizations
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Tim_Porter
- Last Active May 1st 2011

A point of information. These constructions are due to Charles Wells in this particular setting and to Jonathan Leech, (H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975) in the single object case, and McLane introduces the category of factorisations I think. Charlie Wells even pushes things a bit further than Baues. Hans does not seem to have known of that work. (Charles Wells, Extension theories for categories (preliminary report), (available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979. ) I have been meaning to have a go at this entry as I have written up a modern version of Wells especially in the non-Abelian case. There is a very nice interpretation of Natural System as a lax functor. (I will do this some time…. but I can make the notes available to anyone interested.)

- Discussion Type
- discussion topiccharacteristic class of a structure
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by Urs
- Last Active May 1st 2011

New entry characteristic class of a structure to complement characteristic class and historical note on characteristic classes. I did not link to it from outside so far.

- Discussion Type
- discussion topicuniversally closed morphism
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Urs
- Last Active Apr 30th 2011

universally closed morphism and improvements at proper morphism

- Discussion Type
- discussion topicproadjoint
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Tim_Porter
- Last Active Apr 30th 2011

New entry proadjoint.

- Discussion Type
- discussion topicnuclear space
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 30th 2011

New stub nuclear topological vector space with redirect nuclear space. Grothendieck’s reference also at Fredholm operator.

- Discussion Type
- discussion topicprime spectrum
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 30th 2011

New entry prime spectrum with redirect Zariski spectrum.

- Discussion Type
- discussion topicFréchet
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by Andrew Stacey
- Last Active Apr 28th 2011

Urs created Frechet manifold, so I created Frechet space. (We violated the naming conventions too, but I guess it's OK since we have the redirects in.)

- Discussion Type
- discussion topictoposes as theories
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by zskoda
- Last Active Apr 28th 2011

I am trying to begin to coherently add some of the topics of part D of the Elephant into the Lab.

Currently I am creating lots of stub entries, splitting them off from existing entries if necessary, cross-link them appropriately, and then eventually add content to them.

so far I have for instance created new (mostly stub) entries for things like

I have created

and made it a disambiguation page.

I have edited the linked table of contents at Elephant, etc.

(or rather I will have in a few minutes. All my save-windows are currently stalled. Will have to restart the server.)

- Discussion Type
- discussion topicrepresentable fibered category
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by Urs
- Last Active Apr 27th 2011

New entry representable fibered category.

- Discussion Type
- discussion topicstandard site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 27th 2011

created standard site (maybe not a great term, but since I am $n$Labifying the Elephant). Added the theorem that every sheaf topos has a standard site of definition to site

- Discussion Type
- discussion topicZariski site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 27th 2011

have created Zariski site

- Discussion Type
- discussion topicslice category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Apr 27th 2011

added to overcategory the statement about lifs to adjunctions to slices, here

- Discussion Type
- discussion topicsyntactic site
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 27th 2011

a bit of content at syntactic site

- Discussion Type
- discussion topicindexed topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 26th 2011

started indexed topos

- Discussion Type
- discussion topicbase topos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 26th 2011

I have touched indexed category and then started filling some first content into base topos.

- Discussion Type
- discussion topicsequent calculus
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 26th 2011

started adding rules to sequent calculus, but have to interrup now and hunt some food

- Discussion Type
- discussion topiccartesian, regular, coherent, geometric
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 26th 2011

I have created the following web of entries

cartesian category, cartesian functor, cartesian logic, cartesian theory

regular category, regular functor, regular logic, regular theory, regular coverage, regular topos

coherent category, coherent functor, coherent logic, coherent theory, coherent coverage, coherent topos

geometric category, geometric functor, geometric logic, geometric theory

Most of them stubs. Partly just material split off from other entries. But all with the relavent pointers to the Elephant or other literature. To be expanded.

- Discussion Type
- discussion topicHodge to de Rham spectral sequence degeneration
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Andrew Stacey
- Last Active Apr 26th 2011

I have created degeneration conjecture required at Dmitri Kaledin. In my memory, I never heard ofthis

**degeneration conjecture**by precisely*that*name and I do not like it (there are so many degeneration conjectures in other fields, some of which I heard under*that*name). It is usually said the**degeneration of Hodge to de Rham spectral sequence**(conjecture). It has a classical analogue. I put redirect degeneration of Hodge to de Rham spectral sequence.

- Discussion Type
- discussion topicdifferential cohomology in an (∞,1)-topos -- survey
- Category Latest Changes
- Started by David_Corfield
- Comments 18
- Last comment by Urs
- Last Active Apr 25th 2011

In differential cohomology in an (∞,1)-topos – survey, I can’t guess what ’nothing’ should be here:

The curvature characteristic forms / Chern characters in the traditional formulation of differential cohomology take values in abelian $\infty$-Lie algebras and are therefore effectively nothing differential forms with values in a complex of vector spaces

- Discussion Type
- discussion topicstring structure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 23rd 2011

touched string structure. Added some formal discussion, also polsihed layout and added references. But didn’t change the previous informal discussion.

- Discussion Type
- discussion topicordinary differential cohomology
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Apr 21st 2011

following public demand, I created an entry ordinary differential cohomology.

- Discussion Type
- discussion topiccartesian morphism
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by Urs
- Last Active Apr 21st 2011

This is an excerpt I wrote at logical functor:

As far as cartesian morphism there are two different universal properties in the literature, which are equivalent for Grothendieck fibered categories but not in general. In what Urs calls the “traditional definition” (but is in fact a later one) one has for every $x'$, for every $h$, for every $g$ such that … there exist a unique da da da. This way it is spelled in Vistoli’s lectures. This is in fact the strongly cartesian property, stronger than one in Gabriel-Grothendieck SGA I.6. The usual, Grothendieck, or weak property takes for $g$ the identity, and the unique lift is of the identity at $p(x_1)$. Then a Grothendieck fibered category is the one which has cartesian lifts for all morphisms below and all targets, and cartesian morphisms are closed under composition. With the strong cartesian property one does not need to require the closedness under composition. Now a theorem says that in a Grothendieck fibered category, a morphism is strongly cartesian iff it is cartesian.

Now I have made some changes to cartesian morphism, so that the entry is aware of the two variants of the universal property, which are not equivalent in general but are equivalent for Grothendieck fibered categories.

There was also a statement there

In words: for all commuting triangles in Y and all lifts through p of its 2-horn to X, there is a unique refinement to a lift of the entire commuting triangle.

which is too vague and I am not happy with, as it does not involve the essential parameter: the morphism for which we test cartesianess. I made a hack to it, and still it is not something I happy with (I like the idea of

*horn*mentioned, however not the lack of appropriate quantifiers/conditions etc.). It is cumbersome to talk horn. (Maybe we could skip the whole statement in this imprecise form, and just mention*please note the filling of the horn in $X$ with prescribed projection in $Y$*or alike). Here is the temporary hack:In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-horn to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.