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brief category:people entry, for the purpose of hyperlinking references at manifold with boundary and diffeological space
Link for author of article on bifibration of model categories
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
I have split off the section on points-to-pieces transform from cohesive topos and expanded slightly, pointing also to comparison map between algebraic and topological K-theory
Although referred to in a couple of place, it seems we had no entry for spatial topos, so I’ve made a start.
Since I got questions from the audience (here) why I defined (pre-)sheaves on a site, instead of on a topological space “as in the textbooks”, I created this little entry with some basic pointers, which may complement the entry localic topos for the newbie. Could of course be expanded a lot…
for ease of reference, I gave this statement its own entry, to go along with hom-functor preserves limits, adjoints preserve (co-)limits and similar
added pointer to Scholze 17
In the article cocylinder one reads at the bottom:
George Whitehead, Elements of homotopy theory
(This uses the terminology mapping path space.)
(This was added in revision 3 by Mike Shulman.)
However, I was unable to find any occurrence of this terminology in Whitehead’s book.
Indeed, looking at the table on page 141 below Theorem 6.22, we see that Whitehead refers to the dual construction as the mapping cylinder I_f, whereas the original construction is denoted by I^f, but there is no name attached to it.
Furthermore, on page 43 below Theorem 7.31 one reads:
The process of replacing the map f: X→Y by the homotopically equivalent fibration p : I^f→Y
is, in some sense, analogous to that of replacing f by the inclusion map of X into the mapping cylinder of f;
the latter is a cofibration, rather than a fibration.
Pursuing this analogy further, we may consider the fibre T^f of p over a designated point of Y.
We shall call T^f the mapping fibre of f (resisting firmly the temptation to call I^f and T^f the mapping cocylinder and cocone of f!).
I have expanded norm a bit.
Linked to convex set, norm, absolutely convex set
Created algebraic theories in functional analysis. I've recently learnt about this connection and would like to learn more so I've created this page as a place to record my (and anyone else's) findings on this. I probably won't get round to doing much before the new year, though.
copied over to here (from internal hom) statement and proof that also an internal hom-bifunctor preserves ordinary limits in both arguments (now this prop)
for convenience of hyperlinking and disambiguation, I need an entry of this title, in between presheaf and (2,1)-presheaf. Started with a brief Idea-section that just scans the different possible meanings and their relation. Might copy over more material from geometry of physics – categories and toposes once that has stabilized more.
added remark about and pointer to Cech groupoid as co-representing sets of matching families (here)
I have added the actual general definition of the Cech groupoid as presheaf of groupoids, and headlined the definition previously offered here as “Idea”. Then I added detailed statement and proof, that the Cech-groupoid co-represents sets of matching families for set-valued presheaves (now this prop.)
This entry is currently undecided as to whether “full subcategory” inclusion requires the functor to be an injection on objects. It begins by pointing to subcategory which does require this, but before long it speaks about fully faitful functors being full subcategory inclusions.
This will be confusing to newcomers. There should be at least some comments about invariance under equivalence of categories.
Ah, now I see that at subcategory there is such a discussion (here). Hm, there is some room for cleaning-up here.
I have expanded a little at sifted category: added the example of the reflexive-coequalizer diagram, added the counter-examples of the non-reflexive coequalizer diagram, added a references.
Added a redirect for final category, and made a couple of tiny additions.
discovered this old entry. Touched the formatting and added cross-links with terminal category.
did a little bit of reorganization. Removed one layer of subn-sections, moved the lead-in paragraphs to before the table of contents, added cross-links to geometry of physics – categories and toposes at the point where the concept of categories appears.
I have cross-linked with Structured Spaces, indicating that Structured Spaces has morphed into Part I of Spectral Algebraic Geometry, I suppose(?) (both of these are category:reference entries for recording the documents of these titles by Jacob Lurie)
added pointer to tensor product as the categorification of multiplication in monoids.
I have added to locally convex topological vector space the standard alternative characterization of continuity of linear functionals by a bound for one of the seminorms: here
(proof and/or more canonical reference should still be added).
I started a new page regular semicategory - to be continued.
brief category:people entry for the purpose of hyperlinking references at topological twist and at topologically twisted D=4 super Yang-Mills theory
brief categor:people-entry for the purpose of hyperlinking references at topological twist
Mentioned the related notion of parametrix which will soon have a separate page, of course.
brief category:people-entry for hyperlinking references at string field theory, gauge enhancement, BFSS matrix model and IKKT matrix model.
created an extemely stubby stub Weiss topology, just to record pointer to that cool fact which Dmitri Pavlov advertised on MO (here).
I have no time to expand on the entry right now. But maybe somebody else here does? Would be worthwhile.
brief category:people-entry for hyperlinking of references at cobordism ring, cohomotpy and Pontrjagin-Thom construction
Looking back at an old Café thread, I see Neil Strickland telling us about Baas-Sullivan theory.
Various comments:
1) Baas-Sullivan theory allows you to start with a cobordism spectrum R and introduce singularities to construct R-module spectra that can be thought of as R/(x1,…,xn), where xi ∈ π*R.
2) This is computationally tractable when the elements xi form a regular sequence. You can construct connective Morava K-theories from complex cobordism this way, for example. You can also get ordinary homology, as the cobordism theory of complexes that are allowed arbitrary singularities of codimension at least two.
3) The original Baas-Sullivan framework is quite technical, and combinatorially complex. It is now easier to use the framework developed in the book by Elmendorf, Kriz, Mandell and May.
4) This procedure always gives R-modules, so if you start with MU (= complex cobordism) or MSO or MO, you will always end up with something complex orientable. In particular, you cannot get tmf or KO or the sphere spectrum from MU.
5) You can get more things if you do cobordism of manifolds with extra structure, such as a spin bundle or framing, for example. It is probably possible to get kO from MSpin. It might even be possible to get tmf from MString.
6) There is a theorem that I think appears in an old book by Buoncristiano, Rourke and Sanderson, showing that any generalised homology theory is a cobordism theory of manifolds with some kind of extra structure and singularities. I don’t think that they were able to given any nonobvious concrete examples other than ordinary homology, and I don’t think that anyone else has managed to go anywhere with this theory. But perhaps it would be worth taking another look.
Let’s see if any passing expert can help with an entry.
Some minimum, to go along with the entry on membrane triple junctions.
Started some minimum. This configuration was an early source of speculation of higher structures in M-theory, since it generalizes the binary string junctions between D-branes that carry gauge fields to something trinary.
I left a brief remark on that.
But for the moment what I wanted to mainly record is the remark that if one does accept the story of the M-theory lift of gauge enhancement on D6-branes, in terms of M2-branes wrapping 2-spheres that touch along the shape of a Dynkin diagram blowup of an ADE-singularity, then such triple membrane junctions must correspond to blowups of the dihedral ADE-singularities as well as the three exceptional cases, since in these cases the Dynkin diagram has a triple junction node.