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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..
This is a table-for-inclusion into other entries (hence not a stand-alone entry) for convenience of cross-linking the varios flavours of geometry that appear in geometry of physics.
Will also be creating the relevant missing pages now, first as stubs, then they will grow.
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.