Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Added this pointer:
An early discussion of automata via string diagrams in the Cartesian monoidal category of finite sets:
- Günter Hotz, Eine Algebraisierung des Syntheseproblems von Schaltkreisen, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 (part I, part II, compressed and merged pdf)
incuding that new compressed and merged pdf which I produced, as a service to the community.
created Frobenius monoidal functor
Linked to from https://ncatlab.org/nlab/show/orthogonal+factorization+system and https://ncatlab.org/nlab/show/final+functor
Todo: add more proofs of this result.
For some reason the xymatrixes were causing errors so I had to comment them out to submit. Here is an example error:
An error occurred when running pdflatex on the following diagram. \xymatrix@=5em{e \ar[r]^\gamma \ar[dr]_{\gamma’} & GFc \ar[d]^{Gf} \ & GFc’} The error was: Timed out
How can I fix this?
I added to initial object the theorem characterizing initial objects in terms of cones over the identity functor.
I am giving this bare list of references its own entry, so that it may be !include
-ed into related entries (such as topological quantum computation, anyon and Chern-Simons theory but maybe also elsewhere) for ease of updating and synchronizing
added to polynomial functor the evident but previously missing remark why it is called a “polynomial”, here.
the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).
Stub Makkai duality, just recording the most basic references so far; linked from Stone duality.
added pointer to:
am too tired to do it now, but on occasion of an MO discussion:
remind me to insert at smooth manifold the statement and proof that smooth manifolds are equivalently the locally representable sheaves on CartSp (more precisely: the -schemes).
created supergravity
so far just an "Idea" section and a link to D'Auria-Fre formulation of supergravity (which i am busy working on)
I worked on synthetic differential geometry:
I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.
Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.
brief category:people
-entry for hyperlinking references at symmetric function and Schur polynomial
changed higher algebra - contents to algebra - contents in context sidebar
Anonymouse
I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:
Currently it contains the following table, to be further fine-tuned. Comments are welcome.
| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects | |–|–|–|–|–|—-|–|–| | (0,1)-category theory | (0,1)-toposes | | algebraic lattices | Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | | | category theory | toposes | | locally presentable categories | Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | | accessible categories | | model category theory | model toposes | | combinatorial model categories | Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | | | (∞,1)-topos theory | (∞,1)-toposes | | locally presentable (∞,1)-categories | <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | |accessible (∞,1)-categories |
have cleared this entry (formerly “semi-locally simply connected topological space”), since its content has beenmerged into semi-locally simply-connected topological space, following discussion there
added pointer to:
added pointer to
on formal proof and proof assistants in undergaduate mathematics courses
will add this also to formal proof and proof assistant
Stub Frobenius reciprocity.
created website-link page Denis-Charles Cisinski
At enriched category it uses to say that
A Top-enriched category is a topological category.
But then at topological category, which redirects to topological concrete category it says that it
does not mean Top-enriched category.
Of course for many people it does. But to get the Lab entries straight, and to go along with the entry simplicially enriched category, I started an entry
just for completeness (and since I need the material elsewhere).
cleared this entry (which was an accidental duplicate of relative (infinity,1)-limit, as noticed here)
started Brauer group, collecting some references on the statement that/when and moved notes from a talk by David Gepner on -Brauer groups to there.
Created:
Maharam’s theorem states a complete classification of isomorphism classes of the appropriate category of measurable spaces.
In the σ-finite case, the theorem classifies measure spaces up to an isomorphism. Here an isomorphism is an equivalence class of measurable bijections with measurable inverse such that and preserve measure 0 sets.
As explained in the article categories of measure theory, for a truly general, unrestricted statement for non-σ-finite spaces there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and σ-finiteness should be relaxed to a combination of Marczewski-compactness and strict localizibility.
In this unrestricted form, by the Gelfand-type duality for commutative von Neumann algebras, Maharam’s theorem also classifies isomorphism classes of localizable Boolean algebras, abelian von Neumann algebras, and hyperstonean spaces (or hyperstonean locales).
Every object in one of the above equivalent categories canonically decomposes as a coproduct (disjoint union) of ergodic objects. Here an object is ergodic if the only subobjects of invariant under all automorphisms of are and itself.
Furthermore, an ergodic object is (noncanically, using the axiom of choice) isomorphic to , where is 0 or infinite, and is infinite if is infinite. Here the cardinal is known as the cellularity of and is its Maharam type.
In particular, if , we get a classification of isomorphism classes of atomic measure spaces: they are classified by the cardinality of their set of atoms.
Otherwise, is infinite, and we get a classification of isomorphism classes of ergodic atomless (or diffuse) measure spaces: such spaces are isomorphic to , where and are infinite cardinals.
Thus, a completely general object has the form
where runs over 0 and all infinite cardinals, is a cardinal that is infinite or 0 if , and only for a set of .
The original reference is
A modern exposition can be found in Chapter 33 (Volume 3, Part I) of
I created Bishop’s constructive mathematics by moving some material from Errett Bishop and adding some more discussion of what it is and isn’t. Comments and suggestions are very welcome; I’m still trying to figure out the best way to describe the relationship of this theory to other things like topos logic.
I am starting something at six operations.
(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)
Have added more of the original (“historical”) References with brief comments and further pointers.
(not an edit but to create the forum thread) Is the characterization in As an 11-dimensional boundary condition for the M2-brane complete or does one need to further extend by the m5 cocycle?
am starting differential string structure, but not much there yet
the entry group algebra had been full of notation mismatch and also of typos. I have reworked it now.