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The link bilimit used to redirect to 2-limit. With this the reader following this might miss the sense of biproducts.
I have now removed the redirects and instead made bilimit a category:disambiguation-page. Hopkins-Lurie suggest to speak of “ambidextrous diagrams” (spaces) instead, which is maybe an option out of the terminology clash.
So finally at ambidextrous adjunction I have added the case of coinciding limits and colimits as an example.
created finite homotopy type, just for completeness.
This just a distraction when I saw that it was missing,while I was really going to create an entry on truncated homotopy types with finite homotopy groups.
The main problem about them is that nobody agrees on how to call them ;-)
In groupoid cardinality they have been called “tame”, some call them -finite,I suppose, and homological algebra suggests “of finite type”, which in itself is good, however rather badly goes together with the crucially different “finite homotopy type”.
Jim Stasheff asked me to give a list of examples of applications of Kan extension in physics.
Since this shouldn’t be hidden away in private email, I have started a section Kan extension – Examples – Kan extension in physics
There will be many examples, two came immediately to mind, and so for the moment I have added there the following, to be expanded:
We list here some occurences of Kan extensions in physics.
Notice that since, by the above discussion, Kan extensions are ubiquitous in category theory and are essentially equivalent to other standard universal constructions such as notably co/limits, to the extenent that there is a relation between category theory and physics at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly.
In extended quantum field theory on open and closed manifolds, usually the theory “in the bulk” (on closed manifolds) is induced by “extending” that “on the boundary”, and in good cases this extension is explicitly a (homotopy)-Kan extension. This is the case notably for 2d TQFT in the form of TCFT (Costello 04), see at TCFT – Classification for details.
When path integral quantization is formalized in terms of fiber integration in generalized cohomology (as surveyed at _motivic quantization) then the push-forward step, hence the path integral itself, is given by left homotopy Kan extension of parameterized spectra. For explicit details see (Nuiten 13, section 4.1), also (Schreiber 14, section 6.2). By example 6.3 there a special case of this is are the integration formulas via Kan extension in (Hopkins-Lurie 14, section 4).
Urs put a stub at equicontinuous function; I moved this to equicontinuous family of functions, added many many other redirects, and expanded it. It’s still basically just definitions, however.
am starting K(n)-local stable homotopy theory .
stub for semiadditive (infinity,1)-category, for the moment just so as to record the pointer.
The link to Joyal’s Catlab discussion at Cisinski model structure leads to a page on his Lab at which the main link points to a non-existent pdf document at Paris 13. We have the correct link at Cisinski model structure, namely to his Toulouse address (http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf). What is the best way to fix this? I do not seem to have access to Joyal’s Lab to be able to edit that.
I had call to link to stable proposition, so I wrote it. It is possible that this should be combined with regular element.
started a bare minimum at modus ponens, as this came up in another thread.
So as to get rid of a grey unattached link, I created a stub for finitely presented group.
I felt that we had too many gray links to metalanguage, so I gave it a try. But I don’t really have the leisure for it now and not the expertise anyway. Experts please feel invited to take apart what I wrote there and replace with it something better.
In looking at pro-categories and prohomotopy, I find statements in the literature to the effect that every constant pro-object is cosmall, and then (Christensen and Isaksen):Every object of every pro-category is κ-cosmall relative to all pro-maps for some κ. The proof that they give seems to me a bit like reinventing the wheel. Isn’t this something like the dual of the arguments used in looking at locally FP categories and Gabriel-Ulmer duality? Their result is used in a lot of the papers on pro-homotopy theories as then these are (very nearly) fibrantly generated.
I need this for my monograph on profinite homotopy, but we have nothing on cosmall objects and the consequences of the cosmall object argument in the nLab, and intend putting a version of it there afterwards.
Does anyone have thoughts on how to present this in the Lab. (I will have to give more (tedious) detail in the monograph as I do not have LFP categories explained anyway.) I also feel that some of the gory detail given more or less categorical folklore, but have not been able to track down enough to be able to pin that down. (It is almost in SGA4 which is online.) I am hindered by not having access to a library as I work from home. (Oh for universal open access!!!!!)
started a bare minimum at Bloch region
New entry special function, extensions to hypergeometric function, Selberg integral. New entries gamma function, recently also Euler beta function.
Stub for elementary function.
brief entry on the idea of world sheets for world sheets.
In the course of creating this I also started a stub for 2d quantum gravity, but clearly more needs to go there.
I was looking at simnplicial topological group and found mention of -cofibration. This is not provided with a link, and a search for the term did not find anything. What is one of these and where is that explained? (It occured to me that it related to the Strom model category structure on , but I could not find it on the relevant page.)
created some minimum at proof net (long requested by string diagram)
There was some confusion on the separator page in the section on strengthened sorts of separator. I’ve attempted to sort it out.
brief entry for amplimorphism
Bas Spitters had mentioned the following article on the HoTT list. While I suppose the conclusion has to be taken with several grains of salt, I found this discussion interesting and illuminating, and have added it now to the references at foundations of mathematics:
Freek Wiedijk, Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics (pdf)
Abstract This paper presents Automath encodings (which also are valid in LF/P) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.
The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church’s higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo’s extended calculus of constructions, and Martin-Löf predicative type theory) and one foundation based on category theory. The conclusions of this paper are that the simplest system is type theory (the calculus of constructions) but that type theories that know about serious mathematics are not simple at all. Set theory is one of the simpler systems too. Higher order logic is the simplest if one looks at the number of concepts (twenty-five) needed to explain the system. On the other side of the scale, category theory is relatively complex, as is Martin-Löf’s type theory.
Is this here an accurate description of what is meant by the words “computational type theory”:
The term computational type theory is often used generally for intuitionistic type theory, referring to its computational content in view of the propositions-as-types and proofs-as-programs interpretation (e.g. Scholarpedia). More specifically it is used for type theory with inductive types and even more specifically (Fairtlough-Mendler 02) for modal type theory, hence for type theory equipped with a monad (in computer science) which exhibits a kind of computation.
?
While writing this reply on Physics.SE I thought to myself that it is curious that this tight relation between the topics in the title here is rarely made explict in introductions.
Then next occurred to me the observation that, unfortunately, not even the Lab did seem to say this. So therefore I have now briefly copied my reply there also to S-matrix – Formalization and to FQFT – Idea – General.
This deserves to be expanded on much further, of course, but at least it’s a start now.
entry for Bub-Clifton theorem with a rough idea section and some references
gave virtual fundamental class an Idea-section (feel free to improve) and added a bunch of pointers to the literature in the References-section
stub for integral transform (also a highly stubby stub for Fourier-Mukai transform)
started stub entry sheaf of L-infinity algebras, but it is still lacking some evident references
Originally I was going to add a comment on how to axiomatize in differential cohesion a sheaf of -algebras over as a pointed object in which is sent by the reduction modality to an identity. But maybe I’ll better do this tomorrow, when I am more awake (or else whenever that happens again).
I have started an entry on rewriting. It is just a stub for the moment.
Added to natural number a discussion about the fact that constructively, the natural numbers may fail to be (order) complete, as highlighted by Andrej Bauer in a very nice blog post. I quite like this example, because by interpreting a related lemma in the internal language of a certain sheaf topos one obtains a well-known proposition in algebraic geometry almost for free (see entry); but please let me know if stuff like this is too localized for the nLab.
created focal point
I typed at local topos in the section Local over-toposes statement and poof that sufficient for a slice topos to be local is that is tiny .
What are necessary conditions? Is this already necessary?
I added a few words to several complex variables, even though I am out of my depth here. If we have analysts popping by here, hopefully they will get an urge to add more.
stub for Anaxagoras’s Nous.
I wrote analytic function, mostly just a definition. I found a reference that treated the infinite-dimensional case in pretty fair generality (slightly more than I actually did) without making the definition any more complicated (well, except one place where one must insert the word ‘continuous’), so I did that.
I mentioned the intermediate value theorem at pentagon decagon hexagon identity and then began an article on it.
created differentiation and chain rule
brief entry complemented lattice, just to satisfy the link at quantum logic
I added a reference to C-star-system. I propose that we change the name of the page to the C-star dynamical system; this is the standard full term, jargon which is skipping dynamical is confusing for an outsider and not explicative. I can imagine many other things which deserve that name.
I have cross-linked de Morgan duality with Wirthmüller context for the statement that in linear logic
Also I have tried to make more of the links in the tables at de Morgan duality point to something.
The convention, when describing ring extensions, everywhere I’ve seen a convention, is that
I have adjusted four instances of former “at” on three pages that would be, algebraicwise, “away from” (and so they now appear).
Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object -enriched category with morphisms “multiply-by”, the localization-of-the-category “at ” (or its -enriched version, if saying that is necessary) really means the localization-of-the-ring “away from ”.
You all can sort out that contravariance as/if you like, but don’t break the old algebra papers!
created a brief entry membrane matrix model with some commented pointers to the literature
Added to field examples of internal fields: the canonical ring objects of the petit resp. gros Zariski toposes of a scheme.
Started an entry on closed morphisms, containing examples and characterizations using the internal language. Then I noticed that an entry on closed map already exists, but at the moment the nLab is too slow for proper browsing and editing. Will finish later and maybe merge the entries.
I repaired the definition of “unramified morphism” of schemes.
I noticed that the two links : André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf), near the bottom of the entry join of quasi-categories are dead. Does anyone have a more recent link? or if not an alternative reference?
have added some brief Idea-section and lists of references to
Starting a stub Fredholm determinant.
I have added the following reference to Berkovich space. Judging from the abstract this sounds like I nice unifying perspective. But I haven’t studied it yet
We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate’s acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.
I wanted to collect some of the stuff recently added to a bunch of chromatic entries in a way that forms an at least semi-coherent story, so I made an entry
This is built mostly from copy-and-pasting stuff that I had added to dedicated entries, equipped with a bit of glue to make it stick together and form a story.
(Special thanks to Marc Hoyois for general discussion and in particular for working on the text on the Lurie spectral sequence.)
I want to further fine-tune this. But not tonight.
created an entry simplicial object in an (infinity,1)-category and interlinked it a bit. Nothing much there yet, for the moment this is mostly a reminder for me to get back to it later.
am starting spectral sequence of a simplicial stable homotopy type, but right now it’s just a stub.
Have expanded the Lurie spectral sequences – table further:
added a tad more content to infinity-Dold-Kan correspondence
stub for Einstein-Hilbert action
as you may have seen in the logs, I am working on an entry Higher toposes of laws of motion, something like extended talk notes.
I am running a bit out of time, and so the entry is unpolished and turns into just a list of keywords towards the end, for the moment. But in case anyone is wondering about the logs, here is the announcement.
Don’t look at this yet if you feel like just reading. Of course if you feel like joining in with the editing a bit, that’s welcome, as usual.
The characterization of formally étale morphisms of schemes by the infinitesimal shape modality had been scattered a bit through the nLab (at Q-category, at formally etale morphism a bit, at differential cohesion a little).
To make the statement more recognizable, I put it into this new entry here:
I split off an entry dg-geometry from the entry on Hochschild geometry, since it really deserves a stand-alone discussion.
Eventually somebody should add the references by Kapranov et al on dg-schemes etc. And much more.
started topos of laws of motion (lower case!) on the actual notion as such.