Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma 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 foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad 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 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  1. just a personal opinion, which somehow came out reading the thread The nLab in teh eyes of others. something I really like of the nPOV is its being (or at least attempting to be) "socratic". I mean, it tries to answer the question "what is?" rather than "how do I compute?". in my opinion, excellent examples of this are, for instance cohomology or limit in a quasi-category.

    however, in many other cases, the answer to "What is?" is much more something like "Well, it depends... in this context... in this other context..." and the first context to be presented is usually category theory, while one the further context to be considered is infinity-category theory (mainly (infinity,1)-categories. an example of this is limit. in my point of view, this happens because we are still not really adopting nPOV, that is, in most entries an higher category is "something like a category, with some additional structure". but this is not really nPOV; rather it is 1POV on higher structures. and coherently, in 1POV every higher concept has to be described as a generalization of a categorical concept. the nPOV on category theory, should instead rather be something like "a category is a very special kind of infinity-category"; all notions and constructions should be infinity-categorical (or weak-categorical, or whatever we will decide it to be the context in which the theory should be settled) and categorical versions should be specializations of these. an example to explain what I mean could be the following: topological spaces are sets endowed with an additional structure; looking at topological spaces this way is the SetPOV on Top; but one can think of a set as a discrete topological space, and this is the TopPOV on Set. This by no way means that one can speak of topological spaces without having ever introduced the notion of set, but means that in the TopPOV on Set one does not develop Set-constructions (e.g. products, quotients,..) independently of Top-constructions. Rather all constructions in the TopPOV are developed directly in Top, and then specialized to Set.

    coming bact to the nPOV, there are topics which should be looked at as tools for computations, in some sense as basis for vector spaces in linear algebra. maybe a good example of this is the notion of model category, which should move from being a basic structure to be a tool for computations (if I have to work with an (infinity,1)-category, and I'm able to present it in terms of a model category I'm happy..).

    I see this post is too ideological... yet, I'd like to know your opinion on it
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2010
    • (edited Jan 3rd 2010)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <p>After/in vacation I am still not back to full speed. There are a couple of remarks of yours that I would still like to get back to, when I get the time. Here is a quick remark on the above:</p> <blockquote> an example of this is limit. in my point of view, this happens because we are still not really adopting nPOV, that is, in most entries an higher category is "something like a category, with some additional structure". but this is not really nPOV; rather it is 1POV on higher structures. and coherently, in 1POV every higher concept has to be described as a generalization of a categorical concept. the nPOV on category theory, should instead rather be something like "a category is a very special kind of infinity-category"; </blockquote> <p>I see what you mean. We had occasionally some discussion on whether it is wise to start an entry right away with the most abstract nonsense perspective on the subject.</p> <p>I have come to think that good style of an entry is something that roughly follows this pattern:</p> <p>The Idea section should start with mentioning the most basic motivations/examples and then gradually raise the height of the perspective, indicating how one may pass to nPOVs of higher n.</p> <p>With that done well, then the definition section may sensibly go the other way round, and start with stating the most abstract nonsense ooPOV. This should then be unwrapped gradually the other way round, to show how the lower n POVs are included in this.</p> <p>Finally the Examples section reasnoably proceeds in the opposite direction once again and first states the low-brow examples from a low n POV and then gradually comes to discussion of the examples for the full ooPOV.</p> <p>Not many entries currently come close to this ideal structure, of course. But i think eventually we can usefully bring more of them into a form of this sort. You should feel challenged to reorganize for instance <a href="https://ncatlab.org/nlab/show/limit">limit</a> in such a style. (If you agree with this style, of course.)</p> </div>
  2. I absolutely agree with this style: a definition should not come out of nowhere, it should be motivated by basic examples. but then I'd like to see just one simple defnition: if well motivated that could be directly given in the most abstract infinity-nonsense. Then a detailed Examples section should unwrap the definition and contain the infinite zoology of classical examples.

    I'd like to take the challenge of reorganizing limit following this pattern, but since I think the most effective way to so deeply reoganize an entry is to rewrite it from the beginning, and this will need both time and extensive collaboration from the nLab as the entry is developed, I was wondering whether it could be a good idea I create a limit (beta version) which would substitute the original entry only when finished and fixed. clearly the beta version would be canceled after the substitution. is this kind of procedure possible?
    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJan 4th 2010

    Since you have a personal web, you can create your beta version at limit (domenicofiorenza) until you consider it finished and fixed.

  3. good idea! I created it and will start working there soon.
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2010

    sounds good. If and when you include a link to the "beta version" on your personal web to the entry on the main web, it's for all practical purposes as visible as that entry.

  4. I added a link to the beta version (actually still an almost empty page at the moment) on my page on the main web. I also created a subsection "beta versions" where to link all the entries I'll be working on.
    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 6th 2010

    I think I didn't understand what the original post here meant. I think it would be a terrible idea for limit to give only an idea and then start out with a definition for quasi-categories. (In fact, I'd rather that it not discuss quasi-categories at all, put that rather at limit in a quasicategory.) The nPOV doesn't mean that we have to start with the most complicated notions first. Think about all the poor graduate students who are coming to the nLab to learn something about higher category theory, but who don't yet know what a simplicial set is!

    I also disagree with this:

    in most entries an higher category is "something like a category, with some additional structure". but this is not really nPOV; rather it is 1POV on higher structures

    I think that is a perfectly good part of the nPOV. A higher category is like a category, but with more structure; that's just a fact. In most cases, the only way we can get anywhere in higher category theory is by starting with an analogy to 1-categories.