Not signed in (Sign In)

Start a new discussion

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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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).
    • CommentRowNumber1.
    • CommentAuthorAlizter
    • CommentTimeOct 11th 2018

    The HoTT book establishes a way to “informally” discuss HoTT. This is not entirely obvious to outsiders that this is a well defined notion. I would like to create an article, where I can point readers if they are feeling unsure about the informal language. I will probably call it something like “informal HoTT”.

    I am unaware of any literature arguing that this is a good idea (apart from the HoTT book).

    This came up whilst looking at the article functor (homotopytypetheory), I think Urs had added the formal definition. Now I would argue that this is completely pointless because the informal language can be “converted” directly into the formal notion. Not to mention that this whole philosophy is completely irrelevent to someone looking up the definition of a functor in HoTT.

    I would like to hear your opinions on how we might go about this.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 11th 2018

    There’s plenty of literature that follows the HoTT Book in using informal type theory. I don’t know that anyone else has bothered to present an argument that it’s a good idea; I don’t know that anyone has ever disagreed. It could be useful to summarize the informal \to formal translation on a wiki page (although it’s described in the book as well). But I don’t think there’s any need to be antagonistic or dogmatic about it: there’s a long distance between “technically unnecessary” and “completely pointless”. For newcomers it can be helpful to see the translation spelled out in multiple examples (the book does this sometimes too), and even for experts it can save some work.

  1. Out of curiosity: if informal HoTT is a well defined notion that can directly be converted to formal HoTT, would that mean that there is a formalization of informal HoTT that is different from the formalization of HoTT in the appendix of the book? If so I'd find it interesting to read more about it.
    • CommentRowNumber4.
    • CommentAuthorAlizter
    • CommentTimeOct 12th 2018

    @Michael_Bachtold I don’t think it would be different. The “informal” language simply converts into the symbols of the “formal” language. But the correspondance is quite direct.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 12th 2018

    I don’t think that “informal HoTT” is “well-defined” in the sense of a mathematical definition. By definition, informal mathematics cannot be formally defined, otherwise it would be formal mathematics. But one can explain it with English words and by giving examples.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)