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 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 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 limit 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 subobject 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2012

    moving the following old discussion from truth value to here

    +–{.query} Urs: Can’t this be motivated more systematically?

    Mike: There’s a funny weirdness that happens at the bottom with the relationship between sets and posets. I don’t completely understand it, nor do I know any reason why “we should expect” the 00-category of (1)(-1)-groupoids to be a (0,1)(0,1)-category, since the 1-category of 0-groupoids (sets) is not a (1,2)(1,2)-category (except trivially). On the other hand, the 1-category of (0,1)(0,1)-categories is a (1,2)(1,2)-category, so maybe the point is that every (1)(-1)-groupoid is actually a (1,0)(-1,0)-category?

    I don’t really understand why we have separate pages for (1)(-1)-groupoids and (1)(-1)-categories; has anyone ever come up with a way in which they are different?

    Toby: There are separate pages truth value, (-1)-groupoid, (-1)-category and (eventually) 0-poset. Partly it's that (-1)-category was written first, although it's the worst name. But I think that it will be good, given the nature of this wiki, to have pages organised more by term than subject (I will write a bit about this on the Café). But this means that the pages will have different purposes.

    As truth value is the most mundane and ordinary name for the concept, that page will have the most content. There we should discuss what a truth value is in a topos or other sort of category, or what it is to a constructivist or other sort of alternative mathematician; there we should discuss how truth values form a poset, and what structures and properties this poset has; there we should talk about what categories and groupoids enriched over this poset (with either of the two obvious monoidal structures) are.

    At (-1)-category, which exists only because we like nn-categories so much and so wonder what an (1)(-1)-category is, we explain why the concept is a little fishy but the best way to define it is as a truth value. And there we show how this fits the patterns of the periodic table to the extent that it does, and also poitn out those cases where it doesn't.

    At (-1)-groupoid, which again exists because that's our special thing, we explain why a (1)(-1)-groupoid is a truth value, and how that fits the patterns of the periodic table.

    At 0-poset, we'll do exactly the same thing, only thinking about nn-posets rather than nn-groupoids.

    (If this were Wikipedia, we would not organise things this way. I'll discuss that at the Café.)

    In particular, I think that general facts about internalising the concept to categories other than Set\Set work best at truth value, while they would really fit in here only if we wanted to explain how that reproduces (or, conceivably, turns out to be different from) a general method for internalising the concept of nn-groupoid. (Probably we will eventually want to talk about that, but as far as I know there's nothing to say about it now.)

    By the way, here is the answer to Mike's question about what “we should expect”: the \infty-category of (n,r)(n,r)-categories is an (n+1,r+1)(n+1,r+1)-category (which I should discuss on (n,r)-category but have not); accordinly, the \infty-category of nn-groupoids is an (n,1)(n,1)-category. In particular, the \infty-category of (1)(-1)-groupoids is a (0,1)(0,1)-category. =–

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 10th 2012

    Have the punchlines from that discussion been extracted and recorded at the entry? I don’t have time to study this now, but at a glance the discussion looks interesting and deserving of further attention.

  1. Adding a redirect for boolean. Probably should be made into its own page eventually.

    diff, v19, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 30th 2020

    What should be made into its own page? I guess you mean, not the adjective ’boolean’, but “Boolean truth value”?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 30th 2020

    ’Boolean’ is used a noun in programming to mean something of type Bool. I guess this is what Richard means.

    • CommentRowNumber6.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 30th 2020
    • (edited Apr 30th 2020)

    Yes, exactly. Programming languages often have a type called ’bool’ or ’boolean’, and one says things like a method ’returns a boolean’. I think we could have a page called ’boolean type’ or something.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 30th 2020

    Oh, oh, yes of course. Thanks.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2023

    added missing cross-link with boolean domain

    But maybe these entries need to be merged. Currently lattice of truth values and Boolean and variants are redirecting to truth value, while strictly speaking they refer to the type of truth values, hence to boolean domain.

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2023

    Right, the text in this entry definitely wants it to be about the truth values as such (-1-truncated types), not about the type they form. So hereby I am turning at least the following former redirects

      [[!redirects boolean]]
      [[!redirects booleans]]
      [[!redirects Boolean]]
      [[!redirects Booleans]]

    to point instead to boolean domain (as suggested already in comment #3 above)

    diff, v23, current

    • CommentRowNumber10.
    • CommentAuthorYasuaki MORITA
    • CommentTimeNov 10th 2023
    The article says:
    "It is also a complete lattice; in fact, it can be characterised as the initial complete lattice. As a complete Heyting algebra, it is a frame, corresponding to the one-point locale"

    I would like to know the meaning of complete here.