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-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric 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 nforum 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 string string-theory 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).
  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMay 17th 2022

    I may not agree that “concrete” is the right word for this.

    I added a query box requesting the anonymous page creator to come discuss at the nForum.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    Author of this page: I’m not entirely sure what a good name for such a model would be, I don’t really like the current name myself. I chose “concrete” in part because in the categorical semantics of the existence of collections of elements in the model category CC is basically the same as a faithful functor U:CSetU:C \to Set, which is the definition of a concrete category, but mostly because I am terrible at coming up with a better name for this model. So if you have any other suggestions for names feel free to list them out.

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    To elaborate on the “concrete” part, the sets of morphisms of any category are already defined as a functor Mor:C×CSetMor:C \times C \to Set, function composition A,B,C:Mor(B,C)×Mor(A,B)Mor(A,C)\circ_{A,B,C}:Mor(B,C) \times Mor(A, B) \to Mor(A,C) is already defined as a family of functions in SetSet, and a concrete category is defined as a category with a functor U:CSetU:C \to Set, so one could append to the concrete category the structure of function evaluation defined as a family of functions ε A,B:Mor(A,B)×U(A)U(B)\epsilon_{A,B}:Mor(A, B) \times U(A) \to U(B) in SetSet satisfying certain axioms.

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    Clarifying the last sentence in my previous comment: many (but not all) concrete categories (Set being one of them, but also Ab, CRing, Top, Met, Diff, et cetera) have a function evaluation structure ε A,B:Mor(A,B)×U(A)U(B)\epsilon_{A,B}:Mor(A,B) \times U(A) \to U(B) as described in the last sentence of the previous comment, which comes from the fact that group homomorphisms/ring homomorphisms/continuous functions/isometries/differentiable functions are functions of sets which could evaluate an element of a domain to return an element in the codomain.

    Of course, in other concrete categories (such as Rel), such a structure doesn’t exist, since relations aren’t functions.

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    I don’t understand post #5; the faithfulness of U:CSetU : C \to Set means that you lose nothing by reinterpreting morphisms ABA \to B of CC as functions U(A)U(B)U(A) \to U(B).

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022
    Ah, you're right, sometimes I keep on forgetting that the functor is faithful.
    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022
    Anyhow, in line with SEAR I suppose I could call the model "SEAF", but it has choice and doesn't have replacement or unbounded separation, so maybe a more proper name for it could be "Bounded SEAFC", but apart from the explicit inclusion of elements and function evaluation, it is pretty much ETCS.
  2. new temporary name, surely there’s a better name out there.

    Anonymous

    diff, v4, current

  3. somehow this more obvious name never crossed my mind until now.

    Anonymous

    diff, v4, current

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)