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 definitions 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-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory 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 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.
    • CommentAuthorFosco
    • CommentTimeNov 14th 2013
    • (edited Nov 15th 2013)

    I corrected a couple og microscopic typos at k-ary factorization system, and then I noticed that something is unclear in the definition: first of all the family of factorization system is asked to be strong (= uniqueness of solution to any lifting problem) or weak (existence, no uniqueness)? And when the definition says

    M 1M κ1M_1 \subseteq \dots \subseteq M_{\kappa-1} whenever this is meaningful (equivalently, E k1E 1E_{k-1} \subseteq\dots\subseteq E_{1})

    what does it precisely mean? Are we asking that right classes be nested?

    Thirdly, it is my humble opinion that saying

    A discrete category has a (necessarily unique) (1)(-1)-ary factorisation system.

    is formally incorrect: discrete categories are groupoids where the only arrows are identities, so this is a particular kind of 0-ary factorization system.

    Instead, negative thinking suggests that (-1)-ary factorization systems live in non-unital categories, and detect precisely the case where the class of isomorphisms is empty (recall that in a WFS (L,R)(L,R) the intersection LRL\cap R consists of all isomorphisms; if in a 0-ary factorization system we had L=R=LR=Iso(C)L=R=L\cap R=Iso(\mathbf C), morally in a (-1)-ary system the intersection has to be empty, giving a category without identities -i.e. a particular kind of “plot”, in the jargon of this paper which I finally convinced my friend Salvatore to put on the arXiv-, and more precisely an associative, “strongly nonunital” plot).

    This leads to another question: how can be the notion of (W)FS be extended to Mitchell’s semicategories (with empty or partially defined identity function)?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2013

    Of course, we can consider both orthogonal and weak k-ary factorization systems. Perhaps it would be clearest to write the main entry about orthogonal ones, and then mention later that there is a weak analogue?

    I think “whenever this is meaningful” means “whenever i>0i\gt 0 and i+1<ki+1\lt k” so that M iM_i and M i+1M_{i+1} are defined.

    Toby was the one who wrote the bit about the (-1)-ary case, so maybe he can respond to that. Personally I can’t yet see how to make any sense of the (-1)-ary case.

    • CommentRowNumber3.
    • CommentAuthorFosco
    • CommentTimeNov 15th 2013

    In fact I perfectly [well, more or less…] see how one can define (-1)-ary case, but we have to move to the far more general setting of nonunital (or even nonassociative, partially defined) categories. I think that in some sense this is studied in Salvatore’s work (see in particular Example 9 and Remark 16), since after a short discussion he seconds my sensation.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 15th 2013

    I’m not convinced by the mere argument “in the other cases, the interersection is the isomorphisms, and thus in this case the isomorphisms have to be empty”. The general definition of k-ary factorization system lives on a category; I would need more motivation to be convinced that in the particular case k=1k=-1 we have to change the type of the definition.

    Also, I retract my comment in #2 that we can consider weak k-ary factorization systems. We can consider sequences of nested weak factorization systems, but I don’t think I would call them k-ary factorization systems, since they don’t give rise to unique k-ary factorizations of morphisms.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeNov 15th 2013

    Hmm, maybe I have an idea of what Toby had in mind, but it leads me to think that it’s 0-ary factorization systems that should live on discrete categories. Namely, the point of a k-ary factorization system is that every morphism factors uniquely as a composite of k factors, and a composite of 0 factors is just an identity morphism — while I can’t see how to make any sense of “a composite of (-1) factors”.

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)