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 categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory 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 manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural 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 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.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 9th 2013
    • (edited Oct 9th 2013)

    There is a statement at semigroup which looks wrong: that the functor () +:SemigrpMon(-)^+: Semigrp \to Mon that adjoins an identity element to form a monoid is fully faithful. It’s faithful of course, but the monoid morphism S +T +S^+ \to T^+ that factors through the inclusion {e}T +\{e\} \hookrightarrow T^+ is not of the form f +f^+ for any semigroup morphism f:STf: S \to T.

    The exercise mentioned immediately after at semigroup might need to be re-examined, but it reminds me vaguely of a remark out of Categories, Allegories: the category of rings without unit is equivalent to the category of rings over \mathbb{Z} (consider augmentation ideals), an example “where adding in more structure results in less structure”. But I don’t see how to adapt this example to make it fit for semirings without unit, much less semigroups without unit.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2013

    Yes, you’re right. I see you changed the exercise to “describe this structure”. I think one solution would be to say that a semigroup is a monoid equipped with a set of “nonidentity elements” which contains exactly the elements that are not the identity. That might not be describable in such a clever way as in the ring case, though.

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)