Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeJan 14th 2015
   • (edited Jan 14th 2015)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAt [[bounded chain complex ]] should the indexation be for $n\in \mathbb{Z}$ rather than $n\in \mathbb{N}$? This makes no difference for bounded above, but does for bounded below. In fact at [[chain complex]] only complexes graded by the natural numbers are initially considered, yet in many places later on one needs them being $\mathbb{Z}$-graded otherwise the suspension / delooping set up gets very messy. I suggest all should be $\mathbb{Z}$-graded with a note added that $\mathbb{N}$-graded ones form a category equivalent to the full subcategory of $Ch(\mathbf{A})$ given by the non-negatively graded ones.

   At bounded chain complex should the indexation be for $n\in \mathbb{Z}$ rather than $n\in \mathbb{N}$? This makes no difference for bounded above, but does for bounded below.

   In fact at chain complex only complexes graded by the natural numbers are initially considered, yet in many places later on one needs them being $\mathbb{Z}$-graded otherwise the suspension / delooping set up gets very messy. I suggest all should be $\mathbb{Z}$-graded with a note added that $\mathbb{N}$-graded ones form a category equivalent to the full subcategory of $Ch(\mathbf{A})$ given by the non-negatively graded ones.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 15th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> At [[bounded chain complex ]] should the indexation be for $n\in \mathbb{Z}$ rather than $n\in \mathbb{N}$? Yes, that seems right. > In fact at [[chain complex]] only complexes graded by the natural numbers are initially considered, yet in many places later on one needs them being $\mathbb{Z}$-graded It looks to me like the actual "Definition" section takes them to be $\mathbb{Z}$-graded, although the previous "Idea" section had mentioned onl the $\mathbb{N}$-graded ones (presumably so as to be able to refer to Dold-Kan).

   At bounded chain complex should the indexation be for $n\in \mathbb{Z}$ rather than $n\in \mathbb{N}$?

   Yes, that seems right.

   In fact at chain complex only complexes graded by the natural numbers are initially considered, yet in many places later on one needs them being $\mathbb{Z}$-graded

   It looks to me like the actual “Definition” section takes them to be $\mathbb{Z}$-graded, although the previous “Idea” section had mentioned onl the $\mathbb{N}$-graded ones (presumably so as to be able to refer to Dold-Kan).

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeJan 15th 2015
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI have adjusted the wording a bit to straighten this out.

   I have adjusted the wording a bit to straighten this out.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 15th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks!

   Thanks!