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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
   • CommentTimeSep 3rd 2013
   • (edited Sep 3rd 2013)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAt [[nerve]] there is the sentence: >It is in this sense that a simplicial set that is a Kan complex but which does not necessarily have the above pullback property that makes it a nerve of an ordinary groupoid models an ∞-groupoid. This at present is confusing. A bit of punctuation might help, but as 'pullback property' does not occur earlier on that page, all is not clear. (In fact a 'pullback property' is referred to later!) I have changed it to: >It suggests the sense that a Kan complex models an [[∞-groupoid]]. The possible lack of uniqueness of fillers in general gives the 'weakness' needed, whilst the lack of a [[coskeleton|coskeletal]] property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type. I am not 100% happy with that wording however.

   At nerve there is the sentence:

   It is in this sense that a simplicial set that is a Kan complex but which does not necessarily have the above pullback property that makes it a nerve of an ordinary groupoid models an ∞-groupoid.

   This at present is confusing. A bit of punctuation might help, but as ’pullback property’ does not occur earlier on that page, all is not clear. (In fact a ’pullback property’ is referred to later!) I have changed it to:

   It suggests the sense that a Kan complex models an ∞-groupoid. The possible lack of uniqueness of fillers in general gives the ’weakness’ needed, whilst the lack of a coskeletal property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.

   I am not 100% happy with that wording however.

2. • CommentRowNumber2.
   • CommentAuthorjesuslop
   • CommentTimeNov 24th 2018
   • (edited Nov 24th 2018)
   • PermaLink
   Author: jesuslop
   Format: MarkdownItexHi, I am studying this entry, and have a problem at this point: > The collection of all functors between linear orders > $$ > \{ > 0 \to 1 \to \cdots \to n > \} > \to > \{ > 0 \to 1 \to \cdots \to m > \} > $$ > is **generated** from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they > > * map a single generating morphism to the composite of two generating morphisms > $$ > \delta^n_i : [n-1] \to [n] > $$ > > $$ > \delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1)) > $$ > ... It looks to me that here *generating* is being applied at the same time to the monotone maps between the linear orderings (functors), and also to the arrows of the linear orderings themselves qua posetal categories (perhaps wrongly).

   Hi,

   I am studying this entry, and have a problem at this point:

   The collection of all functors between linear orders

   $$\{ 0 \to 1 \to \cdots \to n \} \to \{ 0 \to 1 \to \cdots \to m \}$$

   is generated from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they

   • map a single generating morphism to the composite of two generating morphisms

     $$\delta^n_i : [n-1] \to [n]$$ $$\delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))$$

     …

   It looks to me that here generating is being applied at the same time to the monotone maps between the linear orderings (functors), and also to the arrows of the linear orderings themselves qua posetal categories (perhaps wrongly).

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 24th 2018
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI think that you are complaining about the poor wording at some places. I find the wording heavy and awkward, but would suggest that you try yourself to improve it. Perhaps breaking that part of the definition up into shorter pieces and moving things around a bit might help. Feel free to do this. If it ends up better .... great, if not, you can always rollback to the previous form.

   I think that you are complaining about the poor wording at some places. I find the wording heavy and awkward, but would suggest that you try yourself to improve it. Perhaps breaking that part of the definition up into shorter pieces and moving things around a bit might help. Feel free to do this. If it ends up better …. great, if not, you can always rollback to the previous form.