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.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexPlease prevent me from making a silly mistake on a basic fact (if this is the case). To fix notations, let $N$ the nerve functor from small categories to simplicial sets and $|\,|$ the topological realization functor from simplicial sets to topological spaces. Then a functor $F:\mathcal{C}\to \mathcal{D}$ induces a continuous function $|N(F)|:|N(\mathcal{C})|\to |N(\mathcal{D})|$. Now, my doubt: a natural transformation $\alpha: F\Rightarrow G$ induces a homotopy between $|N(F)|$ and $|N(G)|$ regardless of the fact that $\alpha$ is an isomorphism or not, right?

   Please prevent me from making a silly mistake on a basic fact (if this is the case).

   To fix notations, let $N$ the nerve functor from small categories to simplicial sets and $|\,|$ the topological realization functor from simplicial sets to topological spaces. Then a functor $F:\mathcal{C}\to \mathcal{D}$ induces a continuous function $|N(F)|:|N(\mathcal{C})|\to |N(\mathcal{D})|$.

   Now, my doubt: a natural transformation $\alpha: F\Rightarrow G$ induces a homotopy between $|N(F)|$ and $|N(G)|$ regardless of the fact that $\alpha$ is an isomorphism or not, right?

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexRight, because $N$ doesn't know the difference between a category and its free groupoid.

   Right, because $N$ doesn’t know the difference between a category and its free groupoid.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItex2: After the realization it does not. But $N$ is fully faithful so it does. Are we speaking the same thing ?

   2: After the realization it does not. But $N$ is fully faithful so it does. Are we speaking the same thing ?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexDomenico, you are correct. But Zoran is also right that $N$ can tell the difference between a category and its free groupoid, and this also remains true after realization: the realization of the nerve of a groupoid is always a 1-type, whereas the realization of a nerve of a category can be any weak homotopy type. But maybe Toby meant to say "...and its free $\infty$-groupoid".

   Domenico, you are correct. But Zoran is also right that $N$ can tell the difference between a category and its free groupoid, and this also remains true after realization: the realization of the nerve of a groupoid is always a 1-type, whereas the realization of a nerve of a category can be any weak homotopy type. But maybe Toby meant to say “…and its free $\infty$-groupoid”.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexDomenico, just to add one aspect: this is an important tool for producing homotopies. In particular it means that every adjunction is sent by the nerve to a homotopy equivalence. That's a strong tool for producing homotopy equivalences. It's the theme of Quillen's theorems A and B, see at [[geometric realization of categories]].

   Domenico,

   just to add one aspect: this is an important tool for producing homotopies. In particular it means that every adjunction is sent by the nerve to a homotopy equivalence. That’s a strong tool for producing homotopy equivalences. It’s the theme of Quillen’s theorems A and B, see at geometric realization of categories.

6. • CommentRowNumber6.
   • CommentAuthordomenico_fiorenza
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexphew! :-) thanks!

   phew! :-)

   thanks!

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeMay 21st 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexNever mind what I said.

   Never mind what I said.