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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 sheaves 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.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 4th 2010
   • (edited Jun 4th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThe definition over at [[accessible functor]] doesn't agree with the one I read earlier today. Neither the domain nor codomain has to be accessible. We simply require that the domain has all k-filtered colimits and that F preserves them (according to Cisinski Ast308 definition 1.2.2). Is this usage nonstandard? Can I add a note that "Some authors don't require that the domain and codomain be accessible categories." Perhaps we could come up with the following definition: A category C is weakly k-accessible if it has all k-filtered colimits? I don't know if the name would be confusing or not.

   The definition over at accessible functor doesn’t agree with the one I read earlier today. Neither the domain nor codomain has to be accessible. We simply require that the domain has all k-filtered colimits and that F preserves them (according to Cisinski Ast308 definition 1.2.2). Is this usage nonstandard? Can I add a note that “Some authors don’t require that the domain and codomain be accessible categories.” Perhaps we could come up with the following definition: A category C is weakly k-accessible if it has all k-filtered colimits? I don’t know if the name would be confusing or not.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, I think it's standard that this is not standard. :-) Compare also HTT, remark 5.4.2.6. > Can I add a note that "Some authors don't require that the domain and codomain be accessible categories." I suppose it would be good if you did that.

   Yes, I think it’s standard that this is not standard. :-) Compare also HTT, remark 5.4.2.6.

   Can I add a note that “Some authors don’t require that the domain and codomain be accessible categories.”

   I suppose it would be good if you did that.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 4th 2010
   • (edited Jun 4th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexAh, and Cisinski doesn't require that k be regular. Why should we require that?

   Ah, and Cisinski doesn’t require that k be regular. Why should we require that?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 5th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think I remember that if $\kappa$ is not regular, then $\kappa$-filteredness more or less reduces to $\lambda$-filteredness for some $\lambda$ which is regular, so that regularity not really an assumption but a WLOG. Maybe $\lambda = cf(\kappa)$? I can't remember exactly how this goes, though, and I could be remembering wrong. All the applications I've ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

   I think I remember that if $\kappa$ is not regular, then $\kappa$-filteredness more or less reduces to $\lambda$-filteredness for some $\lambda$ which is regular, so that regularity not really an assumption but a WLOG. Maybe $\lambda = cf(\kappa)$? I can’t remember exactly how this goes, though, and I could be remembering wrong.

   All the applications I’ve ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 5th 2010
   • (edited Jun 5th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItex> All the applications I've ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories? To be honest, I don't know. However, he never defines accessible categories in Ast308. What he does define is the "accessible part" (my term, not his) $Acc_\kappa(\mathcal{C})$ to be the full subcategory of $\kappa$-accessible presheaves.

   All the applications I’ve ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

   To be honest, I don’t know. However, he never defines accessible categories in Ast308. What he does define is the “accessible part” (my term, not his) $Acc_\kappa(\mathcal{C})$ to be the full subcategory of $\kappa$-accessible presheaves.