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 definitions 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 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.
   • CommentAuthorMike Shulman
   • CommentTimeSep 5th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated [[arity class]]. Added links from a few places, but there are probably others I didn't think of.

   Created arity class. Added links from a few places, but there are probably others I didn’t think of.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 6th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexExcellent term! Is it yours?

   Excellent term! Is it yours?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeSep 6th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes. I'm glad you like it!

   Yes. I’m glad you like it!

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeSep 6th 2012
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIs the full subcategory spanned by a class of arities always also a category of arities for the identity monad, in the sense of a [[monad with arities]]? This is certainly true for $\{ 1 \}$ (for trivial reasons), $\aleph_0$ (because filtered colimits preserve finite products), and $\infty$ (because the diagram in question has a terminal vertex). But I'm less certain about the other regular cardinals, especially the finite ones...

   Is the full subcategory spanned by a class of arities always also a category of arities for the identity monad, in the sense of a monad with arities? This is certainly true for $\{ 1 \}$ (for trivial reasons), $\aleph_0$ (because filtered colimits preserve finite products), and $\infty$ (because the diagram in question has a terminal vertex). But I’m less certain about the other regular cardinals, especially the finite ones…

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeSep 6th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexGood question! Let's see, $Set$ is locally $\kappa$-presentable for any infinite regular cardinal $\kappa$, so the $\kappa$-small sets are $\kappa$-presentable and the associated canonical colimits are $\kappa$-filtered. Seems like that's exactly what we need to conclude that the identity monad has the appropriate arities. For the unique finite regular cardinal $2$, the $2$-small sets are 0 and 1, so we only need to worry about whether mapping out of 0 preserves the canonical $2$-colimits. Being 2-filtered means being inhabited, but mapping out of 0 doesn't preserve inhabited colimits (e.g. coproducts). It does, however, preserve connected colimits, and any canonical colimit w.r.t. 2 is connected (the diagram has an initial object). So I think we're good there too. That suggests that maybe there's some sense in which "$2$-filtered" ought to mean "connected"? I can't see how that could be true other than by convention, though. Maybe I'm just confused.

   Good question! Let’s see, $Set$ is locally $\kappa$-presentable for any infinite regular cardinal $\kappa$, so the $\kappa$-small sets are $\kappa$-presentable and the associated canonical colimits are $\kappa$-filtered. Seems like that’s exactly what we need to conclude that the identity monad has the appropriate arities.

   For the unique finite regular cardinal $2$, the $2$-small sets are 0 and 1, so we only need to worry about whether mapping out of 0 preserves the canonical $2$-colimits. Being 2-filtered means being inhabited, but mapping out of 0 doesn’t preserve inhabited colimits (e.g. coproducts). It does, however, preserve connected colimits, and any canonical colimit w.r.t. 2 is connected (the diagram has an initial object). So I think we’re good there too.

   That suggests that maybe there’s some sense in which “$2$-filtered” ought to mean “connected”? I can’t see how that could be true other than by convention, though. Maybe I’m just confused.

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeSep 7th 2012
   • (edited Sep 7th 2012)
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexI'm quite confused as well: as far as I can tell, the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic... (Assuming I haven't made any mistakes, the density of $\Theta \hookrightarrow \mathcal{C}^\mathbb{T}$ can be proven with weaker assumptions.) Regarding 2-filteredness: if we (re)define "small $\kappa$-filtered category $\mathcal{J}$" to mean "$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$", then a 2-filtered category is precisely a connected inhabited category. Since $\kappa$-filtered categories in the traditional sense are connected for $\kappa \ge \aleph_0$, this doesn't seem to be too much to ask for...

   I’m quite confused as well: as far as I can tell, the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic… (Assuming I haven’t made any mistakes, the density of $\Theta \hookrightarrow \mathcal{C}^\mathbb{T}$ can be proven with weaker assumptions.)

   Regarding 2-filteredness: if we (re)define “small $\kappa$-filtered category $\mathcal{J}$” to mean “$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$”, then a 2-filtered category is precisely a connected inhabited category. Since $\kappa$-filtered categories in the traditional sense are connected for $\kappa \ge \aleph_0$, this doesn’t seem to be too much to ask for…

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeSep 7th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic... Well, sure, but that's separate to the question of whether the arity class gives arities to the identity monad for the formal definition of "has arities", right? It's just saying that in the special case of Id, the formal definition of "has arities" is stronger than necessary for the conclusion of the main general theorem about it to hold. > if we (re)define "small $\kappa$-filtered category $\mathcal{J}$" to mean "$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$", then a 2-filtered category is precisely a connected inhabited category. Ah, yes!! And that's the right definition, too, because it [generalizes even further](http://maths.mq.edu.au/~slack/papers/acc.html). (Small point: a connected category is automatically inhabited.)

   the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic…

   Well, sure, but that’s separate to the question of whether the arity class gives arities to the identity monad for the formal definition of “has arities”, right? It’s just saying that in the special case of Id, the formal definition of “has arities” is stronger than necessary for the conclusion of the main general theorem about it to hold.

   if we (re)define “small $\kappa$-filtered category $\mathcal{J}$” to mean “$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$”, then a 2-filtered category is precisely a connected inhabited category.

   Ah, yes!! And that’s the right definition, too, because it generalizes even further.

   (Small point: a connected category is automatically inhabited.)

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 7th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex> a connected category is automatically inhabited Smile. I expect Zhen was just being on the safe side, since there remain some people who think empty things are connected. :-)

   a connected category is automatically inhabited

   Smile. I expect Zhen was just being on the safe side, since there remain some people who think empty things are connected. :-)

9. • CommentRowNumber9.
   • CommentAuthorvarkor
   • CommentTimeApr 21st 2023
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdd a reference. <a href="https://ncatlab.org/nlab/revision/diff/arity+class/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/arity+class/8">v8</a>, <a href="https://ncatlab.org/nlab/show/arity+class">current</a>

   Add a reference.

   diff, v8, current