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.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThere is what I think is an incorrect or misleading statement in the Examples section of [[total category]]: that the category of algebras of a monad $T$ on $Set$ is total "because" it is cocomplete (yes), has a generator (yes), and is well-copowered. The last part is incorrect by considering the category of frames as monadic over $Set$. In particular, the free frame on a countable number of generators has a proper class of (isomorphism classes of) epimorphisms coming out of it. See Johnstone's Stone Spaces, page 57 (corollary of 2.10). I do believe it is a theorem that any category which is monadic over $Set$ is total, but the proof is perhaps somewhat non-trivial. (The case of monads with rank is, I believe, unproblematic -- it's the unbounded case which is harder.) If anyone knows of a good proof of that, I'd like to hear; otherwise I'll try to dig it out and write it up for the Lab.

   There is what I think is an incorrect or misleading statement in the Examples section of total category: that the category of algebras of a monad $T$ on $Set$ is total “because” it is cocomplete (yes), has a generator (yes), and is well-copowered. The last part is incorrect by considering the category of frames as monadic over $Set$. In particular, the free frame on a countable number of generators has a proper class of (isomorphism classes of) epimorphisms coming out of it. See Johnstone’s Stone Spaces, page 57 (corollary of 2.10).

   I do believe it is a theorem that any category which is monadic over $Set$ is total, but the proof is perhaps somewhat non-trivial. (The case of monads with rank is, I believe, unproblematic – it’s the unbounded case which is harder.) If anyone knows of a good proof of that, I’d like to hear; otherwise I’ll try to dig it out and write it up for the Lab.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThere is a proof for categories monadic over Set in Tholen's paper, referenced in the entry, which goes via [[solid functors]]. I've corrected the statement at [[total category]] (I think).

   There is a proof for categories monadic over Set in Tholen’s paper, referenced in the entry, which goes via solid functors. I’ve corrected the statement at total category (I think).

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 29th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks, Mike. I ought to look at that paper.

   Thanks, Mike. I ought to look at that paper.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 2nd 2011
   • (edited Aug 2nd 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe examples section at [[total category]] still looks suspect to me. The edit by Mike looks fine, but the assertion > Any cocomplete, well-copowered category with a generator is total looks dubious to me (even though I don't have a counterexample). I am guessing that what was intended was a corollary of something from Tholen's paper (referenced in [[total category]]) that I *can* see is true: * Any cocompact category with a generating set is total. By definition, a category $C$ is "cocompact" if any functor $G: C \to A$ that is co-admissible (i.e., $A(a, G c)$ is small for any $a \in Ob(A)$, $c \in Ob(C)$), and that preserves any limit which exists in $C$, has a left adjoint. If $C$ has a generating set, then it's not too hard to see that $y: C \to Set^{C^{op}}$ is co-admissible, and of course $y$ preserves any limit that exists in $C$, so under these conditions $y$ would have a left adjoint, i.e., $C$ is total. The special adjoint functor theorem gives conditions under which a category is cocompact: * A locally small category that is complete, well-powered, and has a cogenerating set is cocompact. These conditions are dual to those of the assertion above. So those hypotheses give us that the category is compact. But it is not true that a compact category with a generator need be total. If there's something that I'm missing here, I hope someone can tell me. An example which refutes the iffy assertion would be wonderful. As would a proof of the assertion, but as I say I have my doubts. I'll wait a bit for a response, before I undertake to rewrite the page (which I am more or less prepared to do), and correct some other errors on the Lab that have emanated from the iffy assertion. <b>Edit:</b> The latex is not rendering correctly. To see it, I guess click on Source, because I can't figure out what's wrong. Maybe something to do with the server upgrade?

   The examples section at total category still looks suspect to me. The edit by Mike looks fine, but the assertion

   Any cocomplete, well-copowered category with a generator is total

   looks dubious to me (even though I don’t have a counterexample). I am guessing that what was intended was a corollary of something from Tholen’s paper (referenced in total category) that I can see is true:

   • Any cocompact category with a generating set is total.

   By definition, a category $C$ is “cocompact” if any functor $G: C \to A$ that is co-admissible (i.e., $A(a, G c)$ is small for any $a \in Ob(A)$, $c \in Ob(C)$), and that preserves any limit which exists in $C$, has a left adjoint. If $C$ has a generating set, then it’s not too hard to see that $y: C \to Set^{C^{op}}$ is co-admissible, and of course $y$ preserves any limit that exists in $C$, so under these conditions $y$ would have a left adjoint, i.e., $C$ is total.

   The special adjoint functor theorem gives conditions under which a category is cocompact:

   • A locally small category that is complete, well-powered, and has a cogenerating set is cocompact.

   These conditions are dual to those of the assertion above. So those hypotheses give us that the category is compact. But it is not true that a compact category with a generator need be total.

   If there’s something that I’m missing here, I hope someone can tell me. An example which refutes the iffy assertion would be wonderful. As would a proof of the assertion, but as I say I have my doubts. I’ll wait a bit for a response, before I undertake to rewrite the page (which I am more or less prepared to do), and correct some other errors on the Lab that have emanated from the iffy assertion.

   Edit: The latex is not rendering correctly. To see it, I guess click on Source, because I can’t figure out what’s wrong. Maybe something to do with the server upgrade?

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 3rd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI believe that assertion is proven in Day's short paper "Further criteria for totality", in Cahiers [here](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1987__28_1_77_0).

   I believe that assertion is proven in Day’s short paper “Further criteria for totality”, in Cahiers here.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 3rd 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexWow. Thanks! (How did you happen to know of this paper?)

   Wow. Thanks! (How did you happen to know of this paper?)

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeAug 3rd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWell, I think I was the one who originally added that comment to [[total category]] based on having read that paper. I should have included the reference then, of course. As for how I found the paper originally, a while back I wanted to know all I could about total categories, so I read all the papers I could find. I have no memory of how I found that particular paper.

   Well, I think I was the one who originally added that comment to total category based on having read that paper. I should have included the reference then, of course. As for how I found the paper originally, a while back I wanted to know all I could about total categories, so I read all the papers I could find. I have no memory of how I found that particular paper.