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 nlab noncommutative noncommutative-geometry number-theory object 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.
   • CommentAuthorbarakat
   • CommentTimeAug 23rd 2014
   • (edited Aug 23rd 2014)
   • PermaLink
   Author: barakat
   Format: MarkdownItex@Urs: I do not quite agree with the sentence "This is unrelated to other notions of monads" in [[Beilinson monad]]. One can indeed view the Beilinson monad as the monad of an adjoint equivalence between $\mathfrak{Coh} \mathbb{P}^n$ (interpreted as the heart of $D^b \mathbb{P}^n$ and some category of linear complexes over an exterior algebra (the Koszul dual of the Cox ring of $\mathbb{P}^n$).

   @Urs: I do not quite agree with the sentence “This is unrelated to other notions of monads” in Beilinson monad.

   One can indeed view the Beilinson monad as the monad of an adjoint equivalence between $\mathfrak{Coh} \mathbb{P}^n$ (interpreted as the heart of $D^b \mathbb{P}^n$ and some category of linear complexes over an exterior algebra (the Koszul dual of the Cox ring of $\mathbb{P}^n$).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexAll right, if there is a useful way to make a relation, would be great if you could add explanation in the entry! But it still seems that Beilinson chose the name "monad" in this case entirely uninfluenced by the category theoretic concept of monad, no? (There are few things which may not be expressed in terms of _some_ monad in _some_ way...)

   All right, if there is a useful way to make a relation, would be great if you could add explanation in the entry!

   But it still seems that Beilinson chose the name “monad” in this case entirely uninfluenced by the category theoretic concept of monad, no?

   (There are few things which may not be expressed in terms of some monad in some way…)

3. • CommentRowNumber3.
   • CommentAuthorbarakat
   • CommentTimeAug 23rd 2014
   • (edited Aug 24th 2014)
   • PermaLink
   Author: barakat
   Format: MarkdownItex(Sorry, I should've just added my comment to http://nforum.mathforge.org/discussion/1563/monad/) I was not referring to the historical origin of the naming in my comment. Beilinson generalized the $3$-term Horrocks monad (a complex with $3$ split vector bundles) introduced in G. Horrocks, Vector bundles on the punctured spectrum of a ring, 1964, Proc. London Math. Soc. (3) 14, 689-713 to a complex with $2n+1$ vector bundles on $\mathbb{P}^n$. So the question is whether Horrocks was influenced by the category theory concept back then (btw Horrocks' papers are full of categorical concepts). I don't know when the word monad was used for the very first time instead of Godement's "standard construction" (Benabou in 1967 is 3 years after Horrocks). What I was referring to is probably what you expressed by saying: (There are few things which may not be expressed in terms of some monad in some way…)

   (Sorry, I should’ve just added my comment to http://nforum.mathforge.org/discussion/1563/monad/)

   I was not referring to the historical origin of the naming in my comment.

   Beilinson generalized the $3$-term Horrocks monad (a complex with $3$ split vector bundles) introduced in

   G. Horrocks, Vector bundles on the punctured spectrum of a ring, 1964, Proc. London Math. Soc. (3) 14, 689-713

   to a complex with $2n+1$ vector bundles on $\mathbb{P}^n$. So the question is whether Horrocks was influenced by the category theory concept back then (btw Horrocks’ papers are full of categorical concepts). I don’t know when the word monad was used for the very first time instead of Godement’s “standard construction” (Benabou in 1967 is 3 years after Horrocks).

   What I was referring to is probably what you expressed by saying: (There are few things which may not be expressed in terms of some monad in some way…)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 24th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay. Whatever useful information, such as above, you have to add to the entry, please do. Whether it relates to monads in the sense of category theory or not!

   Okay. Whatever useful information, such as above, you have to add to the entry, please do. Whether it relates to monads in the sense of category theory or not!

5. • CommentRowNumber5.
   • CommentAuthorbarakat
   • CommentTimeAug 25th 2014
   • PermaLink
   Author: barakat
   Format: MarkdownItexOK, I will, probably next week. I will then need to create at least one additional page about the Tate resolutions/complexes in this context.

   OK, I will, probably next week. I will then need to create at least one additional page about the Tate resolutions/complexes in this context.