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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.
   • CommentAuthorzskoda
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: zskoda
   Format: Markdown[[algebrad]] and additions at [[Nikolai Durov]]. The movie starts slow and boring but gets very interesting after a while when the topic develops.

   algebrad and additions at Nikolai Durov. The movie starts slow and boring but gets very interesting after a while when the topic develops.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeApr 7th 2010
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   Author: zskoda
   Format: MarkdownI added the abstract and its translation. Unfortunately, the time limitation made him cut from the most interesting part.

   I added the abstract and its translation. Unfortunately, the time limitation made him cut from the most interesting part.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownAdded query at [[algebrad]].

   Added query at algebrad.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownI guess that's what it must mean, yes. I added some hyperlinks.

   I guess that's what it must mean, yes.

   I added some hyperlinks.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownLook at the video yourself, I may have misunderstood the point.

   Look at the video yourself, I may have misunderstood the point.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeApr 8th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownMike posted a very useful comment. I hope people will gradually extract more definitions from the video.

   Mike posted a very useful comment. I hope people will gradually extract more definitions from the video.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeApr 11th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI corrected slightly the entry.

   I corrected slightly the entry.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeApr 11th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexMore corrections.

   More corrections.

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeJan 7th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI read more of the manuscript of Durov in Russian recently. I have an impression (I am in a hurry right now, but hope to be able to explain that soon) that the fact that the algebrads, i.e. monads in the 2-category of vectoids generalize symmetric operads, non-sigma operads and alike is sort of a manifestation of an idea which is very close in spirit to John Baez's microcosmos-macrocosmos principle. Namely, one uses the cocontinuity of the functor and aprpopriate completeness axiom to get that classification of an object with some kind of structure is reflected in the structure of finite approximations which determine the algebrad. This way one gets series of operations, satisfying some relations, depending which kind of algebaric structures the vectoid classifies. To say it differently, the main examples come from classifying vectoids, they typically classify objects with certain structure. Then the endocell respects this and reads the finite-level structure from there. This gives a combinatorics similar to the combinatorics of operads (category of species for example is an intermediate step in the case of usual operads which come from a classifier for objects).

   I read more of the manuscript of Durov in Russian recently. I have an impression (I am in a hurry right now, but hope to be able to explain that soon) that the fact that the algebrads, i.e. monads in the 2-category of vectoids generalize symmetric operads, non-sigma operads and alike is sort of a manifestation of an idea which is very close in spirit to John Baez’s microcosmos-macrocosmos principle. Namely, one uses the cocontinuity of the functor and aprpopriate completeness axiom to get that classification of an object with some kind of structure is reflected in the structure of finite approximations which determine the algebrad. This way one gets series of operations, satisfying some relations, depending which kind of algebaric structures the vectoid classifies. To say it differently, the main examples come from classifying vectoids, they typically classify objects with certain structure. Then the endocell respects this and reads the finite-level structure from there. This gives a combinatorics similar to the combinatorics of operads (category of species for example is an intermediate step in the case of usual operads which come from a classifier for objects).

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 13th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexInteresting. What is gained by working with vectoids rather than, say, locally presentable categories? Are there important vectoids that are not locally presentable?

    Interesting. What is gained by working with vectoids rather than, say, locally presentable categories? Are there important vectoids that are not locally presentable?

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJan 14th 2011
    • PermaLink
    Author: zskoda
    Format: MarkdownItexRight, this is a good question and I was just trying last about a week to equip myself with understanding of such variants. Hopefully I will be able to answer your question soon.

    Right, this is a good question and I was just trying last about a week to equip myself with understanding of such variants. Hopefully I will be able to answer your question soon.