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algebrad and additions at Nikolai Durov. The movie starts slow and boring but gets very interesting after a while when the topic develops.
I added the abstract and its translation. Unfortunately, the time limitation made him cut from the most interesting part.
Added query at algebrad.
I guess that's what it must mean, yes.
I added some hyperlinks.
Look at the video yourself, I may have misunderstood the point.
Mike posted a very useful comment. I hope people will gradually extract more definitions from the video.
I corrected slightly the entry.
More corrections.
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 Baezâ€™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).
Interesting. What is gained by working with vectoids rather than, say, locally presentable categories? Are there important vectoids that are not locally presentable?
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.
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