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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 29th 2011
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   Author: Mike Shulman
   Format: MarkdownItexJust an idle speculation: is there an $(\infty,1)$-version of the notion of [[AT category]]? If so, presumably it would deal with $(\infty,1)$-pretoposes (those satisfying the exactness conditions in the $\infty$-Giraud theorem) replacing pretoposes and stable $(\infty,1)$-categories replacing abelian categories?

   Just an idle speculation: is there an $(\infty,1)$-version of the notion of AT category? If so, presumably it would deal with $(\infty,1)$-pretoposes (those satisfying the exactness conditions in the $\infty$-Giraud theorem) replacing pretoposes and stable $(\infty,1)$-categories replacing abelian categories?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 31st 2011
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   Author: David_Corfield
   Format: MarkdownItexOn the topic of A versus T, is there a relation with the two conditions Lawvere [points to](http://books.google.co.uk/books?id=e0mC9StvaOIC&pg=PA16) in categories with finite products and finite coproducts, namely linear and distributive. He defines a linear category as such a category where the terminal and coterminal object coincide, giving an natural map, an isomorphism between coproduct and product. He also talks about a distributive category being 'normalized' by the topos it generates. Is his use of 'linear' more like our [[additive category]] than our [[algebroid]]? Can an additive category be similarly 'normalized' by the abelian category it generates?

   On the topic of A versus T, is there a relation with the two conditions Lawvere points to in categories with finite products and finite coproducts, namely linear and distributive. He defines a linear category as such a category where the terminal and coterminal object coincide, giving an natural map, an isomorphism between coproduct and product. He also talks about a distributive category being ’normalized’ by the topos it generates.

   Is his use of ’linear’ more like our additive category than our algebroid?

   Can an additive category be similarly ’normalized’ by the abelian category it generates?