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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 26th 2014
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   Author: Dmitri Pavlov
   Format: MarkdownItexWolff proved that the category of small V-enriched categories is monadic over the category of V-graphs, where V is a cocomplete closed symmetric monoidal category, see Theorem 2.13 in “V-cat and V-graph”. Kelly and Lack proved that the underlying monad is finitely accessible, i.e., preserves filtered colimits, see Theorem 3.3 in “V-Cat is locally presentable or locally bounded if V is so”. I wonder if this monad is also finitely sifted-accessible, i.e., preserves sifted colimits. The category of V-graphs is cocomplete and by the above we already know that filtered colimits are preserved, thus the question boils down to the preservation of reflexive coequalizers. By Theorem 4.5 in op. cit. the category V-cat is locally finitely presentable if V is so. If the above conjecture is true, then V-Cat is in fact an algebraic category, i.e., the category of algebras over some (multisorted) algebraic theory.

   Wolff proved that the category of small V-enriched categories is monadic over the category of V-graphs, where V is a cocomplete closed symmetric monoidal category, see Theorem 2.13 in “V-cat and V-graph”.

   Kelly and Lack proved that the underlying monad is finitely accessible, i.e., preserves filtered colimits, see Theorem 3.3 in “V-Cat is locally presentable or locally bounded if V is so”.

   I wonder if this monad is also finitely sifted-accessible, i.e., preserves sifted colimits. The category of V-graphs is cocomplete and by the above we already know that filtered colimits are preserved, thus the question boils down to the preservation of reflexive coequalizers.

   By Theorem 4.5 in op. cit. the category V-cat is locally finitely presentable if V is so. If the above conjecture is true, then V-Cat is in fact an algebraic category, i.e., the category of algebras over some (multisorted) algebraic theory.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeDec 27th 2014
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   Author: Zhen Lin
   Format: MarkdownItexWell, it's not true for $\mathcal{V} = Set$, because the forgetful functor does not preserve reflexive coequalisers. (If the monad preserves colimits of a certain shape, then so does the forgetful functor.) Indeed, let $\mathbb{2} = \{ 0 \to 1 \}$ and let $\mathcal{R} \subset \mathbb{2} \times \mathbb{2}$ be the non-full subcategory consisting of the four objects and the morphism $(0, 0) \to (1, 1)$. It is not hard to see that $\mathcal{R} \rightrightarrows \mathbb{2}$ is an internal equivalence relation, but the coequaliser in $Cat$ does not agree with the coequaliser in $Grph$.

   Well, it’s not true for $\mathcal{V} = Set$, because the forgetful functor does not preserve reflexive coequalisers. (If the monad preserves colimits of a certain shape, then so does the forgetful functor.) Indeed, let $\mathbb{2} = \{ 0 \to 1 \}$ and let $\mathcal{R} \subset \mathbb{2} \times \mathbb{2}$ be the non-full subcategory consisting of the four objects and the morphism $(0, 0) \to (1, 1)$. It is not hard to see that $\mathcal{R} \rightrightarrows \mathbb{2}$ is an internal equivalence relation, but the coequaliser in $Cat$ does not agree with the coequaliser in $Grph$.