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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 14th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexbrief `category:people`-entry for hyperlinking references <a href="https://ncatlab.org/nlab/revision/Harvey+Wolff/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Harvey+Wolff">current</a>

   brief category:people-entry for hyperlinking references

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 15th 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThis query at [[algebrad]] (aka [[vectoid]]) mentions Wolff > [[Todd Trimble|Todd]]: "Commutes with colimits" must really mean: $F: A^{op} \to Set$ takes colimits in $A$ to limits in $Set$, and the axiom is that such continuous functors are representable. This reminds me of notions of totality in category theory. > [[Mike Shulman]]: Yes, that exact condition has been studied by category theorists under the name of a "compact" category. That's a terrible name, of course, so even the odd-sounding (to me) "vectoid" is better. I think the original reference is Isbell's paper [Small subcategories and completeness](http://www.ams.org/mathscinet-getitem?mr=0224670), and one later one is [Compact and hypercomplete categories](http://www.ams.org/mathscinet-getitem?mr=614376) by Börger, Tholen, Wischnewsky, and Wolff. The property is implied by totality (= the Yoneda embedding has a left adjoint), and implies hypercompleteness (= admits every limit which it could conceivably admit, subject to local smallness).

   This query at algebrad (aka vectoid) mentions Wolff

   Todd: “Commutes with colimits” must really mean: $F: A^{op} \to Set$ takes colimits in $A$ to limits in $Set$, and the axiom is that such continuous functors are representable. This reminds me of notions of totality in category theory.

   Mike Shulman: Yes, that exact condition has been studied by category theorists under the name of a “compact” category. That’s a terrible name, of course, so even the odd-sounding (to me) “vectoid” is better. I think the original reference is Isbell’s paper Small subcategories and completeness, and one later one is Compact and hypercomplete categories by Börger, Tholen, Wischnewsky, and Wolff. The property is implied by totality (= the Yoneda embedding has a left adjoint), and implies hypercompleteness (= admits every limit which it could conceivably admit, subject to local smallness).

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeAug 15th 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexGiven a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$ containing the identities of $A_0$, H. Wolff defines the corresponding theory of [[localization of an enriched category]]. * H. Wolff, _$V$-localizations and $V$-triples_, Dissertation, University of Illinois-Urbana, 1970. * H. Wolff, _$V$-localizations and $V$-monads_, J. Alg. __24__, 405-438, 1973, [MR310041](http://mathscinet.ams.org/mathscinet-getitem?mr=310041), <a href="https://doi.org/10.1016/0021-8693(73)90116-6">doi</a>; _V-localizations and $V$-monads. II_, Pacific J. Math. 63 (1976), no. 2, 579&#8211;589, [MR412253](http://www.ams.org/mathscinet-getitem?mr=412253), [euclid](http://projecteuclid.org/getRecord?id=euclid.pjm/1102867417); _$V$-localizations and $V$-Kleisli algebras_, Manuscripta Math. __16__ (1975), no. 3, 203--228, [MR382383](https://mathscinet.ams.org/mathscinet-getitem?mr=382383), [doi](https://doi.org/10.1007/BF01164425) <a href="https://ncatlab.org/nlab/revision/diff/Harvey+Wolff/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Harvey+Wolff/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Harvey+Wolff">current</a>

   Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$ containing the identities of $A_0$, H. Wolff defines the corresponding theory of localization of an enriched category.

   • H. Wolff, $V$-localizations and $V$-triples, Dissertation, University of Illinois-Urbana, 1970.
   • H. Wolff, $V$-localizations and $V$-monads, J. Alg. 24, 405-438, 1973, MR310041, doi; V-localizations and $V$-monads. II, Pacific J. Math. 63 (1976), no. 2, 579–589, MR412253, euclid; $V$-localizations and $V$-Kleisli algebras, Manuscripta Math. 16 (1975), no. 3, 203–228, MR382383, doi

   diff, v2, current

4. • CommentRowNumber4.
   • CommentAuthorvarkor
   • CommentTimeFeb 13th 2024
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdded a paper of Wolff. <a href="https://ncatlab.org/nlab/revision/diff/Harvey+Wolff/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Harvey+Wolff/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Harvey+Wolff">current</a>

   Added a paper of Wolff.

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the words "On [[free monads]]" before the item. (On the one hand because it's good practice in general, on the other hand because otherwise it looks like you added the item under the heading of localizations of enriched categories.) <a href="https://ncatlab.org/nlab/revision/diff/Harvey+Wolff/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Harvey+Wolff/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Harvey+Wolff">current</a>

   added the words “On free monads” before the item.

   (On the one hand because it’s good practice in general, on the other hand because otherwise it looks like you added the item under the heading of localizations of enriched categories.)

   diff, v4, current