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1. • CommentRowNumber1.
   • CommentAuthormaxsnew
   • CommentTimeDec 9th 2016
   • (edited Dec 10th 2016)
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   Author: maxsnew
   Format: MarkdownItexOften in categorical constructions, the structures are so intricately related that only a small amount is sufficient to define the whole. This is very useful in applications because constructions can be made more efficiently. Also it seems like it's almost necessary in infinite-dimensional structures. Examples I have in mind include [this paper](https://arxiv.org/abs/1112.0094) which shows that an adjunction can be presented without directly giving the action of the right adjoint on arrows, which is a generalization of a well-known folk theorem in abstract interpretation that you don't need to show the right adjoint in a Galois connection is monotone. Another class of examples are where an incoherent system of data is used to present the more desirable coherent data as in: 1. An equivalence of categories can be improved to an adjoint equivalence. 2. In [this paper](https://arxiv.org/abs/1507.03634) Mike Shulman has an example in MLTT where the incoherent "pre-idempotents" can be improved to fully coherent idempotents. Is there a general term for this phenomenon? "Improvement" seems good for the latter examples, but not the first. Also is there any chance of some categorical theory of this sort?

   Often in categorical constructions, the structures are so intricately related that only a small amount is sufficient to define the whole. This is very useful in applications because constructions can be made more efficiently. Also it seems like it’s almost necessary in infinite-dimensional structures.

   Examples I have in mind include this paper which shows that an adjunction can be presented without directly giving the action of the right adjoint on arrows, which is a generalization of a well-known folk theorem in abstract interpretation that you don’t need to show the right adjoint in a Galois connection is monotone.

   Another class of examples are where an incoherent system of data is used to present the more desirable coherent data as in:

   1. An equivalence of categories can be improved to an adjoint equivalence.

   2. In this paper Mike Shulman has an example in MLTT where the incoherent “pre-idempotents” can be improved to fully coherent idempotents.

   Is there a general term for this phenomenon? “Improvement” seems good for the latter examples, but not the first.

   Also is there any chance of some categorical theory of this sort?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 10th 2016
   • (edited Dec 10th 2016)
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   Author: Todd_Trimble
   Format: MarkdownItexMax, you can create a link if you set the format to Markdown+Itex, like so: [ [1]](https://arxiv.org/abs/1112.0094) (click Source to see the syntax). Similarly, [ [2]](https://arxiv.org/abs/1507.03634).

   Max, you can create a link if you set the format to Markdown+Itex, like so: [1] (click Source to see the syntax). Similarly, [2].

3. • CommentRowNumber3.
   • CommentAuthorKeithEPeterson
   • CommentTimeDec 10th 2016
   • (edited Dec 10th 2016)
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   Author: KeithEPeterson
   Format: MarkdownItexIf I understand you correctly: [Lawvere Theory](https://ncatlab.org/nlab/show/Lawvere+theory)

   If I understand you correctly: Lawvere Theory

4. • CommentRowNumber4.
   • CommentAuthormaxsnew
   • CommentTimeDec 11th 2016
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   Author: maxsnew
   Format: MarkdownItexHm, Keith I don't see the connection. Could you put one of my examples in terms of Lawvere theories?

   Hm, Keith I don’t see the connection. Could you put one of my examples in terms of Lawvere theories?