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I have touched the following entries, trying to interlink them more closely by added sentences with cross-links that indicate how they relate to each other:
Also linked for instance to semicategory from category, etc.
Linked also to Delta space, but the entry doe not exist yet.
added to semicategory a little bit of this and that, and in particular a section Relation to categories with some basic remarks.
For the fun of it, I stated a univalence/semi-complete Segal space-style formulation of when a semi-category $\mathcal{C}$ is a category: precisely if
$\array{ Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1 \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}} \\ && \mathcal{C}_0 } \,,$where $Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}$ is the subset of morphisms that are neutral elements in endomorphism semi-monoids.
Added a paragraph, while I thought of it, to semicategory on the left and right adjoints to the forgetful functor from Cat; similar paragraph added at semifunctor.
At some point I want to write a page for semiadjunctions, which are adjunctions in the 2-category of categories, semifunctors and (I think) ordinary transformations; they give you things like models for non-extensional lambda-calculus, and possibly things like weak limits. I don’t have the time right now, but maybe putting this here will remind me.
Thanks!! Nice. I wasn’t actually aware of that relation to idempotent completion, I have to admit. Added a brief pointer there back to semicategories.
On this page, it says that the right adjoint to U (i.e. the Karoubi envelope) is not a 2-functor, but Theorem 7.3 of Hoofman’s The theory of semi-functors states that it is. There’s no justification on the nLab page: is the statement incorrect, or is there a difference in definitions that I haven’t spotted?
Edit: In Hoofman’s paper, the Karoubi envelope is defined as a 2-functor from the 2-category of categories, semifunctors, and semi-natural transformations, rather than semicategories, which explains the disparity.
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