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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.
For Proposition 3.3 it’s not sufficient in Definition 3.2 for endomorphisms to be neutral elements in their endomorphism semimonoids, they need to be neutral w.r.t. composition with non-endomorphisms as well. A counterexample for Proposition 3.3 in its current form would be the semicategory with two objects $A, B$, trivial endomorphism monoids, two morphisms $f, g: A \to B$, and composition defined such that composition of $f$ or $g$ with any endomorphism yields $f$.
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