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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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: * [[semicategory]], [[semi-simplicial set]], [[semi-Segal space]] Also linked for instance to _[[semicategory]]_ from _[[category]]_, etc. Linked also to [[Delta space]], but the entry doe not exist yet.

   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:

   • semicategory, semi-simplicial set, semi-Segal space

   Also linked for instance to semicategory from category, etc.

   Linked also to Delta space, but the entry doe not exist yet.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 23rd 2012
   • (edited Nov 23rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[[semicategory]]_ a little bit of this and that, and in particular a section _[Relation to categories](http://ncatlab.org/nlab/show/semicategory#RelationToCategories)_ 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 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.

3. • CommentRowNumber3.
   • CommentAuthorFinnLawler
   • CommentTimeNov 23rd 2012
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   Author: FinnLawler
   Format: MarkdownItexAdded 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 23rd 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks!! Nice. I wasn't actually aware of that relation to _[[idempotent completion]]_, I have to admit. Added a brief pointer there back to semicategories.

   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.

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeJul 21st 2020
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   Author: nLab edit announcer
   Format: MarkdownItexI've added what, I think, can be a good starting point to read how I use topologically enriched semicategories (that I call flows) in my work in concurrency theory. Philippe Gaucher <a href="https://ncatlab.org/nlab/revision/diff/semicategory/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/semicategory/23">v23</a>, <a href="https://ncatlab.org/nlab/show/semicategory">current</a>

   I’ve added what, I think, can be a good starting point to read how I use topologically enriched semicategories (that I call flows) in my work in concurrency theory.

   Philippe Gaucher

   diff, v23, current

6. • CommentRowNumber6.
   • CommentAuthornLab edit announcer
   • CommentTimeJul 21st 2020
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexI've added what, I think, can be a good starting point to read how I use topologically enriched semicategories (that I call flows) in my work in concurrency theory. Philippe Gaucher <a href="https://ncatlab.org/nlab/revision/diff/semicategory/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/semicategory/23">v23</a>, <a href="https://ncatlab.org/nlab/show/semicategory">current</a>

   I’ve added what, I think, can be a good starting point to read how I use topologically enriched semicategories (that I call flows) in my work in concurrency theory.

   Philippe Gaucher

   diff, v23, current

7. • CommentRowNumber7.
   • CommentAuthorvarkor
   • CommentTimeNov 23rd 2020
   • (edited Nov 23rd 2020)
   • PermaLink
   Author: varkor
   Format: MarkdownItexOn 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](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/theory-of-semifunctors/C68B85097361F39E4050E45C1E99BB93) 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.

   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.