## October 30, 2019

### Weak Infinity-Categories Via Terminal Coalgebras

#### Posted by Tom Leinster

A very quick post to note that Eugenia Cheng and I just published our paper Weak $\infty$-categories via terminal coalgebras in Theory and Applications of Categories. We, um, were not very quick about turning it around: Eugenia presented the results at Category Theory 2008 in Calais, we arXived it in 2012, and it now appears to be 2019.

Shortest possible summary: Carlos Simpson pointed out (as others may also have done) that if you consider the endofunctor $\mathcal{V} \mapsto \mathcal{V}\text{-}\mathbf{Cat}$ on the category of categories with finite products, its terminal coalgebra is $\mathbf{Str}\infty\mathbf{Cat}$, the category of strict $\infty$-categories. Eugenia and I show how to adapt that observation to get weak $\infty$-categories. Enjoy!

Posted at October 30, 2019 10:59 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3153

### Re: Weak Infinity-Categories Via Terminal Coalgebras

Obligatory clarification: when we say “$\infty$-category”, we mean it in the fully general sense, not just $(\infty, 1)$-category.

Posted by: Tom Leinster on October 30, 2019 11:08 AM | Permalink | Reply to this

### Bruce

Congratulations!

Posted by: Bruce Bartlett on November 4, 2019 8:16 PM | Permalink | Reply to this

### Re: Bruce

Thanks Bruce!

By the way, what’s up with this thing that keeps happening on the Cafe these days, whereby Bruce leaves a comment and the subject comes up as Bruce, or Sue leaves a comment and the subject comes up as Sue, etc.? Is it the Cafe software doing something funny, or is it an accidental slip that people are somehow making? As in, did you, Bruce, somehow accidentally type “Bruce” in the subject box?

Posted by: Tom Leinster on November 5, 2019 9:31 AM | Permalink | Reply to this

### Re: Bruce

I’m not sure what happened. It may be that if you click on “Post a New Comment”, and then enter your name in the “Name:” field, it automatically makes that the subject of the comment. Or, I just did an Ed Balls somehow.

Posted by: Bruce Bartlett on November 6, 2019 11:48 AM | Permalink | Reply to this

### Re: Bruce

I just confirmed that that happens for me (at least, with the computer and browser I’m using at the moment). It doesn’t normally come up, because when I comment here the window opens with my name and email address already there. But if I click on the Name field and choose one of drop-down options that the browser provides for auto-filling web forms, the Subject field gets filled with “Mark”.

Posted by: Mark Meckes on November 6, 2019 1:50 PM | Permalink | Reply to this

### Re: Bruce

Sounds to me like a bug with the browser autofill, then.

Posted by: Mike Shulman on November 6, 2019 4:59 PM | Permalink | Reply to this

What’s extra weird is that I went into the back end of the cafe after this happened to me a few weeks ago and couldn’t figure out how to edit the comment subject lines.

Also have we always had comment subject lines? Somehow I’ve never noticed them before.

Posted by: Emily Riehl on November 7, 2019 7:54 PM | Permalink | Reply to this

### Re:

Also, the text entry box for the subject line is under the long description of text filtering options, far from the name and email boxes. So it’s easy to miss that it’s there at all, and easy to miss that an autofill bug has put something weird there.

Posted by: Mark Meckes on November 7, 2019 11:24 PM | Permalink | Reply to this

### Re:

The above was meant to be a reply to one or the other of Mike’s comments (my “also” meant to follow up on what Mike wrote below).

But as long as I’m here maybe I can help set an n-Category Cafe record for least average relevance of comments to a post.

Posted by: Mark Meckes on November 7, 2019 11:28 PM | Permalink | Reply to this

### Re: Bruce

We have always had comment subject lines. But hardly anyone ever edits them from the default, so it’s easy to just tune them out without even thinking about it.

Posted by: Mike Shulman on November 7, 2019 9:42 PM | Permalink | Reply to this

A few people seem to have started deliberately using the comment lines recently - I don’t know why.

Posted by: John Baez on November 8, 2019 1:30 AM | Permalink | Reply to this

### Re: Weak Infinity-Categories Via Terminal Coalgebras

That was my one and only CT! Still remember my awe, especially inspired by Emily and Eugenia. Even got a chance to chat with Steve. A pity, there’s not much love for CT at German universities.

(Apologies for the non-technical blabbering; it’s mainly about providing a new comment, Firefox, about expected behaviour, default subject line.)

Posted by: Tobias Heindel on November 11, 2019 1:39 PM | Permalink | Reply to this

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