## May 5, 2009

### Gerundives

#### Posted by David Corfield

If we were to have a page at nLab on things to be categorified should it be titled categorifAcienda, categorifIcienda or something else?

My suggestions are based on the gerundives formed from verbs such as agenda and Miranda. Concerning verbs more closely resembling ‘categorify’ we have

• Satisfacio (satisfy) - satisfaciendus
• Efficio (bring to pass) - efficiendus

Unfortunately, categorify is a hybrid word, with Greek stem and Latin suffix. I suppose categorize was out of the question.

I believe the facere of ‘satisfacere’, ‘putrefacere’ and ‘liquefacere’ is less usual than the ficare of ‘edificare’, ‘specificare’ and ‘fortificare’ (is it just a question of whether an ‘i’ precedes the ending?), so ‘categorification’ was probably better than ‘categorifaction’.

Anyway, perhaps a suggestion for something to go on that page. Coalgebras, as maps $\alpha: \mathcal{C} \to F(\mathcal{C})$, are of great interest in computer science, e.g., this workshop, capturing the notion of the state of observations on, and transitions within, a system. Now, has anyone looked at coalgebras for (pseudo) (lax) 2-functors between 2-categories, allowing state spaces to be, say, categories?

Posted at May 5, 2009 4:26 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1961

### Re: Gerundives

What would be nice is a page of things where we almost know how to categorify it, and why it would be cool if we did. Because nearly every mathematical structure could be categorified.

And I’ve never heard of satisfacienda (at least, not as an English word), so I’d say that “Things to be categorified” is a much better title than either of the other suggestions.

Posted by: Mike Stay on May 5, 2009 5:27 PM | Permalink | Reply to this

### Re: Gerundives

The page awaits.

Posted by: David Corfield on May 5, 2009 9:47 PM | Permalink | Reply to this

### Re: Gerundives

‘categorification’ (not ‘categorifaction’), so ‘categorificienda’ (not ‘categorifacienda’).

Although ‘things to be categorified’ also works.

Posted by: Toby Bartels on May 5, 2009 5:47 PM | Permalink | Reply to this

### Re: Gerundives

How about “Categorigenda?” (like agenda and corrigenda … who’s to say the verb wasn’t categorigio/categorigere? (apart from being, as you say, at least as much greek as latin)

Posted by: some guy on the street on May 5, 2009 8:27 PM | Permalink | Reply to this

### Re: Gerundives

Categoriata? Categorialia? Categorientals? Precategories? Kittegories? Pussigories? Whose Ox is Catigored?

Posted by: Jonathan Vos Post on May 5, 2009 8:36 PM | Permalink | Reply to this

### Re: Gerundives

I think ‘categorienda’ sounds best, even tho the philology doesn’t work out. Kittegories should be finite categories

Posted by: Avery Andrews on May 6, 2009 1:58 AM | Permalink | Reply to this

### Re: Gerundives

Categoriata nicely parallels desiderata - ‘something desired to be categorified’ and (to yours truly anyway) is the funnest to say.

Posted by: Michigan J. Frog on May 6, 2009 6:00 AM | Permalink | Reply to this

### Re: Gerundives

Except that -(a)ta suffix means that they’ve already been done; when messing up the philology, it’s worse to stuff up the suffix than the stem, I think.

Posted by: Avery Andrews on May 6, 2009 6:19 AM | Permalink | Reply to this

### Re: Gerundives

No, no, kittegories are small categories!

Posted by: Toby Bartels on May 6, 2009 6:01 AM | Permalink | Reply to this

### Re: Gerundives

So the Yoneda embedding of a category $\mathcal{C}$ to $F(\mathcal{C}) = [\mathcal{C}^{op}, Set]$ is a 2-coalgebra for the 2-endofunctor $F$.

Like with the powerset functor there surely can’t be a terminal 2-coalgebra for $F$. What about $G(\mathcal{C}) = 1 + A \times \mathcal{C}$, where $A$ is a fixed category? I suppose the terminal 2-coalgebra would have as objects finite or infinite lists of objects of $A$ with lists of arrows of $A$ as morphisms.

Posted by: David Corfield on May 6, 2009 10:11 AM | Permalink | Reply to this

### Re: Gerundives

By rights, the initial algebra should consist of finite lists, while the terminal coalgebra should consist of finite or infinite lists. (And if you use $A \times \mathcal{C}$ instead, then the inital algebra should be empty, while the terminal coalgebra should consist of infinite lists.)

This seems to me to work just fine, but maybe nobody's ever written it down?

Posted by: Toby Bartels on May 7, 2009 6:33 PM | Permalink | Reply to this

### Re: Gerundives

Categorificanda, from the well known latin verb categorificare. ;)

Posted by: latin mathematician on May 16, 2009 10:39 AM | Permalink | Reply to this
Read the post Coalgebraic Modal Logic
Weblog: The n-Category Café
Excerpt: Putting together ideas of modal logic and coalgebra with ways of modelling first-order theories.
Tracked: September 7, 2009 12:48 PM

### Re: Gerundives

It’s a Golden Oldie, but I like it —

• Categorical Imperatives
Posted by: Jon Awbrey on October 14, 2009 6:16 AM | Permalink | Reply to this

Post a New Comment