### 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!

## 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.