On 8:38 of Session 9: Masterclass in Condensed Mathematics an $\infty$-category is defined as a simplicial set $\mathcal{S}$ (i.e a functor $\Delta^{op}\rightarrow Sets$) such that for every horn $\wedge_{i}^{n}\rightarrow \mathcal{S}$ with $1<i<n$ there is at least a lift $\Delta^{n}\rightarrow \mathcal{S}$. As far as I understand, this definition means that there are "multiple ways" of composing morphisms, as opposed to regular categories, in which the lift is unique. My question regards a somehow dual notion of this. That is, let $\mathcal{S}$ be a simplicial set such that for every horn $\wedge_{i}^{n}\rightarrow \mathcal{S}$ with $1<i<n$ there is at most a lift $\Delta^{n}\rightarrow \mathcal{S}$. So morphisms are not always composable, but when they are the composition is unique. My questions are the following: Does this concept have a name? Is it of interest? If so, what are its applications?

## 1 Answer

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*Partial monoids* (see Definition 2.2) play a useful role in

```
Segal, Graeme
Configuration-spaces and iterated loop-spaces.
Invent. Math. 21 (1973), 213–221.
```