$$A_1\times_{A_0}A_1 \stackrel{\cong}{\to} A_2\stackrel{d_1}{\to}A_1.$$ Associativity is given by $A_3$. The other laws are easy, and illuminating, to check. $\blacksquare$

The condition ‘Segal maps are a bijection’ is closely related to notions of ‘thinness’ as used by Brown and Higgins in the study of crossed complexes and their relationship to $\omega$-groupoids, see, for instance, , and also to Duskin’s ‘hypergroupoid’ condition, [@duskin].

Another result that is sometimes useful, is a refinement of ‘groupoids give Kan complexes’:. The proof is ‘the same’:

 \
Let $A = Ner(\mathbb{C})$, the nerve of a category $\mathbb{C}$.

\(i) Any $(n,0)$-horn $$f : \Lambda^0[n]\rightarrow A$$ for which $f(01)$ is an isomorphism has a filler. Similarly any $(n,n)$-horn $g : \Lambda^n[n]\rightarrow A$ for which $g(n-1~n)$ is an isomorphism, has a filler.

\(ii) Suppose $f$ is a morphism in $\mathbb{C}$ with the property that for any $n$, any $(n,0)$-horn $\phi : \Lambda^0[n]\rightarrow A$ having $f$ in the $(0,1)$ position, has a filler, then $f$ is an isomorphism. (Similarly with $(n,0)$ replaced by $(n,n)$ with the obvious changes.) $\blacksquare$

**Remark**

Joyal in [@joyal] suggested that the name ‘weak Kan complex’, as introduced by Boardman and Vogt, [@boardmanvogt], could be changed to that of ‘quasi-category’ to stress the analogy with categories *per 