The generators $d\_i, s\_i$ for morphisms of the simplicial category satisfy simplicial identities:


$d\_jd\_i = d\_id\_{j−1}$ for $i < j$


$s\_jd\_i = d\_is\_{j−1}$ for $i < j$


$s\_jd\_i = id$ for $i = j$, $i = j + 1$


$s\_jd\_i = d\_{i−1}s\_j$ for $i > j + 1$


$s\_js\_i = s\_is\_{j+1}$ for $i ≤ j$.


Has anyone seen something written about a 2-category with the same objects, the same generators for morphisms, and the simplicial identities replaced with invertible 2-cell which generate the second dimensional structure and satisfy some "coherence conditions"?


I also wonder weather there is any useful 2-category which is obtained by replacing the simplicial identities with non-invertible 2-cells which satisfy some sort of "coherence". In this case one should pay attention to the choice of direction of each of these 2-cell. 



Weakening simplicial identities
Todd's answer to your second question is excellent. As for your first question, you may want to check out Steve Lack's paper *A Coherent Approach to Pseudomonads*. He constructs a 2-category $\Delta'$ which has the same relationship to pseudomonads that the 1-category $\Delta$ has to ordinary monads and the 2-category $\Delta$ has to lax-idempotent 2-monads. His definition is not in terms of generators and relations but rather of an "all diagrams commute" sort, but he then proves a coherence theorem from which one should be able to extract a generators-and-relations description.


