TITLE: Weakening simplicial identities
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: 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.