Weakening simplicial identities

Question

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.

Answer 1

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.

Answer 2

In 2-category theory, there is a notion of "lax idempotent 2-monad" $M$ on a 2-category, for which the multiplication $m: M M \to M$ is left adjoint to the unit $u M: M \to M M$. A typical sort of example is a cocompletion monad with respect to some class of colimits. For example, consider the 2-category $Pos$ of posets; the 2-category $Sup$ of sup-lattices is 2-monadic over $Pos$, where the free sup-lattice generated by a poset $P$ is the sup-lattice of poset maps $P^{op} \to \mathbf{2} = \{0 \leq 1\}$ (equivalently, the sup-lattice of downward closed subsets). In this case the unit of the 2-monad is a Yoneda embedding $y_P: P \to [P^{op}, \mathbf{2}]$, and the 2-monad multiplication is

$$[y_P^{op}, \mathbf{2}]: [[P^{op}, \mathbf{2}]^{op}, \mathbf{2}] \to [P^{op}, \mathbf{2}].$$

What turns out to be true always is that $M u: M \to M M$ is in turn left adjoint to $m: M M \to M$.

Just as the augmented simplicial category $\Delta$ of finite ordinals (including the empty ordinal), as a monoidal 1-category equipped with a monoid = 1-element ordinal, is initial among monoidal categories equipped with a monoid, we can make an analogous statement for $\Delta$ as a 2-category (hom-sets being made into poset categories since ordinal maps are ordered among themselves). Namely, we can construct syntactically the initial monoidal 2-category equipped with a lax idempotent monoid, which will have invertible triangulator isomorphisms as part of the data, with suitable coherence conditions. And what turns out to be true is that this initial structure is equivalent (as a monoidal 2-category) to $\Delta$. Notice that 2-cell isomorphisms of $\Delta$ are automatically identities, meaning that the built-in coherence conditions on the initial structure can be "strictified".

The nLab has some information and references for lax idempotent monads which may be useful to you. I'll try to track down a reference for this 2-categorical fact about $\Delta$, but you can check by hand the presence of a lax idempotent (aka "Kock-Zöberlein") monoid structure on the 1-element ordinal.

Edit: Perhaps a canonical reference for this material is Street's Fibrations in Bicategories, particularly section 2.