How to explicitly describe algebras in a monoidal 2-category?

Question

I essentially understand (I think) how this ought to be done. Algebras in a monoidal 2-category $\mathcal{C}$, on the level of 0-cells and 1-cells, should appear as algebras in the 1-category truncation of $\mathcal{C}$. To lift these 1-level algebras we must of course weaken the usual diagrams and then describe (a zoology of possibly 3-dimensional) diagrams required of these weakening factors. How does one determine which diagrams are the "right" ones?

Answer 1

In a category $\mathscr{C}$ by finite limits you can have inside it all algebraic classical structures, for example about “Monoids” there is a category called “Monoid theory” $Mnd$ (see pioneristic works of Lawvrere) and the category of $\mathscr{C}$- monoids is the category of finite limits preserving $F: Mnd\to \mathscr{C}$, the some for groups, exist the "Groups theory" category “Gr” ecc. (for all one-sort theories the support of any algebric thery category is maked from the dual of $\mathbb{N}$ where any $n\in N$ is the sum of $n$-copies of 1, adding other more morphisms).

How generalizing this to a monoidal category $(\mathscr{C}, \otimes , I)$ ?, simply considering for axample “Mnd” with the canonical monidal structure gived by finite products $\times $ and final object $1$ and considering monoidal functors (strict or no) instead. There is a truble, for the Monoidal or Algebras theory the things go well. This generalization as a big obstrucion, but for example for groups there are the first troubles: for the “inverse propriety" axiom we need to use the finality of $1$ but the monoidal identity $I$ isnt a final object (neither exixt a similar monoidal surrogate). I know that Luca Mauri wrote a Thesi about “Algeraic Theory “ in a Monoidal category.

But you ask abot the supplemetary diagrams about coherence of these laxifications of “Monoidal, algebra ecc. “ objects.

A pseudo/lax-monoid (or pseudo/lax-algebras ecc.) in a monoidal category $\mathscr{C}$ (see it as a bicategory, or more genrally as a tricategory i.e. a braided category) is just a pseudo/lax-functors from $Mod$ to $\mathscr{C}$, and you can find (John W. Gray, Formal category theory: adjointness for 2-categories. Gordon, Power, Street. Coherence for tricategories) in supplement to the structural definition of what a pseudo/lax-funtor is (maked it as maps) also the axioms about the coherence thet these pseudo/lax-funtor (these maps) must satisfing, and these are the coherence axioms you need.

Answer 2

You are looking for the notion of pseudomonoid. The canonical example of a pseudomonoid is a monoidal category (thought of as sitting in the cartesian monoidal 2-category of categories). You can work out the correct axioms for a pseudomonoid from the axioms for a monoidal category: we have a multiplication $m: \mathcal{M} \otimes \mathcal{M} \to \mathcal{M}$, a unit $e: \mathbf{1} \to \mathcal{M}$, an associator $\alpha: m \circ (m \otimes \mathcal{M}) \Rightarrow m \circ (\mathcal{M} \otimes m)$, left and right unit laws $\lambda: m \circ (e \otimes \mathcal{M}) \Rightarrow \mathcal{M}$ and $\rho: m \circ (\mathcal{M} \otimes e) \Rightarrow \mathcal{M}$, and these data must satisfy two diagrams, the pentagon diagram for four-fold products, and the small diagram relating the left and right units.