I’m glad more people are trying to find setups where the microcosm principle becomes a theorem — and thanks for pointing this out, David.

When Jim and I gave an $n$-categorical formulation of the microcosm principle in terms of $n$-categorical algebras of an operad $O$, the key idea was to use *lax* maps from the terminal $n$-categorical $O$-algebra to a fixed $n$-categorical $O$-algebra.

Let me try to explain this, because it’s really not as terrifying as it sounds.

Here’s the classic example of the microcosm principle. We can define a monoid object inside any monoidal category — but a monoidal category is like a categorified monoid! So, for a monoid object to make sense in some category, that category needs to be like a categorified monoid object. The monoid object (the ‘microcosm’) can only thrive in a category (a ‘macrocosm’) that resembles itself.

So, let’s see how this fact is ‘explained’ in terms of lax maps. I’ll try to get you to see that a lax monoidal functor $F : 1 \to D$ from the terminal monoidal category $1$ to a monoidal category $D$ is the same as a monoid object in $D$. So, the concept of monoid object arises naturally from the concept of monoidal category — *if we use lax maps between monoidal categories!*

If you’ve never thought about lax monoidal functors, now’s the time to do so. Mac Lane and the Wikipedia article just call them monoidal functors: what makes them **lax** is that the tensor product is preserved only up to a *morphism*, not necessarily an isomorphism. So, if $F: C \to D$ is our lax monoidal functor, we have morphisms

$\phi_{a,b} : F(a) \otimes F(b) \to F(a \otimes b)$

for every pair of objects $a,b \in C$.

Now, suppose $1$ is the *terminal* monoidal category — the one with a single object, namely $I$, and a single morphism. And suppose $F : 1 \to D$ is a lax monoidal functor from $1$ to another monoidal category $D$. What does this amount to?

Well, we get an object $F(I)$ in $C$, for starters. But why is it a monoid object? Well, it’s equipped with a multiplication

$\phi_{I,I} : F(I) \otimes F(I) \to F(I \otimes I) \cong F(I)$

And, if you examine the coherence laws that $\phi$ must satisfy — read the Wikipedia article — you’ll see they’re precisely what we need to guarantee that this multiplication is associative! Similarly, the lax preservation of the unit object gives the unit for our would-be monoid object $F(I)$.

It’s amazing! You’ve got to do the calculation to see how cool it is.

This idea can be $n$-categorified: for example, a ‘monoidal category object’ makes sense in any monoidal 2-category, and so on. It can also be vastly generalized: what works for monoids works for algebras of any operad. Jim and I explained how for any operad $O$,
we can define an $n$-categorical $O$-algebra *inside* any $(n+1)$-categorical $O$-algebra: it’s just a lax map from the terminal $(n+1)$-categorical $O$-algebra to the given $n+1$-categorical $O$-algebra. For details, go to page 54 of our our paper.

Anyway, all this sounds very similar to Hasuo, Jacobs and Sokolova’s idea of using lax natural transformations $X:1 \to \mathbb{C}$ between functors from a Lawvere theory $\mathbb{L}$ to CAT. They’re only formulating a basic version of the microcosm principle, not an $n$-categorified version. But, they’re doing it for algebras of Lawvere theories, which are more general than algebras of operads! That’s good.

## Re: The Microcosm Principle

Here’s something to test the limits of the microcosm principle:

In its most sweeping version, the microcosm principle says that this concept should make sense. But as far as I can see, it doesn’t.

(I hasten to point out that John and Jim

weren’tthat sweeping: they wrote ‘certainalgebraic structures’.)Explanation:

A

rigor semiring is a riNg without Negatives; in other words, it’s a set equipped with both a monoid structure, called multiplication, and a commutative monoid structure, called addition, with multiplication distributing over addition. So any ring is a rig, and $\mathbb{N}$ is a rig — indeed, a commutative rig — but not a ring.A

rig categoryis a category equipped with both a monoidal structure, called multiplication and written $\otimes$, and a symmetric monoidal structure, called addition and written $\oplus$, with the former distributing over the latter up to coherent isomorphism. The axioms were written down by Laplaza. There are many examples in which $\oplus$ is categorical coproduct, but others in which it is not. For instance, any rig can be regarded as a discrete rig category; the category of finite sets and bijections is a rig category under $\times$ and $\amalg$ (which arenotproduct and coproduct); and the poset $[0, \infty)$ of non-negative reals is a rig category under $\times$ and $+$.You can certainly define ‘rig’ in a finite product category. The rig axioms use the fact that it’s not any old symmetric monoidal category, e.g. the distributivity axiom $x(y + z) = x y + x z$ has a repeated variable on the right-hand side, so its categorical formulation uses the diagonal map $X \to X \times X$ (where $X$ is the underlying object of the rig). But I don’t see a sensible way of defining ‘rig in a rig category’.