## December 19, 2008

### The Microcosm Principle

#### Posted by David Corfield

Furthering my study of coalgebra, I came across slides for a couple of talks (here and here) which put John and Jim’s microcosm principle into a coalgebraic context.

Recall their claim in Higher-Dimensional Algebra III that “certain algebraic structures can be defined in any category equipped with a categorified version of the same structure” (p. 11), as with monoid objects in a monoidal category.

We name this principle the microcosm principle, after the theory, common in pre-modern correlative cosmologies, that every feature of the microcosm (e.g. the human soul) corresponds to some feature of the macrocosm.

Later, in section 4.3, they give a formal treatment of the principle using operads.

On the other hand, in the slides and associated paper, Hasuo, Jacobs and Sokolova give the principle a 2-categorical formulation as a lax natural transformation $X: \mathbf{1} \implies \mathbb{C}$ between functors from a Lawvere theory $\mathbb{L}$ to $\mathbf{CAT}$. I wonder how these treatments compare.

The microcosm principle would make for a good methodological entry for nLab, as would evil. I’ll see about extracting John’s sermon on the latter from old blog files. I must confess to feeling rather guilty for having done so little there, after pushing for it. Perhaps Santa will bring me a generous amount of free time for Christmas.

Posted at December 19, 2008 9:46 AM UTC

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### Re: The Microcosm Principle

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

What is a rig in a rig category?

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’t that sweeping: they wrote ‘certain algebraic structures’.)

Explanation:

A rig or 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 category is 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 are not product 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’.

Posted by: Tom Leinster on December 20, 2008 5:58 PM | Permalink | Reply to this

### Re: The Microcosm Principle

I haven’t really read Hasuo, Jacobs and Sokolova’s talks yet, but they seem to suggest that for any Lawvere theory $L$, and any ‘strict categorified $L$-algebra’, say $C$, we can define an ‘$L$-algebra object in $C$’. If so, this would cover your problematic example: we should be able to define a rig object in a strict rig category.

Here’s how their suggestion seems to go, according to David. $L$ is a category with finite products. A ‘strict categorified $L$-algebra’ is a product-preserving functor

$F : L \to Cat$

An ‘$L$-algebra object in $C$’ is a lax natural transformation

$\phi: 1 \Rightarrow F$

where $1$ is the terminal categorified $L$-algebra.

I don’t know what this gives in your example, but at least it should give something.

Posted by: John Baez on December 20, 2008 6:13 PM | Permalink | Reply to this

### Re: The Microcosm Principle

Yes, their general definition should in particular give a definition of ‘rig in a rig category’. Of course, to make a definition is trivially easy (‘a rig in a rig category is a cheese sandwich’), but to make a sensible or useful definition is another matter.

As far as I can see, the only example of their general definition that they give in their paper is the motivating one, monoids. So I’m far from persuaded that their definition is sensible.

If I get round to working out what happens in the case of rigs, I’ll let you know.

Posted by: Tom Leinster on December 20, 2008 6:30 PM | Permalink | Reply to this

### Re: The Microcosm Principle

OK, here’s what I think their definition does in the case of rigs. I’ve done this only half-rigorously, so don’t trust me too much.

Let $\mathbf{C} = (\mathbf{C}, \otimes, \oplus)$ be a rig category. (I should really mention the multiplicative and additive units too.) They only deal with strict categorical structures, so, for instance, there is an actual equality $A \otimes (B \oplus C) = (A \otimes B) \oplus (A \otimes C)$ for objects $A, B, C$ of $\mathbf{C}$, and similarly for morphisms. (They do make a Remark, 3.3, on how to do it weakly, but I think their suggestion goes wrong in the case of commutative monoids: it gives something stricter than symmetric monoidal categories.)

According to their definition, a rig in $\mathbf{C}$ is, I think, an object $A$ of $\mathbf{C}$ together with maps $\cdot: A \otimes A \to A,      +: A \oplus A \to A$ (and similarly for units, which I’ll continue to ignore). These are required to satisfy axioms:

1. $\cdot$ defines a monoid structure on $A$ (with respect to the monoidal structure $\otimes$ on $\mathbf{C}$)
2. $+$ defines a commutative monoid structure on $A$ (with respect to the strictly symmetric monoidal structure $\oplus$ on $\mathbf{C}$)
3. distributivity: the composite map $A \otimes (A \oplus A) \stackrel{id_A \otimes +}{\to} A \otimes A \stackrel{\cdot}{\to} A$ is equal to the composite map $(A \otimes A) \oplus (A \otimes A) \stackrel{\cdot \oplus \cdot}{\to} A \oplus A \stackrel{+}{\to} A$ (recalling from above that the domains of these two composites are equal), and similarly for distributivity on the other side and for nullary addition.

So, is this a useful notion? I don’t know. Here’s what happens in the examples I mentioned.

If the additive structure $\oplus$ of the rig category $\mathbf{C}$ is coproduct, as it often is, then a rig in $\mathbf{C}$ is merely a monoid in $(\mathbf{C}, \otimes)$. So, for instance, a rig in the rig category $(\mathbf{Set}, \times, +)$ is not an ordinary rig, but a monoid. And as far as I can see, there’s no rig category $\mathbf{C}$ with the property that rigs in $\mathbf{C}$ are just ordinary rigs.

If $\mathbf{C}$ is the discrete rig category corresponding to a rig $R$, then there are no rigs at all in $\mathbf{C}$ unless $R$ is the one-element rig (in which case, there is just one rig in $\mathbf{C}$).

If $\mathbf{C}$ is the rig category of (finite or not) sets and bijections, with the rig structure given by cartesian product and disjoint union of sets, then there are no rigs in $\mathbf{C}$ at all.

If $\mathbf{C}$ is the poset ${}[0, \infty)$ with the reverse of the usual ordering (so that there’s a map $1 \to 0$), and with the rig structure given by $\times$ and $+$, then there’s just one rig in $\mathbf{C}$, namely $0$.

So it doesn’t seem to give anything interesting in any of the examples of rig categories that I can think of. Maybe someone else has other ideas. Or maybe someone would like to try their definition out on the theory of groups?

(Incidentally, David and John, you’ve slightly oversimplified their definition of $\mathbf{L}$-algebra in an $\mathbf{L}$-category — or ‘$\mathbf{L}$-object in an $\mathbf{L}$-category’, as they call it.

First recall that if $\mathbf{L}$ is a Lawvere theory and $\mathbf{F}$ denotes a skeleton of the category of finite sets then there is a canonical functor $\mathbf{F}^{op} \to \mathbf{L}$, which you can think of as the trivial theory sitting inside $\mathbf{L}$. It deals with all the variable relabelling.

Now, if $\mathbf{C}$ is an $\mathbf{L}$-category, i.e. a finite-product-preserving functor $\mathbf{L} \to \mathbf{CAT}$, then an $\mathbf{L}$-algebra in $\mathbf{C}$ is not just any lax transformation $1 \to \mathbf{C}$: it’s one whose restriction to $\mathbf{F}^op$ is a strict transformation.)

Posted by: Tom Leinster on December 20, 2008 8:08 PM | Permalink | Reply to this

### Re: The Microcosm Principle

Tom, I don’t see how to define a rig object in a rig-category either.

But Laplaza also considered the situation where there is only a “lax” distributivity of $\otimes$ over $\oplus$, so that one has for example maps $\delta$ $A\otimes(B\oplus C)\to(A\otimes B)\oplus(A\oplus C)$ In this context there does seem to be a reasonable definition of rig object: you ask that A have a monoid structure $+$ with respect to $\oplus$ and a monoid structure $\times$ with respect to $\otimes$, and then the distributivity of $\times$ over $+$ can easily be expressed using the maps $\delta$ (easily expressed, but not so easy, at least for me, to type the commutative diagram here).

The advantage of allowing the $\delta$s not to be invertible is that now we include the finite products example. If \textbf{C} is a category with finite products, then it has one of these “lax rig-category” structures with $\otimes$ and $\oplus$ both just the product, and with $\delta$ sending $(a,b,c)\in A\times (B\times C)$ to $(a,b,a,c)\in(A\times B)\times(A\times C)$. And now you get the evident notion of rig object in a category with finite products to which you alluded (as well as all the less interesting examples which you enumerated).

Posted by: Steve Lack on December 24, 2008 2:36 AM | Permalink | Reply to this

### Re: The Microcosm Principle

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.

Posted by: John Baez on December 20, 2008 6:03 PM | Permalink | Reply to this

### Re: The Microcosm Principle

But, it’s only good if it actually gives something useful — not a cheese sandwich.

Posted by: John Baez on December 20, 2008 10:02 PM | Permalink | Reply to this

### Re: The Microcosm Principle

Hi, John.

Some notes about useful examples of microcosm principle.

A version of a 2-categorical microcosm principle, which turned out to be very useful, is in my paper
math/0207281 The Eckman-Hilton argument and higher operads, see section on “internal algebras of cartesian monads”.

It works for any cartesian finitary monad (so it is a special case of Hasuo, Jacobs and Sokolova’s approach). I call an internal algebra in a Cat- algebra A of a cartesian monad T a lax-T-algebra morphism 1 –> A. So we can talk about monoids in monoidal categories, \omega-categories inside globular monoidal categories, operads inside categorical operads etc..

But the most useful thing to do is to consider morphisms between monads and internalisation of functors. So, I consider internal algebras of a monad T inside categorical algebras of a monad S. My main example: internal n-operads inside categorical symmetric operads.

The functors

A (a categorical S-algebra) ==> Int_T(A) (the category of internal T-algebra inside A)

turned out to be representable and it is possible to understand the structure of representing object through bar-construction. First, this gives a kind of graphical formalism which allows to express algebras as functors (like classical tree formalism for operads). Second, many important adjunctions become internal Kan extensions and this, (together with graphical formalism) allows to compute these functors explicitly (like symmetrisation from n-operads to symmetric operads.)

One interesting analogy I observed during a recent talk in our Category Seminar by Martin Markl on “Natural differential operators and graph complex.” His main observation is that those natural operators are 0-homology of some graph-complexes (all higher homology are trivial). Up to some extent those graph complexes represent natural differential operators on smooth manifolds. In many respect it looks very similar to what I have done for cartesian monads and internal algebras but there are no cartesian monads around. But it seems to me that we have a microcosm principle at work here. If anybody thought about it?

Posted by: michael on December 22, 2008 12:27 AM | Permalink | Reply to this

### Re: The Microcosm Principle

You remind us:

So, if F:C→D is our lax monoidal functor, we have morphisms

ϕa,b:F(a)⊗F(b)→F(a⊗b)

for every pair of objects a,b∈C.

so why not remind us of the coherence condition? rather than send us to wiki?

Posted by: jim stasheff on December 21, 2008 2:08 PM | Permalink | Reply to this

### Re: The Microcosm Principle

Because I’m a lazy bum who prefers to provide a one-click link to some commutative diagrams rather than spend half an hour redrawing these diagrams in TeX.

Posted by: John Baez on December 21, 2008 11:34 PM | Permalink | Reply to this

### Re: The Microcosm Principle

I was thinking of words - such as:

satisfying a strict 2-cocyle condition

remember my tendency is for higher homotopy conditions

Posted by: jim stasheff on December 23, 2008 1:56 PM | Permalink | Reply to this

### Re: The Microcosm Principle

Further work by the same team + Chris Heunan – Coalgebraic Components
in a Many-Sorted Microcosm
.

Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming.

Posted by: David Corfield on September 24, 2009 1:17 PM | Permalink | Reply to this

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