Monoidal monotone maps

A monoidal monotone map from \((P,\leq_P,I_P,\otimes_P)\) to \((Q, \leq_Q,I_Q,\otimes_Q)\). Also, a strong monoidal monotone map and a strict monoidal monotone map

• A ?monotone map \((P,\leq_P) \xrightarrow{f} (Q,\leq_Q)\) satsifying two conditions:

  1. ?\(I_Q \leq_Q f(I_P)\)

  2. \(\forall p_1,p_2 \in P:\) ?\(f(p_1)\ \otimes_Q\ f(p_2)\ \leq_Q\ f(p_1\ \otimes_P\ p_2)\)

• If the \(\leq\) are replaced with ?\(\cong\), the map is strong, and if replaced with ?\(=\), it is strict.

Linked by

• Natural Real monoidal monotones: referenced
• Exercise 2-43: referenced
• Exercise 2-43: referenced
• Solution: referenced
• Exercise 2-44: referenced
• Exercise 2-45: referenced
• Solution: referenced
• Induced V-categories from monoidal monotones: referenced
• Proof: referenced
• Proof: referenced
• Metric space to preorder: referenced
• Exercise 2-67: referenced
• Exercise 2-68: referenced
• Solution: referenced

• ?Strict monoidal monotone injection \((\mathbb{N},\leq, 0, +) \xhookrightarrow{i} (\mathbb{R},\leq,0,+)\)

• There is also a monoidal monotone \((\mathbb{R}^+,\leq, 0, +) \xrightarrow{\lfloor x \rfloor} (\mathbb{N},\leq,0,+)\)

  • Monotonic: \(x \leq_\mathbb{R} y \implies \lfloor x \rfloor \leq \lfloor y \rfloor\)

  • But it is neither ?strict nor ?strong: \(\lfloor 0.5 \rfloor + \lfloor 0.5 \rfloor \not \cong \lfloor 0.5+0.5 \rfloor\)

Consider a proposed monoidal monotone \(\mathbf{Bool}\xrightarrow{g}\mathbf{Cost}\) with \(g(false)=\infty, g(true)=0\)

• Check that the map is monotonic and that it satisfies both properties of monoidal monotones.

• Is g strict?

Solution(1)

• It is monotonic: \(\forall a,b \in \mathbb{B}: a\leq b \implies g(a)\leq g(b)\)

  • there is only one nonidentity case in \(\mathbf{Bool}\) to cover, and it is true that ?\(\infty\ \leq_\mathbf{Cost}\ 0\).

• Condition on units: ?\(0 \leq_\mathbf{Cost} g(true)\) (actually, it is equal)

• In \(\mathbf{Cost}\): \(g(x) + g(y) \geq g(x \land y)\)

  • if both true/false, ?the equality condition holds.

  • If one is true and one is false, then ?LHS and RHS are \(\infty\) (as \(x \land y = False\)).

• Therefore this is ?a strict monoidal monotone.

• Consider the following functions \(\mathbf{Cost} \xrightarrow{d,u} \mathbf{Bool}\)

  • \(d(x>0)\mapsto false,\ d(0)\mapsto true\)

  • \(u(x=\infty)\mapsto false,\ d(x < \infty) \mapsto true\)

• For both of these, answer:

  1. Is it monotonic?

  2. If so, does it satsify the monoidal monotone properties?

  3. If so, is it strict?

Solution(1)

• The function \(d\) asks ?“Is it zero?”, and the function \(u\) asks ?“Is it finite?”.

• Both of these questions are monotone: ?as we traverse forward in \(\leq_{Cost}\), we traverse towards being zero or being finite.

• The first monoidal monotone axiom is satisifed in both because ?the unit (\(0\)) is mapped to the unit (\(True\)).

• The second monoidal monotone axiom holds for both because ?addition preserves both things being zero (or not) and both things being finite (or not).

• These are strict because, in \(Bool\), ?equality and congruence coincide.

Linked by

• Metric space to preorder: referenced
• Solution: referenced

1. Is \((\mathbb{N},\leq,1,*)\) a symmetric monoidal preorder?

2. If so, does there exist a monoidal monotone \((\mathbb{N},\leq,0,+) \rightarrow (\mathbb{N},\leq,1,*)\)

3. Is \((\mathbb{Z},\leq,*,1)\) a symmetric monoidal preorder?

Solution(1)

1. Yes. Monotonicity holds, and multiplication by 1 is unital. The operator is symmetric and associative.

2. Exponentiation (say, by \(2\)) is a strict monoidal monotone.

   • \(1 = 2^0\) and \(2^x * 2^y = 2^{x+y}\)

3. No: monotonicity does not hold (multiplying negative numbers gives a larger number).