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Stream: learning: questions

Topic: ✔ universal property of Alg(T) for a monoidal theory

Matt Earnshaw (Sep 20 2022 at 15:25):

The category of algebras $\text{Alg}(T)$ for a Lawvere theory has a (2-) universal property: colimit preserving functors $Alg(T) \to B$ correspond essentially uniquely to finite coproduct preserving functors $T^{\text{op}} \to B$. What is the universal property of the category of algebras for a monoidal theory (pro)?

Nathanael Arkor (Sep 20 2022 at 17:23):

If you're taking an algebra for a PRO $\mathcal V$ to be a monoidal functor $\mathcal V^{\mathrm{op}} \to \mathbf{Set}$, then the category of algebras is the free monoidal cocompletion, and so a cocontinuous monoidal functor $\mathbf{Alg}(\mathcal V) \to \mathcal V'$ is equivalent to a monoidal functor $\mathcal V \to \mathcal V'$ (where $\mathcal V'$ is a cocomplete monoidal category).

Nathanael Arkor (Sep 20 2022 at 17:25):

(The universal property for algebraic theories is then the specialisation of the universal property for monoidal structure to cocartesian monoidal structure.)

Matt Earnshaw (Sep 21 2022 at 09:45):

Ah ok, sounds rather sensible. Now I see the general result as 4.51 in Kelly... gives me some reading to do. Cheers

Notification Bot (Sep 21 2022 at 12:07):

Matt Earnshaw has marked this topic as resolved.