Tensor Product of Lawvere Theories

The tensor product 𝒯 1 ⊗ 𝒯 2 tensor-product subscript 𝒯 1 subscript 𝒯 2 \mathcal{T}_{1}\otimes\mathcal{T}_{2} of two Lawvere Theories is the Lawvere theory formed by considering parallel pairs of maps f 1 ⊗ f 2 : m 1 × m 2 → n 1 × n 2 : tensor-product subscript 𝑓 1 subscript 𝑓 2 → subscript 𝑚 1 subscript 𝑚 2 subscript 𝑛 1 subscript 𝑛 2 f_{1}\otimes f_{2}:m_{1}\times m_{2}\to n_{1}\times n_{2} , quotiented to force the following diagram to commute:

m1×m2subscript𝑚1subscript𝑚2{{m_{1}\times m_{2}}}m1×n2subscript𝑚1subscript𝑛2{{m_{1}\times n_{2}}}n1×m2subscript𝑛1subscript𝑚2{{n_{1}\times m_{2}}}n1×n2subscript𝑛1subscript𝑛2{{n_{1}\times n_{2}}}mathrmidifx..⋄lsei⊗f2\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes f_{2}}f1⊗mathrmidifx..⋄lsei\scriptstyle{f_{1}\otimes mathrm{id}{ifx..lse\par i}}f1⊗mathrmidifx..⋄lsei\scriptstyle{f_{1}\otimes mathrm{id}{ifx..lse\par i}}mathrmidifx..⋄lsei⊗f2\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes f_{2}}

Properties