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In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times M\longrightarrow M. \end{equation}

For example, in the definition of group object, the multiplication is given by the later one while the multiplication in the definition of monoid object is given by the first one.

In the category of sets, these two definitions are coincide. But this is not the case in general monoidal category.

For example, in the category of abelian groups $\mathbf{Ab}$, we have an alternative tensor product structure than Cartesian product -- the usual tensor product. Then we have two choices of definition of monoid object in $\mathbf{Ab}$: one is given by tensor product, whose result are the usual rings, the other is given by Cartesian product, which gives an alternative algebraic sructure.

The Lawvere theory gives a categorical approach to general algebra, but they are set-theoric algebras. Where a model of an algebraic theory is a product-preserving functor to $\mathbf{Set}$.

I knew that rings are monoid objects in $\mathbf{Ab}$, thus I thought they should be models of the theory of monoid in $\mathbf{Ab}$. However, if we define a model in $\mathbf{Ab}$ as a product-preserving functor to $\mathbf{Ab}$, then a model of the theory of monoid is not a ring but the alternative algebraic structure obtained above.

My questions are:

1, When do people use Cartesian product and when tensor product? Is it just a custom?

2, What is a suitable definition of models of an algebraic theory in a monoidal category under suitable conditions?

One way to do this is to replace all the things(theory, category of sets and functors) by enriched ones. But I want to keep the theory.

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    $\begingroup$ It is not appropriate to interpret Lawvere theories in a general monoidal category – they are defined with respect to the cartesian product. $\endgroup$ – Zhen Lin Jun 2 '14 at 4:45
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    $\begingroup$ Your alternative algebraic structure is just abelian groups again, by the Eckmann-Hilton argument. Anyway, as Zhen says, Lawvere theories are only to be interpreted in categories with finite products. The thing that models monoids in arbitrary monoidal categories isn't a Lawvere theory. You can think of it as a monoidal category generated under tensor product by one object. $\endgroup$ – Qiaochu Yuan Jun 2 '14 at 6:08
  • $\begingroup$ @ZhenLin How about define a model of a Lawvere theory as a monoidal functor rather than a product-preserving functor? $\endgroup$ – S.Gau at Math Jun 2 '14 at 12:54
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    $\begingroup$ That is what I mean by "interpret [...] in a general monoidal category", and I still say it is inappropriate. It does not produce the correct notion of monoid, for example; instead you get something like a bimonoid, because everything in a cartesian monoidal category is naturally a comonoid. You should look at operads if you want a monoidal version of Lawvere theories. $\endgroup$ – Zhen Lin Jun 2 '14 at 13:08
  • $\begingroup$ @ZhenLin Thanks! I see the point. $\endgroup$ – S.Gau at Math Jun 2 '14 at 13:29

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