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I am looking for examples of locally finitely presentable categories which admit a symmetric monoidal structure, such that the tensor product preserves colimits in each variable, but the unit is not finitely presentable, and/or there is a tensor product of finitely presentable objects which is not finitely presentable.

Are there examples which appear in practice?

(The correct definition of a locally finitely presentable tensor category is one where the unit object is finitely presentable, and the tensor product of two finitely presentable objects is finitely presentable; see for instance this this paper by Kelly. But I wonder if this is automatic - probably not.)

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One can take the category of modules over a Laurent polynomial ring in one variable $\textrm{Mod}\;k[t,t^{-1}]$ and think of $k[t,t^{-1}]$ as the group algebra of $\mathbb{Z}$. The corresponding cocommutative Hopf algebra structure provides a symmetric monoidal structure $(\otimes_k, k)$ on $\textrm{Mod}\;k[t,t^{-1}]$ which certainly preserves colimits in each variable. The unit $k$ is finitely presented, but for instance $$ k[t,t^{-1}] \otimes_k k[t,t^{-1}] $$ is a tensor product of finitely presented objects which is not finitely presented.

One gets plenty of other examples along these lines.

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  • $\begingroup$ Thank you. One can also take $k[t]$ for instance. One gets more example by considering functor categories $[I,\mathsf{Mod}(k)]$ for suitable small categories $I$, where the tensor product is defined pointwise? For instance, when $I$ is infinite discrete, the unit object is not finitely presentable, right? $\endgroup$ – Martin Brandenburg May 31 '17 at 6:01
  • $\begingroup$ In general, $[I,\mathsf{Mod}(k)]$ might not be locally finitely presentable? Anyway, it is the case if $I$ is discrete or has only one object. $\endgroup$ – Martin Brandenburg May 31 '17 at 9:09
  • $\begingroup$ Concerning the second comment, perhaps I'm missing something but won't the representable functors always give you a set of finitely presented projective generators? Concerning the first comment, isn't the unit object in that example actually finitely presented? The colimits and maps are all pointwise, so it seems to me the fact that $I$ has lots of objects doesn't matter much. One should be able to obtain examples where the unit isn't finitely presented by looking at something like categories of complete modules over an adically complete ring though. $\endgroup$ – Greg Stevenson May 31 '17 at 12:11
  • $\begingroup$ 1) For locally finitely presentable we need that every object is a filtered colimit of finitely presentable objects. 2) When $I$ is an infinite set, the unit object in $\mathsf{Mod}(k)^I$ is just $(k)_{i \in I}$ and therefore an infinite direct sum of copies of $k$ placed in degree $i$. This is not finitely generated. $\endgroup$ – Martin Brandenburg Jun 1 '17 at 17:04

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