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Every topos $\mathbf{H}$ is canonically a cartesian monoidal category $(\mathbf{H}, \times)$, with the tensor product being the cartesian product. If in addition to this canonical product $\mathbf{H}$ is equipped with another noncartesian monoidal category structure $\otimes$, then $(\mathbf{H}, \otimes)$ might be called a monoidal topos.
Of course, a topos may carry more than one non-cartesian monoidal category structure, and in some of the interesting examples, the various monoidal structures will be interlinked in some way. A typical example is the phenomenon of plethysm.
For $(\mathcal{C}, \otimes)$ a small monoidal category, the presheaf topos $[\mathcal{C}, Set]$ is naturally a monoidal topos with respect to the Day convolution tensor product.
For $\mathbf{H}$ a topos and $\mathbb{G} \in \mathbf{H}$ a monoid object, the slice topos $\mathbf{H}_{/\mathbb{G}}$ inherits a non-cartesian monoidal structure $\otimes_G$ induced from the product structure on $\mathbb{G}$:
Such monoidal toposes appear as categories of local action functionals in the context of motivic quantization.
The classifying topos for the theory of objects which is the presheaf topos $[FinSet, Set]$ on the opposite category of FinSet or equivalently the category of finitary endofunctors $End_f(Set)$ is naturally a monoidal topos under composition of endofunctors. A standard textbook reference is (Johnstone D3.2). The enriched category theory over this monoidal category is discussed in (Garner 13).
There are several variations on the theme of the previous example , where one considers presheaves on the “$\underline{\;\;\;\;\;}$” monoidal category $M$ generated by a single object, where the “blank” may be filled in with some structure-connoting adjective. In the previous example, we had that $FinSet^{op}$ is the free cartesian monoidal category generated by a single object, but the blank might be filled in with “symmetric” ($M$ is the permutation category $\mathbb{P} = Core(FinSet)$), or “braided” (the braid category $\mathbb{B}$), or “semicartesian symmetric monoidal” ($Fin_{inj}$: finite sets and injections between them), or “strict” (the discrete category $\mathbb{N}$), and so on. In each of these cases, the presheaf topos $E = [M^{op}, Set]$ acquires a Day convolution structure, but more interestingly, we have an equivalence
where the right side is the category of cocontinuous $\underline{\;\;\;\;\;}$ monoidal endofunctors. By transferring the endofunctor composition across the equivalence, $E$ acquires another monoidal product structure. In the classical symmetric case, this goes under names like plethystic monoidal product, and in each such case the notion of monoid therein gives a notion of operad. See also generalized multicategory. For more on this see at operad – A detailed conceptual treatment.
The category of dendroidal sets is a topos (the presheaf topos on the tree category) and naturally carries the non-cartesian Boardman-Vogt tensor product.
In (Dolan 11) is indicated a category of “toric quasicoherent sheaves” which is a topos and hence under tensor product of quansicoherent sheaves a (semi?)-monoidal topos.
For references on Day convolution see there.
The monoidal classifying topos for the theory of objects is discussed for instance in
A monoidal classifying topos of “toric quasicoherent sheaves” is indicated in
Last revised on July 7, 2018 at 10:28:16. See the history of this page for a list of all contributions to it.