monoidal topos


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Every topos H\mathbf{H} is canonically a cartesian monoidal category (H,×)(\mathbf{H}, \times), with the tensor product being the cartesian product. If in addition to this canonical product H\mathbf{H} is equipped with another noncartesian monoidal category structure \otimes, then (H,)(\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 (𝒞,)(\mathcal{C}, \otimes) a small monoidal category, the presheaf topos [𝒞,Set][\mathcal{C}, Set] is naturally a monoidal topos with respect to the Day convolution tensor product.


For H\mathbf{H} a topos and 𝔾H\mathbb{G} \in \mathbf{H} a monoid object, the slice topos H /𝔾\mathbf{H}_{/\mathbb{G}} inherits a non-cartesian monoidal structure G\otimes_G induced from the product structure on 𝔾\mathbb{G}:

[X 1 χ 1 𝔾] 𝔾[X 2 χ 2 𝔾][X 1×X 2 (χ 1,χ 2) 𝔾×𝔾 𝔾]. \left[ \array{ X_1 \\ \downarrow^{\mathrlap{\chi_1}} \\ \mathbb{G} } \right] \otimes_{\mathbb{G}} \left[ \array{ X_2 \\ \downarrow^{\mathrlap{\chi_2}} \\ \mathbb{G} } \right] \;\; \coloneqq \;\; \left[ \array{ X_1 \times X_2 \\ \downarrow^{\mathrlap{(\chi_1, \chi_2)}} \\ \mathbb{G} \times \mathbb{G} \\ \downarrow^{\mathrlap{\cdot}} \\ \mathbb{G} } \right] \,.

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][FinSet, Set] on the opposite category of FinSet or equivalently the category of finitary endofunctors End f(Set)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 3, where one considers presheaves on the “̲\underline{\;\;\;\;\;}” monoidal category MM 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 opFinSet^{op} is the free cartesian monoidal category generated by a single object, but the blank might be filled in with “symmetric” (MM is the permutation category =Core(FinSet)\mathbb{P} = Core(FinSet)), or “braided” (the braid category 𝔹\mathbb{B}), or “semicartesian symmetric monoidal” (Fin injFin_{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]E = [M^{op}, Set] acquires a Day convolution structure, but more interestingly, we have an equivalence

E{E,E}E \simeq \{E, E\}

where the right side is the category of cocontinuous ̲\underline{\;\;\;\;\;} monoidal endofunctors. By transferring the endofunctor composition across the equivalence, EE 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

  • James Dolan, tannakian correspondence for toric varieties (sketch for a doctoral thesis), December 2011 (web)

Last revised on July 7, 2018 at 10:28:16. See the history of this page for a list of all contributions to it.