nLab monoidal topos

Idea

Every topos $\mathbf{H}$ is canonically a cartesian closed monoidal category $(\mathbf{H}, \times)$, with the tensor product being the cartesian product “$\times$”. But if, in addition to this canonical product, $\mathbf{H}$ is equipped with another (hence necessarily non-cartesian) monoidal tensor product structure “$\otimes$”, then $(\mathbf{H}, \otimes)$ might be called a monoidal topos.

If the extra monoidal structure is also closed, one also speaks, generally, of a doubly closed monoidal category-structure (with internal logic a bunched logic). Hence instead of “monoidal topos” it might make sense to speak of “doubly 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.

Examples

Example

For $(\mathcal{C}, \otimes)$ a small monoidal category, the presheaf topos $[\mathcal{C}, Set]$ is naturally a doubly closed monoidal topos with respect to the Day convolution tensor product.

Example

(slice topos over a monoid object)
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 $\mathbb{G}$:

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

Such monoidal toposes are always biclosed and appear as categories of local action functionals in the context of motivic quantization.

Remark

This Ex. overlaps with the previous Ex. in the case where the given topos is $Set$ and the small category $\mathcal{C}$ is the discrete category on a small set $X$. In that case, the monoidal category structure is just a monoid structure on $X$ and the two definitions of a monoidal structure agree up to isomorphism.

Example

The classifying topos for the theory of objects – namely 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).

Example

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

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

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.

Example

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.

Example

In [Dolan 2011] is indicated a category of “toric quasicoherent sheaves” which is a topos and hence under tensor product of quansicoherent sheaves a (semi?)-monoidal topos.

Example

Given an $\infty$-topos $\mathbf{H}$, its tangent $\infty$-topos $T \mathbf{H}$ carries the external smash product of spectra.

Similarly, given a stable $\infty$-category $\mathcal{A}$ (or more generally a Joyal locus, see there for more), the (infinity,1)-category $\int_{\mathcal{X} \in Grpd_\infty} Func(\mathcal{X}, \mathcal{A})$ of $Grpd_\infty$-parameterized $\mathcal{A}$-objects (is an $\infty$-topos and) should carry the corresponding external tensor product.

Related pages

• Day convolution

• doubly closed monoidal category

• bunched logic

References

For references on Day convolution see there.

The monoidal classifying topos for the theory of objects is discussed for instance in

• Peter Johnstone, section D3.2 in Sketches of an Elephant

• Richard Garner, Lawvere theories, finitary monads and Cauchy-completion (arXiv:1307.2963)

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)