Cartesian and cocartesian monoidal structures

Every \(\infty \)-category \(C\) with finite products admits a cartesian monoidal structure in which the tensor product of two objects is their cartesian product; the higher coherences required to make this an honest symmetric monoidal structure are uniquely determined by the universal property of the product. Dually, if \(C\) admits finite coproducts, it admits a cocartesian monoidal structure. The goal of this chapter is to construct these two symmetric monoidal structures.

As a first step, we reformulate the problem in terms of the construction of coherent adjoints. Recall that \(C\) admits finite products if and only if for every finite set \(I\) the diagonal functor \(C \to C^I\) admits a right adjoint; for \(I = \emptyset \) this encodes the terminal object, while for \(I = \{1,2\}\) this returns the cartesian product. As a consequence, the restriction functors \(C^I \to C^J\) have right adjoints for all morphisms of finite sets \(J \to I\). Our task in constructing the cartesian monoidal structure on \(C\) is then to assemble both the contravariant restrictions as well as their covariant right adjoints into a single functor \[ \Span (\Fin ) \to \Cat _{\infty }, \qquad I \mapsto C^I. \] The cocartesian monoidal structure should be constructed similarly by extending by left adjoints instead.

The general construction that allows us to carry this out is due to Barwick (2017), and is known as unfurling. Its input is a contravariant functor whose restriction functors admit left or right adjoints satisfying a certain compatibility known as the Beck–Chevalley condition. Its output is a functor on a span category, which on backwards legs restricts to the original functor and on forward legs implements the adjoints. Applying the unfurling construction to \(I \mapsto C^I\) then produces the two monoidal structures we are after.

If \(C\) is not assumed to admit coproducts, the first steps in the unfurling procedure still make sense and produce an \(\infty \)-operad \(\OpCocart _C\), which we refer to as the cocartesian operad of \(C\). The reason for this is that the universal property of a coproduct \(x_1 \sqcup \dots \sqcup x_n\) tells us how to map out of it: we need to specify maps out of each object \(x_i\). Even if the coproduct does not actually exist, the formula \[ \OpCocart _C(x_1,\dots ,x_n;y) \simeq \prod _{i=1}^n\Hom _C(x_i,y) \] can be used to define the multimorphism animae of an \(\infty \)-operad.

The algebras over \(\infty \)-operads of the form \(\OpCocart _C\) are surprisingly simple: they correspond to \(C\)-indexed families of commutative algebras. To see why this is the case at an intuitive level, note that the cocartesian operad for the terminal \(\infty \)-category \(*\) is precisely the commutative operad \(\Comm \). The assignment \(C \mapsto \OpCocart _C\) is functorial, and so every object of \(C\) determines an operad map \(\Comm \to \OpCocart _C\), and hence evaluating an \(\OpCocart _C\)-algebra at this object produces a commutative algebra. We will show in Section 15.2 that an \(\OpCocart _C\)-algebra is completely captured by the resulting \(C\)-indexed family of commutative algebras.

The classification of algebras over cocartesian operads directly leads to a classification of cocartesian operads themselves: an \(\infty \)-operad \(\Oo \) is of the form \(\OpCocart _C\) for some \(C\) if and only if the multimorphism anima in \(\Oo \) from a disjoint union of unordered tuples is the product of the individual multimorphism animae out of each of the constituents. This then gives a direct classification of the cocartesian monoidal structures, resulting in an equivalence of \(\infty \)-categories \[ \Cat _{\infty }^{\mathrm {coprod}} \iso \Cat _{\infty }^{\otimes ,\cocart } \] between \(\infty \)-categories with finite coproducts and cocartesian symmetric monoidal \(\infty \)-categories. The analogous result for cartesian monoidal structures follows by duality.

Here and below, \(\Cat _{\infty }^{\mathrm {coprod}}\) denotes the \(\infty \)-category of \(\infty \)-categories with finite coproducts and finite-coproduct-preserving functors.

We end the chapter by proving a characterization of algebras in cartesian monoidal structures: for every \(\infty \)-category with finite products \(C\) and every \(\infty \)-operad \(\Oo \), we construct an equivalence of \(\infty \)-categories \[ \Alg _{\Oo }(C, \times ) \iso \Mon _{\Oo }(C) \] between \(\Oo \)-algebras and \(\Oo \)-monoids, i.e. finite-product-preserving functors \(\Oo ^{\otimes } \to C\). In particular, commutative algebras in \((C,\times )\) are nothing but the commutative monoids studied in Chapter 5. The proof of this equivalence proceeds by constructing an alternative model for the cartesian monoidal structure on \(C\).

The unfurling theorem is the technical input for the chapter. On a first reading, the proofs in Subsection 15.1.1 may be skipped. Several statements established there will be used later, notably the criterion for cocartesian morphisms in a span category and the adequate triple associated with a cartesian fibration.

Generated from the authoritative LaTeX source.