nLab
monoidal derivator

Contents

Context

(,1)(\infty,1)-Category theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Definition

Monoidal prederivators

Let DiaDia be a suitable 2-category of diagram shapes, as used to define derivators. The 2-category PDerPDer of prederivators is the functor 2-category [Dia op,CAT][Dia^{op},CAT]. As such, it is a naturally cartesian monoidal 2-category.

A monoidal prederivator is simply a pseudomonoid in PDerPDer. This is equivalent to saying that it is a pseudofunctor Dia opMonCatDia^{op} \to MonCat, where MonCatMonCat consists of monoidal categories, strong monoidal functors, and monoidal natural transformations. We may similarly define braided and symmetric monoidal prederivators.

A monoidal semiderivator is a monoidal prederivator which is a semiderivator.

Monoidal derivators

We could define a monoidal derivator to be simply a monoidal prederivator which is a derivator. This is what Groth does. However, we might also want to ask that the tensor product be well-behaved with respect to homotopy Kan extensions, and the natural requirement is that it preserve homotopy colimits in each variable.

Precisely, let DD be a monoidal prederivator which is a derivator; we say that its tensor product preserves homotopy colimits in each variable separately if for any ADiaA\in Dia, the adjunction r !:D(A)D(*):r *r_! \colon D(A) \leftrightarrows D(*) : r^* is a Hopf adjunction, where r:A*r\colon A\to * is the unique functor to the terminal category. (Note that this is automatically a comonoidal adjunction, since r *r^* is strong monoidal.

Explicitly, this means that for any diagram XD(A)X\in D(A) and any object YD(*)Y\in D(*), the canonical transformations

hocolim A(Xr *Y)(hocolim AX)Y hocolim^A (X \otimes r^* Y) \to (hocolim^A X) \otimes Y
hocolim A(r *YX)Y(hocolim AX) hocolim^A (r^* Y\otimes X) \to Y\otimes (hocolim^A X)

are isomorphisms. In this case we say that DD is a monoidal derivator.

Properties

Preservation of certain homotopy Kan extensions

The above preservation condition, though stated only for colimits, also applies to certain homotopy Kan extensions. If p:BAp\colon B\to A is any opfibration, then we can conclude that p !p *p_! \dashv p^* is also a Hopf adjunction as follows. It suffices to show that for any aAa\in A, XD(B)X\in D(B), and YD(A)Y\in D(A), the induced transformation

a *p !(Xp *Y)a *(p !XY) a^* p_! (X\otimes p^* Y) \to a^* (p_! X \otimes Y)

is an isomorphism (and dually). But since pp is an opfibration, the square

p 1(a) g B q p * a A \array{ p^{-1}(a) & \overset{g}{\to} & B \\ ^q\downarrow & & \downarrow^p \\ * & \underset{a}{\to} & A }

is homotopy exact. Therefore, a *p !q !g *a^* p_! \cong q_! g^*, and (since the square commutes) g *p *q *a *g^* p^* \cong q^* a^*, so using the fact that q !q *q_! \dashv q^* is Hopf, we have

a *p !(Xp *Y)q !g *(Xp *Y)q !(g *Xg *p *Y)q !(g *Xq *a *Y)q !g *Xa *Ya *p !Xa *Ya *(p !XY). a^* p_! (X\otimes p^* Y) \cong q_! g^* (X \otimes p^* Y) \cong q_! (g^* X \otimes g^* p^* Y) \cong q_! (g^* X \otimes q^* a^* Y) \cong q_! g^* X \otimes a^* Y \cong a^* p_! X \otimes a^* Y \cong a^* (p_! X \otimes Y).

We leave it to the reader to verify that this composite isomorphism is, in fact, the transformation in question.

Examples

  • Any representable prederivator represented by a monoidal category is a monoidal prederivator. If the monoidal category is complete and cocomplete, and its tensor product preserves colimits on each side, then this is a monoidal derivator.

  • The homotopy derivator of any monoidal model category is a monoidal derivator.

References

  • Moritz Groth, Derivators, pointed derivators, and stable derivators (pdf)

Last revised on June 14, 2011 at 06:57:22. See the history of this page for a list of all contributions to it.