monoidal derivator


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

Monoidal categories



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.


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.


  • 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.


  • Moritz Groth, Derivators, pointed derivators, and stable derivators (pdf)
Revised on June 14, 2011 06:57:22 by Urs Schreiber (