Functoriality of infinite loop space machines?

Question

If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \to BD$ is not a map of algebras over that operad (unless all the morphisms $(Fx) \otimes (Fy) \to F(x \otimes y)$ are identities).

Because of this I struggle to associate a spectrum map between the two spectra arising from $BC$ and $BD$ by infinite loop space theory, I can only see how to get a zig-zag where the wrong-way map is a weak equivalence. For most practical purposes that's just as good, but nevertheless I wonder: are any of the "well known" infinite loop space machines functorial on the nose, with respect to symmetric monoidal functors?

Answer 1

Here is a nice gentle old-fashioned answer. Symmetric monoidal categories are functorially equivalent as symmetric monoidal categories to permutative (symmetric strict monoidal) categories, and those are functorially equivalent (essentially the same as) algebras over a certain $E_{\infty}$ operad $\mathcal{P}$, known nowadays as the categorical Barratt-Eccles operad, in Cat. Since $B$ is product preserving it gives a functor from $\mathcal P$-algebras in Cat to $B\mathcal P$-algebras in Top. That gives $B$ as a functor from symmetric monoidal categories to algebras over an $E_\infty$ operad. That goes back, at least in outline, to my 1974 paper ``$E_{\infty}$ spaces, group completions, and permutative categories".

Again in outline, by two recent papers, the same argument works equivariantly for (genuine) symmetric monoidal $G$-categories, which give genuine $G$-spectra for finite groups $G$ via infinite loop $G$-space machines. See Equivariant iterated loop space theory and permutative $G$-categories http://www.math.uchicago.edu/~may/PAPERS/GM3.pdf and Symmetric monoidal $G$-categories and their strictification http://www.math.uchicago.edu/~may/PAPERS/AddCat1.pdf}.