The n-Category Café

According to 1.11 of Joyal’s notes here - https://www.math.uchicago.edu/~may/IMA/Joyal.pdf , the functor $\tau_1: S \to Cat$ preserves finite products by a result of Gabriel and Zisman, but I don’t know what the exact reference is. Maybe it’s really old?

I just checked in Gabriel-Zisman and can’t find it. I doubt there’s an older reference, so it might be that none exists. Depending on what you need, it seems to me like the best option might be to cite GZ or some newer reference for the explicit description of the left adjoint, then check the claim from there.

A published reference (and one which places the result in a general framework) can be found in the theory of test categories, for example Proposition 1.6.14 in La théorie de l’homotopie de Grothendieck, e.g. at this link.

But I am sure there are much older ones.

Okay, here’s a nice proof of this fact and many generalizations thereof:

• P. Karazeris and G. Protsonis, Left Kan extensions preserving finite products, Jour. Pure Appl. Alg.216 (2012), 2014-2028.

They refer to this fact in the abstract and call it “well-known”.

In Quillen’s Higher Algebraic K-Theory I he cites a paper of Milnor for this fact.

How about this? Start with $i: \Delta \to \mathbf{Cat}$ (Definition 3.4 in the nLab page you linked to) which preserves finite products. Note that this induces realization = $\vert \cdot \vert = F \dashv N$ = nerve. (I like Thm. 4.51 in Kelly BCECT for that result.) Note that $F = \text{Lan}_Y i$ (4.52 in Kelly), where $Y \colon \Delta \to \Delta\mathcal{P}$. Now apply Street’s result that Lan’s of FPPF are FPPF (he gives many antecedents for that result). Street says “This kind of result goes back at least to Bill Lawvere’s thesis and some 1966 ETH notes of Fritz Ulmer.”

I realize a reference is wanted, not a proof. But the proof is a lot easier than the references make it sound.

Consider categories $C,D$ which have colimits and finite products. Suppose products distribute over colimits in both, i.e., ${-}\times X$ is colimit preserving for any object $X$ in these categories: this is automatic if $C$ and $D$ are cartesian closed (e.g., $\mathrm{Cat}$ and $\mathrm{sSet}$) because ${-}\times X$ is then a left adjoint.

Let $F\colon C\to D$ be a colimit preserving functor, and suppose there is a class $S$ of objects in $C$ such that $F(A\times B)\xrightarrow{\sim} F A\times F B$ for all objects $A,B\in S$. Then the same remains true for $A,B$ which are colimits of diagrams of objects of $S$, by an elementary argument: $$F(\mathrm{colim}_i A_i\times \mathrm{colim}_j B_j) \approx F(\mathrm{colim}_{i j} A_i\times B_j)$$ $$\qquad\qquad\qquad\approx \mathrm{colim}_{i j} F(A_i\times B_j) \approx \mathrm{colim}_{i j} F A_i\times F B_j$$ $$\qquad\qquad\qquad\qquad\qquad\qquad\approx \mathrm{colim}_i F A_i \times \mathrm{colim}_j F B_j \approx F(\mathrm{colim}_i A_i) \times F(\mathrm{colim}_j B_j).$$ So to prove that $F$ preserves binary products, it suffices to find such an $S$ which generates $C$ under colimits.

In our situtation, $C=\mathrm{sSet}$, $D=\mathrm{Cat}$, and $F$ is the left adjoint to the nerve functor $N\colon D\to C$; note that $N$ is fully faithful. $S$ is the collection of standard simplices (representable functors on $\Delta$), and $F$ preserves products of such objects because $S$ is contained in the essential image of the fully faithful right adjoint $N$, and $F N$ is equivalent to the identity functor (and thus product preserving), again because $N$ is fully faithful.

(I think that in the special case of $C$ a presheaf category and $D$ a Grothendieck topos, this is exactly 4.5 of the Karazaris and Protzonis article.)