## August 31, 2019

### From Simplicial Sets to Categories

#### Posted by John Baez

There’s a well-known nerve of a category, which is a simplicial set. This defines a functor

$N \colon Cat \to sSet$

from the category of categories to the category of simplicial sets. This has a left adjoint

$F \colon sSet \to Cat$

and this left adjoint preserves finite products.

Do you know a published reference to a proof of the last fact? A textbook explanation would be best, but a published paper would be fine too. I don’t want you to explain the proof, because I think I understand the proof. I just need a reference.

Posted at August 31, 2019 6:49 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3136

### Re: From Simplicial Sets to Categories

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?

Posted by: h on August 31, 2019 4:57 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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.

Posted by: Kevin Carlson on August 31, 2019 6:14 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Thanks! I feel silly writing up a proof since 1) the paper I need this result for is about theoretical computer science; it’s not really the optimal place for a proof of an old piece of homotopy theory folklore, 2) I can’t believe yet that nobody has published a proof already!

Maybe for now I’ll just sketch the argument, say it’s folklore and cite Joyal. But I’m still hoping someone here can find a published proof!

Posted by: John Baez on September 1, 2019 4:26 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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.

Posted by: Richard Williamson on September 1, 2019 8:04 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Thanks! The proposition says that $\Delta$ is a ‘strict test category’. I guess if I dug around enough I’d see that implies what I want?

Posted by: John Baez on September 3, 2019 2:55 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Yes, although only up to homotopy a priori.

Posted by: Richard Williamson on September 3, 2019 11:11 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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

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

Posted by: John Baez on September 3, 2019 4:40 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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

Posted by: Tim Campion on September 3, 2019 6:00 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Sorry that’s wrong. I was thinking of geometric realization preserving finite products.

Posted by: Tim Campion on September 3, 2019 6:18 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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.”

Posted by: Keith Harbaugh on September 3, 2019 5:20 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

But $\Delta$ doesn’t have products (?).

Posted by: Todd Trimble on September 3, 2019 11:47 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Touché!

Posted by: Keith Harbaugh on September 4, 2019 4:02 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Well, I didn’t mean it as a gotcha! Sometimes in these situations an adjustment can be made; my comment meant I didn’t know how that particular idea would be rescued. (Meanwhile Alexander Campbell pointed to Joyal’s proof, which I consider just the type of simple argument one should shoot for.)

Posted by: Todd Trimble on September 5, 2019 1:03 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

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.)

Posted by: Charles Rezk on September 3, 2019 5:54 PM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

A reference for this argument (in the case of interest) is Proposition B.0.15 of Joyal’s oft-cited lecture notes The theory of quasi-categories and its applications.

Posted by: Alexander Campbell on September 4, 2019 10:00 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Hey, great! I read this just after I submitted my paper with Christian Williams for publication, but we can add this reference later! Heck, I’ll add it now:

That’s indeed a nice proof, Charles.

Posted by: John Baez on September 6, 2019 4:39 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

For some variation in an orthogonal direction, i.e., not the enriching category but the kind of algebraic theory, there’s the recent

• Soichiro Fujii, A unified framework for notions of algebraic theory, (arXiv:1904.08541),

looking to unify clones seen as “more or less equivalent to Lawvere theories” with

several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads.

Posted by: David Corfield on September 9, 2019 8:57 AM | Permalink | Reply to this

### Re: From Simplicial Sets to Categories

Expressing the argument at an intermediate level of abstraction, it seems to show that if $D$ is cocomplete and cartesian closed with a small dense subcategory $S\hookrightarrow D$, then the left adjoint to the fully faithful restricted Yoneda embedding $D \hookrightarrow [S^{op},Set]$ preserves products.

Posted by: Mike Shulman on September 6, 2019 10:23 AM | Permalink | Reply to this

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