5
$\begingroup$

In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely presentable ($1$ object $*$, $0$ generating $1$-morphisms and $1$ generating $2$-morphism on $\mathrm{id}_*$) and it is not homotopy equivalent to a finite complex. Is this a mistake in the book?

UPD. Judging by this entry, my presentation is still described by $S^2$, and not by $K(\mathbb{Z}, 2)$. Then my question is: where can I read the formal definition of the presentation of $\infty$-categories by generators and relations (and in particular, understand in these terms where higher morphisms arise from here). I don't see any details from Lurie yet.

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ I'm no expert, but I believe a simplicial set presents an $\infty$-category $C$ if they are weakly equivalent in the Joyal model structure on simplicial sets. Similarly, a simplicial set will present an $\infty$-groupoid $X$ if they are weakly equivalent in the standard model structure on simplicial sets (i.e. the one that models homotopy types). From this one can see that $\infty$-groupoids with infinite homological dimension cannot be finite presented (meaning represented by a finite simplicial set). $\endgroup$
    – Connor Malin
    Commented Nov 10, 2023 at 18:06
  • $\begingroup$ Really! And in fact, this is discussed further. It was a stupid question, I am sorry. $\endgroup$
    – Arshak Aivazian
    Commented Nov 10, 2023 at 18:10
  • 1
    $\begingroup$ A big warning is in order: there is a standard notion of a finitely presentable object in a category. Later in the book, Lurie treats the generalization of this notion to $\infty$-categories, calling it a "compact" object. Beware that not every compact object in the $\infty$-category $Spaces$ of spaces is "finitely presentable" in the sense of 1.2.14.2, by the Wall finiteness obstruction. What is true is that every compact object in $Spaces$ is a retract of a "finitely presentable" object in the sense of 1.2.14.2. $\endgroup$
    – Tim Campion ♦
    Commented Nov 14, 2023 at 5:30
  • $\begingroup$ @TimCampion Thanks for this comment! This is a really important clarification (and it helped me now). $\endgroup$
    – Arshak Aivazian
    Commented Dec 9, 2023 at 18:02

1 Answer 1

Reset to default
3
$\begingroup$

I answer the question "where can I read the formal definition of the presentation of ∞-categories by generators and relations?"

You can read about this in the Unicity paper by Barwick and Schommer-Pries. This is in the context of $(\infty,n)$-categories but works also for $n=1$. The idea of gaunt $n$-categories was helpful to me when I was a student, and I think the authors do a good job of spelling out where the higher morphisms come from. The $n = 1$ case was also handled earlier by Toen (but, this paper is in French).

Improve this answer
$\endgroup$