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First some notation: Let $\mathscr{F}_* $ be the category of finite pointed sets and pointed maps between them. Then $\Gamma^{op}$ is the full subcategory of $\mathscr{F}_* $ with objects the sets $k_+$ of numbers from $0$ to $k$ pointed at $0$.

If $X$ is a simplicial set then the elements of $X_0$ are pictured as points, the elements in $X_1$ are pictured as line segments and so on.

Main question: Is there a corresponding interpretation of $\Gamma$-sets? (i.e., presheaves on $\Gamma$)

Bonus questions: If so, is there an associated homotopy theory?

There seem to be quite a few of these "combinatorial categories", like $\Delta$, $\Gamma$ and $\Box$. Which of these have good geometric interpretations?

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Did you mean to say that $\Gamma^{op}$ is the opposite of the full subcategory you mention? –  Kevin Walker Sep 3 '11 at 19:13
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Hmmm... it would seem that $\Gamma$ (or $\Gamma^{op}$, with Moi over Walker) is what you might call the star-graph category. Maybe you thought of that already; if not, the motivation isn't difficult. My unfounded intuition is that star graphs might be good for modeling thin homotopy, but not any nontrivial homotopy. But I don't have much reason for that, beyond that star graphs are trees, and hence very thin. –  some guy on the street Sep 3 '11 at 19:47
    
@Kevin: I'm following the conventions I could find in the literature (papers of Segal, Lydakis and Schwede). Anyway, what I want is (covariant) functors from a skeletal subcatory of finite pointed sets. @some guy: I didn't think of that, I'll look into it. –  K.J. Moi Sep 3 '11 at 20:36
    
I got confused by double "op"s. Segal's category $\Gamma$ is the opposite of the category of sets you mention, which is of course why you defined $\Gamma^{op}$ to be that category. Sorry about the mix-up. –  Kevin Walker Sep 9 '11 at 19:44
    
You should look at Cisinski's work (math.univ-toulouse.fr/~dcisinsk/ast.pdf). He studies the so-called test categories: these are categories such that presheaves over them have a Quillen model structure whose homotopy category is equivalent to the homotopy category of spaces. $\Gamma^{op}$ is one of these test categories –  Geoffroy Horel Sep 14 '11 at 14:32

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