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?

oppositeof the full subcategory you mention? – Kevin Walker Sep 3 '11 at 19:13