Segal's category $\Gamma$ is the skeleton of the category $\text{FinSet}_{\ast}$ of pointed finite sets. It is used to write down $\Gamma$-spaces, which are functors $\Gamma \to \text{Top}$ satisfying some conditions, and which model infinite loop spaces. I would like to be able to tell myself a story about this category which would explain in some sense why one might have come up with it as a candidate to be part of a delooping machine.

For comparison, here is the analogous story about $\text{FinSet}$: equipped with disjoint union, it is the free symmetric monoidal category on a commutative monoid. (I guess all of my stories are universal properties.) Since infinite loop spaces are in particular supposed to be like homotopy coherent commutative monoids I can see how one might have come up with $\text{FinSet}$ as a candidate to model infinite loop spaces, but not $\text{FinSet}_{\ast}$.

Riffing off of the above, it seems like $\text{FinSet}_{\ast}$, equipped with wedge sum, is the free symmetric monoidal category on something like a "copointed" commutative monoid; that is, a commutative monoid together with a map $\varepsilon : M \to 1$. The idea is that $\text{FinSet}_{\ast}$ can equivalently be thought of as the category of sets and partial functions, and throwing in a map $\varepsilon : M \to 1$ lets us model partial functions by using $\varepsilon$ to throw away the points at which a partial function isn't defined.

Why is this, and not $\text{FinSet}$, a reasonable category to use to model infinite loop spaces? (The inclusion of $\varepsilon : M \to 1$ is particularly strange because in any cartesian monoidal category, such as $\text{Top}$, it is unique because $1$ is the terminal object.) Does something go horribly wrong if we try to reproduce the $\Gamma$-space story using $\text{FinSet}$ instead, and if so, what is it?

Segal's category $\Gamma$ is the skeleton of the category $\text{FinSet}_{\ast}$ of pointed finite sets. It is used to write down $\Gamma$-spaces, which are functors $\Gamma \to \text{Top}$ satisfying some conditions, and which model infinite loop spaces. I would like to be able to tell myself a story about this category which would explain in some sense why one might have come up with it as a candidate to be part of a delooping machine.

For comparison, here is the analogous story about $\text{FinSet}$: equipped with disjoint union, it is the free symmetric monoidal category on a commutative monoid. (I guess all of my stories are universal properties.) Since infinite loop spaces are in particular supposed to be like homotopy coherent commutative monoids I can see how one might have come up with $\text{FinSet}$ as a candidate to model infinite loop spaces, but not $\text{FinSet}_{\ast}$.

Riffing off of the above, it seems like $\text{FinSet}_{\ast}$, equipped with wedge sum, is the free symmetric monoidal category on something like a "copointed" commutative monoid; that is, a commutative monoid together with a map $\varepsilon : M \to 1$. The idea is that $\text{FinSet}_{\ast}$ can equivalently be thought of as the category of sets and partial functions, and throwing in a map $\varepsilon : M \to 1$ lets us model partial functions by using $\varepsilon$ to throw away the points at which a partial function isn't defined.

Why is this, and not $\text{FinSet}$, a reasonable category to use to model infinite loop spaces? (The inclusion of $\varepsilon : M \to 1$ is particularly strange because in any cartesian monoidal category, such as $\text{Top}$, it is unique because $1$ is the terminal object.) I guess $\varepsilon$ is needed to get an inclusion of $\Delta^{op}$ into $\Gamma$ so we can define the geometric realization of a $\Gamma$-space, but now I don't understand why there should be such an inclusion; the universal properties don't suggest it. The augmented simplex category $\Delta_a$ is the free monoidal category on a monoid, so the universal properties suggest instead a monoidal functor $\Delta_a \to \Gamma$.