[Segal's category](http://ncatlab.org/nlab/show/Segal%27s%20category) $\Gamma$ 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 $\Delta\_a \to \Gamma$.



Where does Segal's category come from?
Infinite loop spaces and spectra are intrinsically pointed, and the purpose of the basepoint is to build in basepoints, which give the units for the associated products. Let $T\_\*$ be the category of based objects in any cartesian monoidal category T. For an object $X$ of $T\_\*$, a covariant functor $X^\*: F\_\* \longrightarrow T\_\*$ that sends the based set $n=\{0,1, \cdots, n\}$ with basepoint $0$ to $X^n$ is precisely a commutative monoid in $T$ with its unit element equal to the basepoint of $X$. Use of basepoints like this long precedes Segal. When $T$ is spaces, the James construction $JX$ is the free monoid on $X$ with unit the basepoint of $X$ and the infinite symmetric product $NX$ on $X$ is the free commutative monoid on $X$ with its unit element the basepoint of $X$, both suitably topologized. 


In more detail, the morphisms of $F$ are generated under composition by injections, projections, and the based maps $\phi\_n \colon n\to 1$ that send $i$ to $1$ for $1\leq i\leq n$. (Using the wedge sum, only $\phi\_2$ need be added). The morphisms $\pi\colon m\to n$ such that $\phi^{-1}(i)$ has $0$ or $1$ element give a subcategory $\Pi$ of $F$, and the functor $X^\*$ has underlying functor $\Pi\to T\_\*$ given by the injections (determined by the basepoint), projections, and permutations that are given by the assumption than $n \mapsto X^n$. The map $\phi\_2$ gives a product $X\times X\longrightarrow X$ and, more generally, the $\phi\_n \colon X^n\longrightarrow X$ give the unique $n$-fold product determined by $\phi\_2$.


The point of infinite loop space theory is to build in the axioms of a commutative monoid up to "all higher coherence homotopies", and the genius of Segal was to see that the evident maps $\delta\_i\colon n\longrightarrow 1$ that send $i$ to $1$ and $j$ to $0$ for $j\neq i$ can be used to build in these homotopies. Taking $T$ to be spaces, for a functor $Y\colon F\_\* \longrightarrow T\_\*$, we have based spaces $Y\_n$, and the $\delta\_i$ determine the Segal maps $\delta^n\colon Y\_n \longrightarrow Y\_1^n$. Requiring these maps to be homotopy equivalences for all $n\geq 0$ makes $Y\_1$ a "commutative monoid up to all higher coherence homotopies", with canonical zigzag product $Y\_1^2 \longleftarrow Y\_2 \longrightarrow Y\_1$ determined by $\delta^2$ and $\phi\_2$. 


I could go on forever about this. But maybe I'll just give the definition of the functor $K\colon \Delta^{op}\longrightarrow F\_\*$. I agree that its interpretation may not be obvious. Of course, $Kn = n$. For a map $\phi\colon n\longrightarrow m$ in $\Delta$, define 
$K\phi\colon m\longrightarrow n$ by sending $i$ to $j$ whenever $\phi(j-1) < i \leq \phi(j)$,
where $1\leq j\leq n$, and sending $i$ to $0$ if there is no such $j$. Thus
$$ (K\phi)^{-1}(j) = \{ i | \phi(j-1) < i \leq \phi(j)\} \ \ \text{for} \ \ 1\leq j\leq n. $$


Please excuse the lousy ad hoc notation (F= finite sets).


