Where does Segal's category come from?

Just a remark about the functor $\Delta^{op}\to \Gamma$. One way to think about it is this:

An object of $\Delta$ is a nonempty ordered finite set $S$. To such a thing, associate another finite ordered set whose elements are all possible "cuts" in $S$: all ways of dividing $S$ into a left part and a right part, including the two extremes where either the left or the right part is empty. This establishes an equivalence of categories between

$\Delta=$(nonempty ordered finite sets, monotone maps)

and the opposite of

(ordered finite sets with two endpoints, monotone end-point-preserving maps).

Now map the latter to

(based finite sets, based maps)

by the functor that identifies the two endpoints.

Not an answer really, just a few random remarks.

Segal's category $\Gamma$ can be described as the one with

• objects: finite sets $S$,
• morphisms $S\to T$: pairs $(T_0\subseteq T,f\colon T_0\to S)$.

Composition is like composition of spans. This is basically the definition Segal gives (Segal's description is a bit more obscure than this.)

As Peter comments, $\mathrm{FinSet}_*$ is essentially opposite category of Segal's $\Gamma$, and $\Gamma$-spaces are presheaves on $\Gamma$. The convention seems to have shifted to write $\Gamma$ for $\mathrm{FinSet}_*$ itself; I think that turns out to be a bad choice.

Any abelian group $A$ gives you a presheaf $F$ on $\Gamma$ by: $F(S)=\{\text{functions $S\to A$}\}$, with $F(T)\to F(S)$ defined by "restrict to $T_0$, then integrate along $f$".

The functor $\Delta\to \Gamma$ is a bit funny to describe, and it isn't actually an inclusion of a subcategory (there's only one map $\{\}\to \{1\}$ in $\Gamma$, but two maps $[0]\to [1]$ in $\Delta$.)

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).