category object in an (∞,1)-category, groupoid object
A $\Gamma$-space is a model for an ∞-groupoid equipped with a multiplication that is unital, associative, and commutative up to higher coherent homotopies: they are models for groupal E-∞ spaces / infinite loop spaces / abelian ∞-groups.
The notion of $\Gamma$-space is a close variant of that of Segal category for the case that the underlying (∞,1)-category happens to be an ∞-groupoid, happens to be connected? and is equipped with extra structure.
Therefore a $\Gamma$-space can be delooped infinitely many times to produce a connective spectrum.
$\Gamma$-spaces differ from operadic models for $E_\infty$-spaces, such as in terms of algebras over an E-∞ operad, in that their multiplication is specified “geometrically” rather than algebraically.
Let $\Gamma^{op}$ denote Segal's category: the skeleton of the category of finite pointed sets. We write $\underline{n}$ for the finite pointed set with $n$ non-basepoint elements. Then a $\Gamma$-space is a functor $X\colon \Gamma^{op}\to Top$ (or to simplicial sets, or whatever other model one prefers).
We think of $X(\underline{1})$ as the “underlying space” of a $\Gamma$-space $X$, with $X(\underline{n})$ being a “model for the cartesian power $X^n$”. In order for this to be valid, and thus for $X$ to present an infinite loop space, a $\Gamma$-space must satisfy the further condition that all the Segal maps
are weak equivalences. We include in this the $0$th Segal map $X(\underline{0}) \to *$, which therefore requires that $X(\underline{0})$ is contractible. Sometimes the very definition of $\Gamma$-space includes this homotopical condition as well.
Note that we have a functor $\Delta\to\Gamma$, where $\Delta$ is the simplex category, which takes $[n]$ to $\underline{n}$. Thus, every $\Gamma$-space has an underlying simplicial space. This simplicial space is in fact a special Delta-space which exhibits the 1-fold delooping of the corresponding $\Gamma$-space.
The topos $\Set^{\Gamma^{op}}$ of $\Gamma$-sets is the classifying topos for pointed objects (MO question).
A model structure on $\Gamma$-spaces can be found in Bousfield and Friedlander below.
The notion goes back to
The model category structure on $\Gamma$-spaces (a generalized Reedy model structure) was established in
See also
C. Balteanu, Z. Fiedorowicz, R. Schwanzl and R. Vogt, Iterated Monoidal Categories, Advances in Mathematics (2003).
B. Badzioch, Algebraic Theories in Homotopy Theory, Annals of Mathematics, 155, 895–913 (2002).
Discussion of $\Gamma$-spaces in the broader context of higher algebra in (infinity,1)-operad theory is around remark 2.4.2.2 of