A neutral Tannakian category over field \(k\) is a rigid \(k\)-linear abelian tensor category \(\mathcal{C}\)
whose unit \(1\) satisfies \(\mathrm{End}(1)\cong k\)
equipped with an exact and faithful (does exact faithful = faithfully exact?) tensor functor that is called the (neutral) fiber functor
\[\omega:\mathcal{C}\to {\bf Vect}_k\]
into the category of finite-dimensional \(k\)-vector spaces.
The category \(\mathrm{Rep}_k(G)\) of finite-dimensional representations of an affine group scheme \(G\) over \(k\), equipped with the forgetful functor \(\mathrm{Rep}_k(G)\to {\bf Vect}_k\), is a neutral Tannakian category.
One interesting question people in representation theory have thought about is, how do we recover the group \(G\) from the category \(\mathrm{Rep}_k(G)\)? This is the essence of the Tannakian formalism.
The first possibility one can try is to consider some symmetry group on some of the structures of the category itself, if we consider the automorphism group of the fiber functor \(\omega\), we get
\[\mathrm{Aut}(\omega) = (kG)^\times,\]
where \(kG\) is the group algebra of \(G\). The result is nearly \(G\) but contains more terms like what we get in a polynomial.
If we add a ‘multiplicative’ requirement, we can get the ‘monomials’ back from \((kG)^\times\), i.e. we require the automorphisms to respect the tensor structure, asking a subset \(\mathrm{Aut}_\otimes(\omega)\subset \mathrm{Aut}(\omega)\) for which \(\alpha_{M\otimes N} = \alpha_M\otimes \alpha_N\) and \(\alpha_{1} = 1\). This time we get
\[\mathrm{Aut}_\otimes(\omega) = G,\]
which is the group we started with owo