In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral:
\begin{equation} 
\int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ) \psi \right] 
\end{equation} 
a functional determinate,
\begin{equation} 
\det \left( i \gamma_\mu D^\mu - m \right) 
\end{equation} 
I've never heard this way of thinking about it. Why would the generating functional be a functional determinate?