Definition: Field spaces, frozen

$\mathfrak{F}$ is the space of multiplets $(\gamma ,\varphi ,c,\bar{c},b)$ with $\gamma ,\varphi ,b$ smooth on a neighbourhood of $M_T$, no decay imposed, and $c,\bar{c}\in {C_c}^{\infty }({M_T}^{\circ };TM)\otimes {\Lambda }^{\mathrm{odd}}$ for a finite Grassmann algebra $\Lambda$, together with antifields ${\Phi }_A^{\ast }\in {C_c}^{\infty }({M_T}^{\circ })\otimes \Lambda$ of parity opposite to ${\Phi }^A$ and ghost number $-1-\mathrm{gh}{\Phi }^A$. The relative space is