The Stacks project

Definition 82.26.1. In Situation 82.2.1 let $f : X \to Y$ be a morphism of good algebraic spaces over $B$. Let $p \in \mathbf{Z}$. A bivariant class $c$ of degree $p$ for $f$ is given by a rule which assigns to every morphism $Y' \to Y$ of good algebraic spaces over $B$ and every $k$ a map

where $X' = Y' \times _ Y X$, satisfying the following conditions

1. if $Y'' \to Y'$ is a proper morphism, then $c \cap (Y'' \to Y')_*\alpha '' = (X'' \to X')_*(c \cap \alpha '')$ for all $\alpha ''$ on $Y''$,

2. if $Y'' \to Y'$ a morphism of good algebraic spaces over $B$ which is flat of relative dimension $r$, then $c \cap (Y'' \to Y')^*\alpha ' = (X'' \to X')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$,

3. if $(\mathcal{L}', s', i' : D' \to Y')$ is as in Definition 82.22.1 with pullback $(\mathcal{N}', t', j' : E' \to X')$ to $X'$, then we have $c \cap (i')^*\alpha ' = (j')^*(c \cap \alpha ')$ for all $\alpha '$ on $Y'$.

The collection of all bivariant classes of degree $p$ for $f$ is denoted $A^ p(X \to Y)$.