I am reading The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces, and I came across a usage of the words "covariant" and "contravariant" that I did not understand.
Here is the relevant passage:
Generalizing the construction of $C_0(X)$, we define, for a $C^∗$-algebra $A$ and a locally compact space $X$, the $C^∗$-algebra $C_0(X, A)$ to consist of all continuous functions $f$ from $X$ to $A$ that vanish at infinity, equipped with the pointwise algebraic and $∗$-operations. This construction is covariant in $A$ with respect to $∗$-homomorphisms and contravariant in $X$ with respect to proper continuous maps, by means of composition of maps.
I wonder if the categorical meaning of the terms is intended, but as these words are so overloaded, I found it quite difficult to Google. Maybe $C_0(\cdot, A)$ is being considered as a functor from some category of topological spaces to that of $C^*$-algebras, and similarly for $C_0(X, \cdot)$?
As someone illiterate in category theory, here is what the statement says to me.
When $\rho:A\to B$ is a $*$-homomorphism, this induces a $*$-homomorphism $\tilde\rho:C_0(X,A)\to C_0(X,B)$ by $\tilde\rho:f\longmapsto \rho\circ f$. This is "covariant", because it goes the "same way" ($C_0(X,A)$ on the left, $C_0(X,B)$ on the right).
When $h:X\to Y$ is proper continuous, this induces a $*$-homomorphism $\tilde h:C_0(Y,A)\to C_0(X,A)$ by $\tilde h:g\longmapsto g\circ h$. This is "contravariant", because it "reverses the order" ($C_0(Y,A)$ on the left, $C_0(X,A)$ on the right).