I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; what happens as they change?

• What if I have an injective function of finite sets $m\to n$; this should induce a map ${\mathcal M}_{g,n}(X,\beta)\to{\mathcal M}_{g,m}(X,\beta)$, assuming both exist, and I want a reference for what I can expect of that map, or of the inverse limit of such maps over $n\in{\bf Fin}^{Inj}$.

• What if I have a morphism $X\to Y$ of ambient spaces. What can I know about the relationship between the associated moduli spaces?

• What if I add or multiply cohomology classes, e.g. $\beta\cap\beta'$? Anything to see here?

• Is there any reasonable notion of changing the genus $g$ so that a map between moduli spaces might be induced?

I'd like a single reference where I can find such answers, and I'd like it to have a category-theoretic flavor, phrasing as much as possible in terms of functorial relationships. If that can't be found, any help on the above types of questions would also be appreciated.

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; what happens as they change?

• What if I have an injective function of finite sets $m\to n$; this should induce a map ${\mathcal M}_{g,n}(X,\beta)\to{\mathcal M}_{g,m}(X,\beta)$, assuming both exist, and I want a reference for what I can expect of that map, or of the inverse limit of such maps over $n\in{\bf Fin}^{Inj}$.

• What if I have a morphism $X\to Y$ of ambient spaces. What can I know about the relationship between the associated moduli spaces?

• What if I add cohomology classes, e.g. $\beta+\beta'$? Anything to see here?

• Is there any reasonable notion of changing the genus $g$ so that a map between moduli spaces might be induced?

I'd like a single reference where I can find such answers, and I'd like it to have a category-theoretic flavor, phrasing as much as possible in terms of functorial relationships. If that can't be found, any help on the above types of questions would also be appreciated.

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; what happens as they change?

• What if I have an injective function of finite sets $m\to n$; this should induce a map ${\mathcal M}_{g,n}(X,\beta)\to{\mathcal M}_{g,m}(X,\beta)$, assuming both exist, and I want a reference for what I can expect of that map, or of the inverse limit of such maps over $n\in{\bf Fin}^{Inj}$.

• What if I have a morphism $X\to Y$ of ambient spaces. What can I know about the relationship between the associated moduli spaces?

• What if I add or multiply cohomology classes, e.g. $\beta\cap\beta'$? Anything to see here?

• Is there any reasonable notion of changing the genus $g$ so that a map between moduli spaces might be induced?

I'd like a single reference where I can find such answers, and I'd like it to have a category-theoretic flavor, phrasing as much as possible in terms of functorial relationships.

I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; what happens as they change?

• What if I have an injective function of finite sets $m\to n$; this should induce a map ${\mathcal M}_{g,n}(X,\beta)\to{\mathcal M}_{g,m}(X,\beta)$, assuming both exist, and I want a reference for what I can expect of that map, or of the inverse limit of such maps over $n\in{\bf Fin}^{Inj}$.

• What if I have a morphism $X\to Y$ of ambient spaces. What can I know about the relationship between the associated moduli spaces?

• What if I add or multiply cohomology classes, e.g. $\beta\cap\beta'$? Anything to see here?

• Is there any reasonable notion of changing the genus $g$ so that a map between moduli spaces might be induced?

I'd like a single reference where I can find such answers, and I'd like it to have a category-theoretic flavor, phrasing as much as possible in terms of functorial relationships. If that can't be found, any help on the above types of questions would also be appreciated.