Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that $$f(x\_1,\dots,x\_n)=-x\_1^2-x\_2^2-\dots-x^2\_{i}+x^2\_{i+1}+\dots+x\_n^2.$$ 
There is an interpretation of Morse functions as "generic" functions. More precisely, one gives a stratification of $J^2(M,\mathbb{R})$---the second jet space of maps from $M$ to $\mathbb{R}$, and Morse functions are those functions whose second jet is transverse to the strata. By the Thom transversality theorem, the resulting space of functions is dense and open in $C^\infty(X,\mathbb{R})$.


The space of Morse functions is not connected. To make it connected one can introduce generalized Morse functions, where the critical points have a local description as above or as follows
$$f(x\_1,\dots,x\_n)=-x\_1^2-x\_2^2-\dots-x^2\_{i}+x^2\_{i+1}+\dots+x\_{n-1}^2+x\_n^3.$$
We call the latter a birth-death singularity. The result in this case is that a "generic" smooth homotopy $M\times I\rightarrow \mathbb{R}$ is a Morse function at the endpoints and it is a Morse function at all values of the parameter, $t\in I$, except for finitely many values, where the function is a generalized Morse function.


What I am interested in is a generalization of Cerf's result, where the parameterizing space $I$ is replaced by an n-simplex. Here is my question.



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> Is there a "nice" description of a generic family of smooth functions of the form $M\times \Delta^n\rightarrow \mathbb{R}$?
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By "nice" I mean having finitely many critical points and the critical points have local descriptions similar to the ones given above.


The goal roughly is to construct a simplicial set, where $n$-simplices are the generic smooth maps $M\times\Delta^n\rightarrow \mathbb{R}$. The hope is that given the transversality results one should be able to show that the simplicial set is contractible, since it would amount to extending a generic function $M\times\partial\Delta^n\rightarrow \mathbb{R}$ to a generic functions $M\times\Delta^n\rightarrow \mathbb{R}$, which we would do by first extending by arbitrary smooth function (the space of all smooth functions is contractible) and then deform it something generic relative to a neighborhood of the boundary.



Higher Cerf Theory
This is what catastrophe theory does, at least for small $n$, $n\leq 10$. Volume 1 of the book by Arnold, Gussein-Zade and Varchenko on singularities has a nice description of this theory; see especially Part 2 of that book.


