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By analogy, we can consider presenting a GAT where the declarations are arranged in a well founded poset, or define infinitary versions of GAT where the arguments of each operation are arranged in this way. It makes writing them dowm so mu…
In the paper, we use an analogy/duality between dependent stuff and cell complexes: A depending on B is like saying the "boundary" of A is B; and building up a context/telescope is like building a cell complex by attaching cells one by one…
This makes it difficult if we want to have infinitely many things depending on each other, for both kinds of dependencies: We want to define GATs with infinitely many declarations, and infinitely many variables (infinitary) in a declaratio…
The second is that the different declarations (sorts/operations/equations) also depend on declarations coming before them. This already occurs in a restricted way for ordinary algebraic theories, where to declare an operation, you obviousl…
The first is that the input variables can have dependent sorts (and in the case of an operation, the output sort can depend on the input). In the definition of a category, composition takes five variables: three objects X,Y,Z and two arrow…
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