Homotopy Type Theory net > history (Rev #11)


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A net is a function a:IAa: I \to A from a directed type II to a type AA. II is called the index type, the terms of II are called indices (singular index), and AA is called the indexed type.


Let II be a preordered type. Given a term i:Ii:I, the positive cone of II with respect to ii is defined as the type

I i + j:IijI^+_i \coloneqq \sum_{j:I} i \leq j

with monic function f:I i +If:I^+_i \to I such that for all terms j:Ij:I, if(j)i \leq f(j).

Given a net a:IAa: I \to A and a net b:JAb:J \to A, we say that bb is a subnet of aa if bb comes with a function f:IJf:I \to J and a dependent function g: i:IJ f(i) +Ig:\prod_{i:I} J^+_{f(i)} \to I such that for every dependent term j(i):J f(i) +j(i):J^+_{f(i)}, there is a dependent identification p(i,j(i)):a j(i)=b g(i)(j(i))p(i, j(i)): a_{j(i)} = b_{g(i)(j(i))}.

ba i:I k:J(f(i)k)×[ l:I(il)×(a k=b l)]b \subseteq a \coloneqq \prod_{i:I} \prod_{k:J} (f(i) \leq k) \times \left[\sum_{l:I} (i \leq l) \times (a_k = b_l)\right]


See also

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