A Homology Theory for Hybrid Systems: Hybrid Homology - CiteSeerX

90A.D. Ames and S. SastryFor all a ∈∧(H) there are two and only two morphisms (which are not theidentity) α, β ∈ Mor id/ (H) such that a = s(α) anda = s(β), so we denote thesemorphisms by α a and β a . Conversely, given a morphism γ ∈ H (which is notthe identity), there exists a unique a ∈∧(H) such that γ = α a or γ = β a .Thesymbol ∧ is used because every object a ∈∧(H) sits in a diagram of the form:a = s(α a ) = s(β a )b = t(α a )✛ α aβ a✲c = t(β a )Note that giving all diagrams of this form (of which there is one for each a ∈∧(H)) gives all the objects in H, i.e., every object of H is the target of α a or β a ,or their source, for some a ∈∧(H). In particular, we can define ∨(H) =(∧(H)) cwhere (∧(H)) c is the complement of ∧(H) in the set Ob(H).Definition 1. A categorical H-space is a pair H cat =(H, S H ) where H is anH-small category and S H : H → Top is a functor such that for every diagram ofthe formA ✛ α E β ✲ Bin H in which α and β are not the identity, either S H (α) or S H (β) is an inclusion.Theorem 1. There is an injective correspondence{Classical H − spaces, H class }−→{Categorical H − spaces, H cat }This is a bijective correspondence if H has a finite number of objects.Example 2. The categorical hybrid space for the water tank, H W cat =(H W , S W H )is defined by the following diagram:b✛ αa✛βda S W H (a) = GW e 1dβ a✲α d✲cS W H✲ SWH (b) = D W 1✛ SW H (α a) = id✛S W H (β d) = idS W H (d) = G W e 2S W H (β a) = id ✲S W H (c) = D W 2✲S W H (α d) = idNote that the H-small category H W is defined by the diagram on the left togetherwith the identity morphism on each object, while the functor S W H is defined by