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  • A Calculus of Substitution for Dinatural Transformations, I
    We also define a notion of horizontal composition for dinatural transformations, extending the well known version for natural transformations, and prove it is associative and unitary Horizontal composition embodies substitution of functors into transformations and vice-versa, and is intuitively reflected from the string-diagram point of view
  • Composing Dinatural Transformations: Towards a Calculus of Substitution
    This work represents the first, fundamental steps towards a substitution calculus for dinatural transform- ations as sought originally by Kelly, with the intention then to apply it to describe coherence problems
  • Dinatural Transformations by - Springer
    e = (ec) : K ---) K is a dinatural transformation Suppose QAH --)K is dinatural Whenever (S) commutes in A, the following diagram commutes in B It follows that there exist a unique «a,b: H(a,b) --)K(a,b) for each (a,b) E AOPx A such that the square H(a"b) a a,b H(u,v ) H(c,c}----__ K(c,c} gc commutes for all commutative diagrams (S) in A
  • dinatural transformation in nLab
    By a yoneda-like argument, dinatural transformations α: F → • G \alpha : F \xrightarrow{\bullet} G are in bijection with natural transformations η x, y: hom (x, y) → hom (F (y, x), G (x, y)) \eta_{x,y} : \hom(x, y) \to \hom(F(y,x), G(x,y)) The corresponding transformations are related by
  • Composing dinatural transformations: Towards a calculus of substitution . . .
    This work represents the first, fundamental steps towards a substitution calculus for dinatural transformations as sought originally by Kelly, with the intention then to apply it to describe coherence problems abstractly
  • Extranatural transformations are a special case of dinatural . . .
    A dinatural transformation from $F$ to $G$ is a collection of morphisms $\alpha_c: F(c,c) \to G(c,c)$ such that for every morphism $f: c\to c'$ in $C$, we have $$G(1_c,f)\circ \alpha_c\circ F(f,1_c) = G(f,1_{c'}) \circ \alpha_{c'}\circ F(1_{c'},f)$$
  • Towards a Calculus of Substitution for Dinatural Transformations
    These results provide the first steps for a full calculus of dinatural transformations that would fulfil a project started by Kelly in the 1970’s centred around the idea of substitution as a generalisation of composition
  • (PDF) a review of McCusker, Guy; Santamaria, Alessio Composing . . .
    In this thesis we show how acyclicity of certain graphs associated to these transformations is a sufficient and essentially necessary condition that ensures that the composite of two arbitrary
  • Composing Dinatural Transformations: Towards a Calculus of Substitution
    We also define a notion of horizontal composition for dinatural transformations, extending the well known version for natural transformations, and prove it is associative and unitary Horizontal composition embodies substitution of functors into transformations and vice-versa, and is intuitively reflected from the string-diagram point of view
  • Example of two dinatural transformations between finite categories that . . .
    In particular, let's let $G(1,0)$ be initial in $\mathbb D$ and $G(0,1)$ be terminal so that $\alpha$ and $\beta$ will be dinatural no matter what other choices we make





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