The interval in algebraic topology
M. V. Mielke · Illinois Journal of Mathematics · 1981
corresponding X-notion or X-construction. Categorical preliminariesIf V is a complete, symmetric, closed monoidal category and C is a small V-category [2, p. xiii] then the V-functor category B V cp also has the struc- ture of a V-category [2, p. 150] and the right Yoneda functor R: C -B given by R(c) C(-, c) is a V-full and faithful V-functor [2, p. 152].The image of C under R consists of the representable functors.If A is a tensored V-category [2, p. 20] and T: C -, A is a V-functor then the left Kan extension of T along R, LanR T: B-A, is given in terms of coend and tensor by L r= LanR T c B(R(c), -) (R)A T(c)([2], dual of 1.43, p. 52).Since R is V-full, Lr may be assumed to satisfy Lr R T [2, dual of 1.4.5, p. 56].If Lr is pointwise (e.g. if A is tensored and cotensored [2, dual 1.4.4 p. 55]) then for F 6 B, Lr(F) f B(R(c), F)(R)a Tc f Fc (R)a Tc since B(R(c), F)= Fc [2, IV.I.1 p. 152].Hence if Lr is pointwise it is V-left adjoint to the singular V-functor U r: A -, B given by U r(a)= A(T(-), a) as the following calculation shows" A(Lr(F), a) A FC (R) A TC, a A(Fc (R) A TC, a)V (Fc, a(Tc, a))= B(F, A(T(-), a))= B(F, Ur(a)).