Cross effects and calculus in an unbased setting

Abstract

We study functors F : C f → D where C and D are simplicial model categories and C f is the category consisting of objects that factor a fixed morphism f : A → B in C. We define the analogs of Eilenberg and Mac Lane's cross effect functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of Deriving calculus with cotriples (by the second and third authors) from the setting of functors from pointed categories to abelian categories to that of functors from C f to S, a suitable category of spectra, to produce a tower of functors · · · → Γ n+1

Unbased Calculus for Functors to Chain Complexes

“…are an adjoint pair of functors, with the forgetful functor U + being the left adjoint. This is Theorem 3.14 of [1]. Hence, from (1) one obtains another adjoint pair by composition:…”

“…The tower can be considered as a Taylor series approximation to F . In our case, we seek to approximate F by functors that are degree n. In this section, we review the construction of [1] for universal degree n approximations to homotopy functors.…”

“…The construction of Γ n F in [1] depends on establishing that ⊥ n is a cotriple on Fun(C η , Ch(R)). To prove that ⊥ n forms a cotriple, a sequence of adjunctions is constructed…”

“…In this section, we also outline the proof of this result. With the exception of the proof that t is a cotriple, as described above, the proofs used in [1] carry over to this context of functors to chain complexes. In Section 3 we describe our models for iterated fibers, starting first with an explicit model for homotopy fibers in the category of chain complexes of modules over a fixed commutative ring.…”

Spectral Algebra Models of Unstable $$v_n$$-Periodic Homotopy Theory

“…Note that the second of these functors is not finitary (Φ K(n) is actually finitary, as long as the corresponding homotopy colimit is taken in the category Sp K(n) ). In 9 Yet another general treatment of homotopy calculus can be found in [BJM15], but at present this approach only applies to functors which take values in spectra. the case of Φ K(n) , it is fairly easy to see that its Goodwillie tower is closely related to the Goodwillie tower of the identity functor Id : Top * → Top * .…”