Kan Extensions Research Articles

Homology theories and Kan extensions

Homology theories and Kan extensions

Coherent extensions and relational algebras

The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained from the latter by weakening that of 2functor and replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra for a monad in Sets is precisely a lax algebra for the lax monad induced in Rel. Similar considerations allow us to recover the T-categories of Burroni as well. These are all examples of lax adjoints of the “normalized” sort and the universal property they satisfy can be expressed by the requirement that certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providing a simple illustration of our somewhat involved constructions.

Localization, Homology and a Construction of Adams

In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory ${\tilde H_ \ast }( - ;{{\mathbf {Z}}_P})$ is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.

Localization, homology and a construction of Adams

In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory H ~ ∗ ( − ; Z P ) {\tilde H_ \ast }( - ;{{\mathbf {Z}}_P}) is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.

On filtered systems of groups, colimits, and kan extensions

On filtered systems of groups, colimits, and kan extensions

On Kan extensions of cohomology theories and Serre classes of groups

On Kan extensions of cohomology theories and Serre classes of groups

So, what is a derived functor?

In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In particular, canonicity of this passage becomes obvious. 2nd version: added comparison to Deligne's definition (SGA4) and a discussion of diagrams of derived functors. Introduction rewritten and references added. 3rd version: description of Kan extensions in terms of correspondences more detailed. 4th version: the final version accepted to HHA.