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3. Path homology

    1. Algebraic K-theory of digraphs

          Based on the work of Carranza-Doherty-Kapulkin-Opie-Sarazola-Wong, there is a cofibration category structure on the category of digraphs whose weak equivalences are digraph maps that induce isomorphisms on all path homology groups. One can use it to define a Waldhausen category on (pointed) digraphs, which can then be used to take algebraic K-theory.

      Add remark

      Problem 3.1.

      [Brandon Doherty] Compute $K_0(*)$ and check if it agrees with what one might expect from spaces, i.e., an isomorphism $K_0(*) \cong \mathbb{Z}$, given by taking a class $[X] \in K_0(*)$ to the Euler characteristic of $X$.

      Can other K-groups be computed?
          Carranza, Doherty, and Kapulkin defined a map $K_0(*) \rightarrow \mathbb{Z}$ and verified that it is well-defined and injective.

      Several assumptions need to be added to have a well-defined map, e.g., requiring graphs to have bounded homology.

          Cite this as: AimPL: Discrete and combinatorial homotopy theory, available at http://aimpl.org/combhomotop.