In this paper, we prove that the zeroth and first path cohomology groups of a finite directed graph are Brown functors.

Contents
Acknowledgements 3
1 Introduction 4
2 Path cohomology of digraphs with coefficients in Z 7
2.1 Digraphs and homotopy of digraphs 7
2.2 Path chain complex on finite sets 10
2.3 Path homology of digraphs 12
2.4 Path cohomology of digraphs 15
3 Path cohomology as a Brown functor 19
3.1 Brown functor and main theorem 19
3.2 Proof of main theorem 21
References 27

References
[1] J. F. Adams. “A variant of E. H. Brown’s representability theorem”. In: Topology 10 (1971), pp. 185–198. doi: 10.1016/0040-9383(71)90003-6.
[2] Alexander Grigor’yan, Yuri Muranov, and Shing-Tung Yau. “On a cohomology of digraphs and Hochschild cohomology”. In: J. Homotopy Relat. Struct. 11.2 (2016), pp. 209–230. doi: 10.1007/s40062-015-0103-1.
[3] Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. “Cohomology of digraphs and (undirected) graphs”. In: Asian J. Math. 19.5 (2015), pp. 887–931. doi: 10.4310/AJM. 2015.v19.n5.a5.
[4] Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. “Homologies of path complexes and digraphs”. In: (July 2012). doi: 10.48550/ARXIV.1207.2834. arXiv: 1207.2834 [math.CO].
[5] Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. “Homotopy theory for digraphs”. In: Pure Appl. Math. Q. 10.4 (2014), pp. 619–674. doi: 10.4310/PAMQ.2014.v10. n4.a2.
[6] Zachary McGuirk and Byungdo Park. “Brown representability for directed graphs”. In: (Mar. 2020). doi: 10.48550/ARXIV.2003.07426. arXiv: 2003.07426 [math.CT].
[7] E. Riehl. Category theory in context. Aurora: Dover modern math originals. Dover Publications, 2017. isbn: 978-0-486-82080-4