Some aspects of cosheaves on diffeological spaces

Document Type: Research Paper


1 Department of Math. Yazd University Yazd, Iran

2 School of Mathematics, Iran University of Science and Technology, Narmak,Tehran, 16846--13114, Iran


We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-\v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.


[1] Artin, M., Grothendieck, A., and Verdier, J.L., "Theorie des Topos et Cohomologie Etale des Schemas" (SGA4), Lecture Notes in Math., Springer-Verlag, 1972.
[2] Baez, J.C. and Hoffnung, A.E., Convenient categories of smooth spaces, Trans. Amer. Math. Soc. 363(11) (2011), 5789-5825.
[3] Bredon, G.E., Cosheaves and homology, Pacific J. Math. 25 (1968), 1-32.
[4] Bredon, G.E., "Sheaf Theory", Graduate Texts in Mathematics 170, Springer-Verlag, 1997.
[5] Chen, K.T., Iterated path integrals, Bull. Amer. Math. Soc. 83(5) (1977), 831-879.
[6] Christensen, J.D., Sinnamon, J.G., and Wu, E., The D-topology for diffeological spaces, Pacific J. Math. 272(1) (2014), 87-110.
[7] Christensen, J.D. and Wu, E., Tangent spaces and tangent bundles for diffeological spaces, Cah. Topol. Geom. Differ. Categ. 57(1) (2016), 3-50 (to appear: arXiv : 411:5425v2).
[8] Curry, J.M., "Sheaves, Cosheaves and Applications", Ph.D. Thesis, University of Pennsylvania, 2014.
[9] Dehghan Nezhad, A. and Ahmadi, A., A novel approach to sheaves on diffeological spaces, Topology Appl. 263 (2019) 141-153.
[10] Haraguchi, T., "A Homotopy Theory of Diffeological and Numerically Generated Spaces", Ph.D. Thesis, Okayama University, 2013.
[11] Hector, G. and Macias-Virgos, E., Diffeological groups, Res. Exp. Math. 25 (2002), 247-260.
[12] Iglesias-Zemmour, P., "Fibrations Diffeologiques et Homotopie", These de Doctorat Es-sciences, L' Universite de Provence, 1985.
[13] Iglesias-Zemmour, P., "Diffeology", Math. Surveys Monogr. 185, Amer. Math. Soc., 2013.
[14] Kashiwara, M. and Schapira, P., "Categories and Sheaves", Grundlehren Math. Wiss. 332, Springer-Verlag, 2006.
[15] Mac Lane, S., "Categories for the Working Mathematician", Springer-Verlag, 1998.
[16] Mac Lane, S. and Moerdijk, I., "Sheaves in Geometry and Logic: A First Introduction to Topos Theory", Springer-Verlag, 1992.
[17] May, J.P., "Concise Course in Algebraic Topology", Chicago Lectures in Mathematics, University of Chicago Press, 1999.
[18] Prasolov, A.V., Cosheafification, Theory Appl. Categ. 31(38) (2016), 1134-1175.
[19] Souriau, J.M., Groupes differentiels, in Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836 (1980), 91-128.
[20] Stacey, A., Comparative smootheology, Theory Appl. Categ. 25(4) (2011), 64-117.
[21] Wu, E., "A Homotopy Theory for Diffeological Spaces", Ph.D. Thesis, University of Western Ontario, 2012.