We continue the study of the category of functors ${ℱ}_{quad}$, associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in ${ℱ}_{quad}$; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in ${ℱ}_{quad}$: $P_{H₀}$ and $P_{H₁}$. As an application of these two decompositions, we give a complete description of the polynomial functors...

The homology theory of colored posets, defined by B. Everitt and P. Turner, is generalized. Two graph categories are defined and Khovanov type graph cohomology are interpreted as Ext* groups in functor categories associated to these categories. The connection, described by J. H. Przytycki, between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained from the point of view of homological algebra in functor categories.