In this PhD thesis we investigate a representation theorem for small abelian categories and a representation theorem for left proper, enriched model categories, with the purpose of describing them concretely in terms of specific well-known categories.

For the abelian case, we study the constructivity issues of the Freyd-Mitchell Embedding Theorem, which states the existence of a full embedding from a small abelian category into the category of modules over an appropriate ring. We point out that a large part of its standard proof doesn't work in the constructive set theories IZF and CZF and in the logical system IHOL. Working constructively, we then define an embedding from a small abelian category into the category of sheaves of modules over a ringed space.

In the context of enriched model categories, we define homotopy enriched tiny objects and we prove that any left proper, enriched model category which is generated by these objects under weak equivalences, homotopy tensor products and homotopy colimits is, under certain extra hypothesis, Quillen equivalent to the enriched presheaf category over these objects. As we show, from our result it is possible to derive Elmendorf's Theorem for equivariant spaces and the Schwede-Shipley Theorem for spectral model categories.

Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University, 2023. p. 254