symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In the generality of higher algebra:
Let $R$ be an A-∞ ring. Write $R Mod^{perf} \hookrightarrow R Mod$ for the smallest stable (∞,1)-category inside that of all ∞-modules which contains $R$ and is closed under retracts. An object in $R Mod$ is called a perfect $R$-module .
Let $R$ be an A-∞ ring. The (∞,1)-category of ∞-modules $R Mod$ is a compactly generated (∞,1)-category and the compact objects coincide with the perfect modules, def. 1
If $R$ is commutative (E-∞) then the perfect modules (and hence the compact objects) also coincide with the dualizable objects.
The first statement is (HA, prop. 8.2.5.2), the second (HA, prop. 8.2.5.4). For perfect chain complexes this also appears as (BFN 08, lemma 3.5).
geometry | monoidal category theory | category theory |
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perfect module | (fully-)dualizable object | compact object |
For perfect chain complexes see the references there.
In the general context of higher algebra perfect modules are discussed in
For the properties of perfect modules in derived algebraic geometry, see
David Ben-Zvi, John Francis, David Nadler, section 3.1 of Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 (arXiv:0805.0157)
B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.
Dennis Gaitsgory, Notes on geometric Langlands: Quasi-coherent sheaves.