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Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines maps $$ c^\sharp : V^* \to W, \quad (\alpha : V \to \mathbb K) \mapsto (\alpha \otimes \mathrm{id}_W)\circ c, \\ c^\flat: W^* \to V, \quad (\beta : W \to \mathbb K) \mapsto (\mathrm{id}_V \otimes \beta)\circ c.$$ The elementary fact is that the images of these maps $\mathrm{im}(c^\sharp) \subseteq W$ and $\mathrm{im}(c^\flat)\subseteq V$ are in duality. By this I mean that $c$ factors through $\mathrm{im}(c^\flat) \otimes \mathrm{im}(c^\sharp)$ and is the copairing of a duality: there is also a pairing $\mathrm{im}(c^\sharp) \otimes \mathrm{im}(c^\flat) \to \mathbb K$, so that in particular $\mathrm{im}(c^\flat) \cong \mathrm{im}(c^\sharp)^*$ and $\mathrm{im}(c^\sharp) \cong \mathrm{im}(c^\flat)^*$.

My question is whether anything like this holds in the $(\infty,1)$-category of chain complexes, or more generally in (well enough behaved) stable $\infty$-categories. A naive translation leads immediately to problems, since normally "image" is defined as, perhaps, "cokernel of kernel" or "kernel of cokernel", but the natural $\infty$-versions of these are "cocone of cone" or "cone of cocone", which do not give the correct answers.

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    $\begingroup$ "Cokernel of kernel" and "kernel of cokernel" are extremely misleading definitions; e.g. they don't suggest the correct form of the 1-categorical nonlinear generalizations, which are "coequalizer of the kernel pair" and "equalizer of the cokernel pair" (which are distinct in general). In the $\infty$-setting the kernel and cokernel pairs generalize to the standard simplicial resp. cosimplicial objects you can build out of a morphism by taking repeated pullbacks resp. pushouts and the coequalizer resp. equalizer generalizes to the geometric realization resp. totalization. $\endgroup$ – Qiaochu Yuan Apr 29 '15 at 5:06

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