ScalingStacks

0NWD
  1. (1)

    Definition 3.3. An object MM of a stable โˆž\infty-category ๐’ž\mathcal{C} is said to be compact if Hom๐’žโก(M,โˆ’)\Hom_{\mathcal{C}}(M,-) commutes with all coproducts (equivalently, with all colimits).

  2. (2)

    An object MM of a stable symmetric monoidal โˆž\infty-category ๐’ž\mathcal{C} is said to be (strongly) dualizable if there is an object MโˆจM^{\vee} and unit and trace maps

    1\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}u\scriptstyle{u}MโŠ—Mโˆจ\textstyle{M\otimes M^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ฯ„\scriptstyle{\tau}1\textstyle{1}

    such that the composite map

    M\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}uโŠ—id\scriptstyle{u\otimes\operatorname{id}}MโŠ—MโˆจโŠ—M\textstyle{M\otimes M^{\vee}\otimes M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}idโŠ—ฯ„\scriptstyle{\operatorname{id}\otimes\tau}M\textstyle{M}

    is the identity.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5