### Abstract

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.

Original language | English |
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Journal | Journal of Pure and Applied Algebra |

DOIs | |

Publication status | Accepted/In press - Jul 31 2016 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Pure and Applied Algebra*. https://doi.org/10.1016/j.jpaa.2017.02.006

**The heart of the Banach spaces.** / Wegner, Sven Ake.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The heart of the Banach spaces

AU - Wegner, Sven Ake

PY - 2016/7/31

Y1 - 2016/7/31

N2 - Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.

AB - Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel-cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.

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U2 - 10.1016/j.jpaa.2017.02.006

DO - 10.1016/j.jpaa.2017.02.006

M3 - Article

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -