Quillen [Qui73] は, algebraic \(K\)-theory の定義のために exact category の概念を導入したが, その後,
様々な目的のために様々な方向に一般化されている。
まず, algebraic \(K\)-theory のためには, Waldhausen category (category with cofibrations and
weak equivalences) がある。
Huayi Chen は, [Che10] で exact category を一般化した arithmetic exact category
という概念を定義している。ベクトル束の Harder-Narasimhan filtration を考えているなど, Bridgeland の意味の
stability とどのように関係があるのか興味深い。
一般化としては, Bazzoni と Crivei の one-sided exact category [BC13] もある。そこでは,
Rosenberg の preprint や Rump の [Rum10] などが挙げられている。
- left exact category と right exact category
Additive でない category への一般化として, Dyckerhoff と Kapranov [DK19] が proto-exact
category とその Waldhausen \(S\)-construction や Hall algebra を導入している。Hekking の master’s
thesis [Hek17] や Eppolito, Jun, Szczesny の pointed matroid と strong map を調べた
[EJS20] で使われている。
Nakaoka と Palu [NP19] は exact category と triangulated category の共通の一般化として
extriangulated category という概念を導入している。
Barwick [Bar15; Bar]は \((\infty ,1)\)-categoryへの一般化を考え て, その algebraic \(K\)-theoryを調べ
ている。
- exact \((\infty ,1)\)-category
他には, 次のような一般化や変種がある。
- CGW category [CZ]
- ACGW category [CZ]
- ECGW category [SS]
- \(n\)-exact category [Jas16]
- weakly exact category [Jaf]
- relative exact category [HM]
- complicial exact category [Sch11]
- A.I.S exact category [HR]
- semi-exact category [DGG17]
- exact category with duality [Sch10]
- exact \(\infty \)-category [Bar15]
References
-
[Bar]
-
C. Barwick. On the \(Q\) construction for exact quasicategories. arXiv:
1301.4725.
-
[Bar15]
-
Clark Barwick. “On exact \(\infty \)-categories and the theorem of the heart”.
In: Compos. Math. 151.11 (2015), pp. 2160–2186. arXiv: 1212.5232.
url: https://doi.org/10.1112/S0010437X15007447.
-
[BC13]
-
Silvana Bazzoni and Septimiu Crivei. “One-sided exact categories”.
In: J. Pure Appl. Algebra 217.2 (2013), pp. 377–391. arXiv: 1106.
1092. url: https://doi.org/10.1016/j.jpaa.2012.06.019.
-
[Che10]
-
Huayi Chen. “Harder-Narasimhan categories”. In: J. Pure Appl.
Algebra 214.2 (2010), pp. 187–200. arXiv: 0706 . 2648. url:
https://doi.org/10.1016/j.jpaa.2009.05.009.
-
[CZ]
-
Jonathan A. Campbell and Inna Zakharevich. Devissage and
Localization for the Grothendieck Spectrum of Varieties. arXiv:
1811.08014.
-
[DGG17]
-
Jérémy Dubut, Eric Goubault, and Jean Goubault-Larrecq.
“Directed homology theories and Eilenberg-Steenrod axioms”.
In: Appl. Categ. Structures 25.5 (2017), pp. 775–807. url:
https://doi.org/10.1007/s10485-016-9438-y.
-
[DK19]
-
Tobias Dyckerhoff and Mikhail Kapranov. Higher Segal spaces.
Vol. 2244. Lecture Notes in Mathematics. Springer, Cham, 2019,
pp. xv+218. isbn: 978-3-030-27122-0; 978-3-030-27124-4. arXiv:
1212.3563. url: https://doi.org/10.1007/978-3-030-27124-4.
-
[EJS20]
-
Chris Eppolito, Jaiung Jun, and Matt Szczesny. “Proto-exact
categories of matroids, Hall algebras, and K-theory”. In: Math.
Z. 296.1-2 (2020), pp. 147–167. arXiv: 1805 . 02281. url:
https://doi.org/10.1007/s00209-019-02429-z.
-
[Hek17]
-
J. Hekking. Segal Objects in Homotopical Categories & \(K\)-theory of
Proto-exact Categories. 2017. url: https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/hekking_master.pdf.
-
[HM]
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Toshiro Hiranouchi and Satoshi Mochizuki. Delooping of relative
exact categories. arXiv: 1304.0557.
-
[HR]
-
Souheila Hassoun and Sunny Roy. Admissible intersection and sum
property. arXiv: 1906.03246.
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[Jaf]
-
Amir Jafari. Weakly exact categories and the snake lemma. arXiv:
0901.2372.
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[Jas16]
-
Gustavo Jasso. “\(n\)-abelian and \(n\)-exact categories”. In: Math.
Z. 283.3-4 (2016), pp. 703–759. arXiv: 1405 . 7805. url:
https://doi.org/10.1007/s00209-016-1619-8.
-
[NP19]
-
Hiroyuki Nakaoka and Yann Palu. “Extriangulated categories, Hovey
twin cotorsion pairs and model structures”. In: Cah. Topol. Géom.
Différ. Catég. 60.2 (2019), pp. 117–193. arXiv: 1605.05607.
-
[Qui73]
-
Daniel Quillen. “Higher algebraic \(K\)-theory. I”. In: Algebraic \(K\)-theory,
I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle,
Wash., 1972). Berlin: Springer, 1973, 85–147. Lecture Notes in
Math., Vol. 341.
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[Rum10]
-
Wolfgang Rump. “Flat covers in abelian and in non-abelian
categories”. In: Adv. Math. 225.3 (2010), pp. 1589–1615. url:
http://dx.doi.org/10.1016/j.aim.2010.03.027.
-
[Sch10]
-
Marco Schlichting. “Hermitian \(K\)-theory
of exact categories”. In: J. K-Theory 5.1 (2010), pp. 105–165. url:
http://dx.doi.org/10.1017/is009010017jkt075.
-
[Sch11]
-
Marco Schlichting. “Higher algebraic
\(K\)-theory”. In: Topics in algebraic and topological \(K\)-theory. Vol. 2008.
Lecture Notes in Math. Springer, Berlin, 2011, pp. 167–241. url:
https://doi.org/10.1007/978-3-642-15708-0_4.
-
[SS]
-
Maru Sarazola and Brandon Shapiro. A Gillet-Waldhausen Theorem
for chain complexes of sets. arXiv: 2107.07701.
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