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On (co)ends in \(\infty\)-categories. (English) Zbl 1470.18029
The principal objective in this paper is to lift the familiar equivalence of the definitions of (co)ends via twisted arrow categories \[ \mathrm{Tw}^{l}(\mathcal{C})\rightarrow\mathcal{C}^{\mathrm{op}}\times\mathcal{C} \] and via categories of simplices \[ \Delta_{/\mathcal{C}}\rightarrow\mathcal{C}^{\mathrm{op}}\times\mathcal{C} \] to \(\infty\)-caterories. It is also shown that weighted (co)limits, which can be defined as certain (co)ends, can altertively be described as (co)limits over left and right fibrations, respectively.

18N60 \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
Full Text: DOI
[1] Anel, Mathieu; Lejay, Damien, Exponentiable higher toposes (2018), available at
[2] Barwick, Clark, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304, 646-727 (2017), available at · Zbl 1348.18020
[3] Cordier, Jean-Marc; Porter, Timothy, Homotopy coherent category theory, Trans. Amer. Math. Soc., 349, 1, 1-54 (1997) · Zbl 0865.18006
[4] Gepner, David; Haugseng, Rune; Nikolaus, Thomas, Lax colimits and free fibrations in ∞-categories, Doc. Math., 22, 1225-1266 (2017), available at · Zbl 1390.18021
[5] Glasman, Saul, A spectrum-level Hodge filtration on topological Hochschild homology, Sel. Math. New Ser., 22, 3, 1583-1612 (2016), available at · Zbl 1371.18011
[6] Haugseng, Rune; Melani, Valerio; Safronov, Pavel, Shifted coisotropic correspondences (2019), available at
[7] Loregian, Fosco, (Co)end Calculus, London Mathematical Society Lecture Note Series, vol. 468 (2021), Cambridge University Press: Cambridge University Press Cambridge, 308 p · Zbl 07344233
[8] Loregian, Fosco, A Fubini rule for ∞-coends (2019), available at
[9] Lurie, Jacob, Higher Topos Theory, Annals of Mathematics Studies, vol. 170 (2009), Princeton University Press: Princeton University Press Princeton, NJ, available from · Zbl 1175.18001
[10] Lurie, Jacob, Higher algebra (2017), available at · Zbl 1175.18001
[11] Mac Lane, Saunders, Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0906.18001
[12] Mazel-Gee, Aaron, On the Grothendieck construction for ∞-categories, J. Pure Appl. Algebra, 223, 11, 4602-4651 (2019), available at · Zbl 1428.18046
[13] Rovelli, Martina, Weighted limits in an \((\infty, 1)\)-category (2019), available at
[14] Shah, Jay, Parametrized higher category theory and higher algebra: exposé II - indexed homotopy limits and colimits (2018), available at
[15] Yoneda, Nobuo, On Ext and exact sequences, J. Fac. Sci., Univ. Tokyo, Sect. I, 8, 507-576 (1960) · Zbl 0163.26902
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