# zbMATH — the first resource for mathematics

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.

##### MSC:
 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:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.