Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

So, what is a derived functor?

Pages: 279 – 293

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a18


Vladimir Hinich (Department of Mathematics, University of Haifa, Mount Carmel, Haifa, Israel)


We rethink the notion of derived functor in terms of correspondences, that is, functors $\mathcal{E} \to [1]$. While derived functors in our sense, when they exist, are given by Kan extensions, their existence is a strictly stronger property than the existence of Kan extensions. We show, however, that derived functors exist in the cases one expects them to exist. Our definition is especially convenient for the description of a passage from an adjoint pair $(F,G)$ of functors to a derived adjoint pair $(\mathbf{L}F, \mathbf{R}G)$. In particular, canonicity of such a passage is immediate in our approach. Our approach makes perfect sense in the context of $\infty$-categories.


derived functor, $\infty$-category

2010 Mathematics Subject Classification

18G10, 18G55

Copyright © 2020, Vladimir Hinich. Permission to copy for private use granted.

Received 20 January 2019

Received revised 26 July 2019

Accepted 27 January 2020

Published 6 May 2020