symmetric monoidal (∞,1)-category of spectra
In homological algebra, what came to be known [Bass (1963)] as the Eilenberg swindle (in honor of Samuel Eilenberg) is an argument (originally due to Mazur (1961)) proving that the Grothendieck group (the 0th algebraic K-theory group $K_0$) of many abelian categories is trivial.
The content of the “swindle” is essentially the following Hilbert hotel-type argument:
In an abelian category $A$ with countable direct sums, we have for any object $X \in A$ an isomorphism
which implies $[X]=0$ in $K_0(A)$.
This is the reason that, for instance, one has to restrict oneself to categories of finitely generated (projective) modules (which lack infinite direct sums) in defining (nontrivial) algebraic K-theory groups $K_0$ of a ring.
The earliest known appearance of Eilenberg’s swindle in writing is in Lemma 3 of
The earliest attribution to Samuel Eilenberg as well as the use of the term “swindle” is in §1 of
Last revised on April 8, 2023 at 19:15:50. See the history of this page for a list of all contributions to it.