Abstract
There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension. Here we establish its significance in an algebraic context. Specifically, in the representation theory of an associative algebra A, a central role is played by the indecomposable projective Amodules, which form a category enriched in vector spaces. We show that the magnitude of that category is a known homological invariant of the algebra: writing chi_A for the Euler form of A and S for the direct sum of the simple Amodules, it is chi_A(S, S).
Original language  English 

Number of pages  8 
Journal  Theory and Applications of Categories 
Publication status  Published  7 Jan 2016 
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Tom Leinster
 School of Mathematics  Personal Chair of Category Theory
Person: Academic: Research Active