### Who’s Your Friend?

#### Posted by David Corfield

Strange. No sooner had I set down ‘my friend Colin McLarty’ in the previous post, than I find him repaying the compliment here. Being an erudite man, however, he prefers to do so in Latin:

There is no category theory today without equality. And I don’t think there ever will be. Amicus <David> Corfield sed magis amica veritas. (p. 39)

This is a variant of Amicus Plato sed magis amica veritas (Plato is my friend, but truth is a better friend).

No detail is given there of the need for equality, but in his paper ‘There is no ontology here’, Chap 14 of The Philosophy of Mathematical Practice (preprint), Colin writes

Gelfand and Manin overstate an important insight when they call isomorphism ‘useless’ compared to equivalence:

“Contrary to expectations [isomorphism of categories] appears to be more or less useless, the main reason being that neither of the requirements $G F = 1_C$ and $F G = 1_D$ is realistic. When we apply two natural constructions to an object, the most we can ask for is to get a new object which is canonically isomorphic to the old one; it would be too much to hope for the new object to be identical to the old one.” (Gelfand and Manin, 1996, p. 71)

This is actually not true even in Gelfand and Manin’s book. Their central construction is the

derived category$D(A)$ of any Abelian category $A$. Given $A$, they define $D(A)$ up to a unique isomorphism (1996, §III.2). They use the uniqueness up to isomorphism repeatedly. The notion of isomorphic categories remains central. Yet for many purposes equivalence is enough.

Gelfand and Manin’s book is *Methods of Homological Algebra*.

Can it really be that the derived category construction is usefully defined up to isomorphism?

Posted at September 12, 2011 5:06 PM UTC
## Re: Who’s Your Friend?

I like to make a distinction between “technical usefulness” and “moral usefulness”. I doubt that any “truly meaningful” mathematical theorems depend on defining the derived category up to isomorphism, rather than equivalence. But it could easily be that some “truly meaningful” theorems become

easier to proveif one works with a version of the derived category which is defined up to isomorphism.We see this phenomenon a lot in 2-category theory. Strict Cat-enriched limits and colimits (of which a derived category is, of course, one – a coinverter) are often much easier to work with than “fully weak” 2-dimensional limits, and in good situations can be used in place of the latter. This is not so different from using model categories or other strict models for $(\infty,1)$-categories.