# What did Gelfand mean by suggesting to study “Heredity Principle” structures instead of categories?

Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the following intriguing statement (Page 7, section 1.3, end of the first paragraph):

Maybe instead of categories one should study structures with the "Heredity Principle".

In the same paragraph, Gelfand gives an example of a "Heredity Principle" satisfied by the Quasideterminants. Here is the full paragraph (emphasis and sub-paragraphs are mine):

An important problem both in pure and applied mathematics is how to deal with block-matrices. Attempts to find an adequate language for this problem go back to Frobenius and Schur. My colleagues and I think that we found an adequate language: quasideterminants.

Quasideterminants do not possess the multiplicative property of determinants but unlike commutiative determinants they satisfy the more important "Heredity Principle": let $A$ be a square matrix over a division algebra and $(A_{ij})$ a block decomposition of $A$. Consider $A_{ij}$'s as elements of a matrix $X$. Then the quasideterminant of $X$ will be a matrix $B$, and (under natural assumptions) the quasideterminant of $B$ is equal to a suitable quasideterminant of $A$.

Maybe, instead of categories one should study structures with the "Heredity Principle".

Quasideterminants were introduced in

where in Section 3, page 24, a notion of a predeterminant $D_{I,J}(A)$ is introduced, where $A$ is a $n\times n$ matrix and $I,J$ are orderings of $\{1, \ldots, n\}$, followed by the statement:

From the “categorical point of view” the expressions $D_{I, \tilde I}(A)$, where $I=(i_1,i_2,\ldots, i_n)$, $\tilde I=(i_2,i_3,\ldots,i_n,i_1)$, are particularly important.

Earlier in that paper, in a remark on page 11, the authors suggest to generalize quasideterminants to matrices having as entries morphisms in an additive category.

However, no general "Heredity Principle" structures seem to be suggested in any of these sources.

Hence my questions:

1. What Gelfand could have meant by his suggestion?
2. Any work had been done by anyone along these lines?
• With due respect to Gelfand, I don't think he really had a precise definition in mind, or even any example other than quasideterminants. I think he had the impression that there must be something more general beyond the quasideterminant heredity phenomenon (which I agree with), and left finding it as an exercise to the reader. To my knowledge, no one has solved this exercise so far. (I don't even know of a proper theory of quasideterminants which deals with foundational issues in a reasonable way...) – darij grinberg Nov 25 '17 at 20:28
• @darijgrinberg I have updated the questions by including possible other work in that direction. – Dmitri Zaitsev Nov 26 '17 at 13:14
• Perhaps he just meant to draw an analogy: "determinant : category theory :: quasideterminant : ???". In other words, the determinant interacts nicely with composition: what do quasideterminants interact nicely with? It seems like some kind of higher order composition law (subdividing into blocks), which perhaps would correspond to multicategories or something. Perhaps Tom Leinster could help us here? – Steven Gubkin Dec 1 '17 at 17:57
• @darijgrinberg I wonder what evidences led you to this conclusion? And to exclude that Gelfand has something specific in mind that he did not possibly consider ready to publish? Also when he spoke about "adequate language", didn't he allude to any foundational issues? – Dmitri Zaitsev Dec 18 '17 at 10:12
• @StevenGubkin That sounds very interesting. Building a matrix sounds like a fundamental operation (a functor?), and building a matrix of matrices is like composing this operation with itself. In a similar fashion, a list is a functor, and a list of lists is its composition with itself, that can be flattened into a list. And quasideterminant seems to be a natural transformation from this functor to the identity. Could he mean some kind of axiomatic formalization of this structure, without referring to functors? – Dmitri Zaitsev Dec 18 '17 at 10:22