# SUM

From: F. W. Lawvere [wlawvere at acsu.buffalo.edu], Sent: February 25, 2001, To: prayudu, Subject: Re: co-product

1) Interpret “same kind” to mean a definite category; this is important since the definition of sum is a universal mapping property, and in fact often changes concretely when the category is enlarged to another one.

2) The addition and multiplication of quantities having physical dimension is faithfully represented in the branch of algebra called “graded” where tensor products of graded modules can act as the domain for product maps; the quantities themselves are morphisms, just as in ordinary linear algebra and the category is also linear in the sense that the finite sums are isomorphic to the finite products (as is briefly dealt with later in the book). The more profound question of realizing these graded algebras of quantities as an abstraction of the Cantor-Grothendieck kind apply to a category of categories of physical objects is still largely unexplored.

Best wishes,

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F. William Lawvere

Mathematics Department, State University of New York

HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere

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On Sun, 25 Feb 2001, prayudu wrote:

Dear Prof. William Lawvere,

I am studying category theory from your Conceptual Mathematics textbook. I found your exposition of categorical product and sum very helpful, especially the way you relate to layman’s notion of product and sum (3+2=5). I am wondering whether categorical definitions also accommodate physicist’s conception of product and sum. By physicist’s conception, I am referring to the fact that we cannot add two different things (‘acceleration + mass’ is not allowed) but we can multiply them (F = ma).

Within the categorical definition of sum, is there a requirement that the two or more objects that are to be added should be of the same kind and the sum also will also be of the same kind.

Would you be kind enough to answer my question (ill-formulated?)

thanking you,

sincerely,

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