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I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory.

I quote Mac Lane's book "Category theory for working mathematicians":

"...An adequate treatment of the natural isomorphisms occurring for such limits was a major motivation of the first Eilenberg-Mac Lane paper on category theory [The general theory of natural equivalence]..."

Here's my question:

What are these natural isomorphisms that Mac Lane were referring to?

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    "I didn't invent categories to study functors; I invented them to study natural transformations." - Mac Lane Just some context.2015-02-21

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Eilenberg and MacLane published "General theory of equivalences" in 1945. The initial idea of the two mathematicians was to provide an autonomous framework for the concept of natural transformation, which they came across while working on analogies between group extensions and homology groups and whose generality, pervasiveness and usefulness became soon clear to both of them. For this reason, their initial project turned into that one of devising an axiomatic system in which natural transformations would arise naturally from a stable and self-consistent theoretic grounding (the subject later called category theory, of course). After they realized that a natural transformation is nothing but a family of maps providing a sort of "deformation" between two "collections of interrelated entities" within a given structure, they introduced what are now called functors, which played the role of the "deformation" of a natural transformation. The collection of interrelated entities" was soon formalized through the definition of category. (by the way, notice anyway that MacLane and Eilenberg explicitly avoided using a set-theoretical terminology and notation!!!) You can find more information on papers like "The history of categorical logic: 1963-1977" by Marquis and Reyes (which is extremely recent, by the way), and of course in Kroemer's book.

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    By the way: “I recall that, at the 1963 meeting devoted to Logic, Methodology and Philosophy of Science in Jerusalem, Bill Lawvere proposed basing mathematics on categories rather than sets. Alfred Tarski, who was in the audience, objected: what is a category if not a set of objects and a set of arrows? Lawvere replied: set theory deals with the binary relation of membership, category theory with the ternary relation of composition. Apparently, Tarski was satisfied with the answer.” It's worth mentioning that in 1963 Lawvere was nothing more than a PhD student!!!2017-10-02