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Reading about category theory a category is a collection of objects "types" that have morphisms "transformations" with two properties:

  • Associativity: for every object all its morphisms should be associative when composed together.
  • Identity: for every object must have one identity morphism which doesn't change any of his other morphisms when it is composed with them.

So associativity seems like a reasonable property that ensures correct composition. but why is identity important? what purpose does it serve? and why is it called "identity" and not just "neutral morphism"? And why do some categories like "magma, semigroup" don't have identity or associativity but are still called categories?

anony mous
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  • Not really the right forum for this question. Try either [math.stackexchange.com/](https://math.stackexchange.com/) or if you want a postgrad level answer [mathoverflow.net/](https://mathoverflow.net/). – Salix alba Feb 01 '20 at 09:55
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    I'm voting to close this question as off-topic because it should either be on math.stackexchange or mathoverflow.net. – Salix alba Feb 01 '20 at 10:09
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    Cf. [this question](https://stackoverflow.com/q/21504350/1346276) for its usages in programming. – phipsgabler Feb 01 '20 at 10:38
  • @phipsgabler: Coffee is also used in programming, but questions about making coffee are always off-topic here. – President James K. Polk Feb 04 '20 at 01:00
  • @JamesReinstateMonicaPolk I certainly agree that this is better suited to math(overflow). But the coffee analogy isn't quite working, IMHO. I'd rather compare it with a question like "why do we care about a language being context-free vs. regular", or "what's the use of concept X in relational algebra". It's just that category theory is much less known outsidy of FP and type theory circles. – phipsgabler Feb 04 '20 at 07:14
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    @phipsgabler: I was certainly exaggerating ;) – President James K. Polk Feb 04 '20 at 12:26

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I've found the answer here The Answer

it is possible to define isomorphisms without identities, but it is a bit more complicated and less elegant. I think you're reading far too much into Eilenberg and Mac Lane: they weren't trying to make some "universal" definition that applies to every situation imaginable; they were just making a natural-seeming definition that works in all the relevant examples they could think of.

Considering "associative" operations which don't have identities is thus analogous to considering finite sets but not allowing the empty set. Of course, this is sometimes useful to do, but unless you have a good reason to, it is probably not the natural thing to do.

anony mous
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