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So, a category is made of objects and morphisms with some axioms on the morphism. I want to ask, what exactly does it mean for two morphisms to be equal? (eg: given in associatvity property in a category)

tryst with freedom
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  • Here I got an FOL description of it but I can't seem to answer the above with it.. – tryst with freedom Apr 19 '22 at 18:55
  • Hey Buraian. How are you defining a category? and in which sense are you asking if it makes sense? why wouldn't it make sense? How would you state the associativity and identity axioms without equality of morphisms? – Jackozee Hakkiuz Apr 19 '22 at 18:58
  • Hallo @JackozeeHakkiuz , same this post. Sense of "something analogous to function equality be defined" – tryst with freedom Apr 19 '22 at 19:01
  • what is 'equality' of functions? – Asinomás Apr 19 '22 at 19:01
  • @Asinomás domain, and codomain agree, values on all points of domain agree etc. Here – tryst with freedom Apr 19 '22 at 19:01
  • Well, as I said earlier, without a predicate for equality of functions you cannot state the identity and associativity axioms, can you? – Jackozee Hakkiuz Apr 19 '22 at 19:02
  • Good point, I had forgotten that. I have rephrased my question @JackozeeHakkiuz – tryst with freedom Apr 19 '22 at 19:04
  • I remember that in the book "categories for the working mathematician" the author defines "meta categories" just by mentioning some axioms,and then he says a category is "an interpretation of the category axioms within set theory". I think one can somewhat interpret this to mean we shouldn't really worry about this sort of stuff when trying to understand category theory. – Asinomás Apr 19 '22 at 19:11
  • Now I suppose meta categories is another can of worms. Could you explain the last sentence @Asinomás – tryst with freedom Apr 19 '22 at 20:03
  • My point's just that you're worrying about stuff that isn't what is trying to be explained when one learns category theory. – Asinomás Apr 19 '22 at 20:54
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    Whenever we have a set $X$ and elements $a,b\in X$, it makes sense to ask whether $a=b$. For all objects $c$ and $d$ in a category, we have a set of arrows $\mathrm{Hom}(c,d)$, and we can ask whether two arrows are equal. There's nothing to worry about here! – Alex Kruckman Apr 19 '22 at 22:00
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    Hahahha how was this so simple @AlexKruckman Please write an answer, I'll accept it – tryst with freedom Apr 20 '22 at 03:25

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This question is actually trickier than it seems! The foundations-agnostic answer is that every category comes equipped with an equivalence relation on its morphisms, which composition must respect — what it means for two morphisms to be equal is something we must decide when constructing a category.

When working in traditional, set-theoretic foundations (i.e. what 99% of mathematicians are implicitly doing), every category has a canonical choice of equivalence for morphisms: Since $f, g \in \mathrm{Hom}(A,B)$ and $\mathrm{Hom}$ is a set, we can ask whether $f = g$.

In different foundations, we must explicitly keep track of how morphisms are being identified. For example, in weak type theories (e.g. intensional Martin-Löf type theory without function extensionality), what is internally called a "category" is what a traditional mathematician would call an $\mathscr{E}$-category, as in the paper "Category theoretic structure of setoids" by Kinoshita and Power (link on ScienceDirect — note: an $\mathscr{E}$-category is a category enriched over the ex/lex completion of Sets). Explicitly, the data of such a category consists of

  • A type of objects,
  • For each pair of objects $A, B$, a type $\mathrm{Hom}(A, B)$,
  • For each $A, B$, an equivalence relation on the type $\mathrm{Hom}(A, B)$
  • The usual identities and compositions,
  • Such that the composition map respects $\approx$, i.e. $f \approx f'$ and $g \approx g'$ imply $(f \circ g) \approx (f' \circ g')$.

The same approach would be (implicitly) applied if formalising category theory within the context of Bishop's constructive mathematics, since there, a "set" by definition comes equipped with an equivalence relation (as opposed to "pre-sets", which are just stuff), and "functions" must respect this relation (as opposed to "operations", which are just mappings of stuff).

Amélia Liao
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