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May
5
revised How to define topology in terms of subobjects?
edited small typos
May
5
suggested approved edit on How to define topology in terms of subobjects?
May
5
answered Thin categories: up to isomorphism Vs up to equivalance
May
5
revised Thin categories: up to isomorphism Vs up to equivalance
edited catlab -> ncatlab
May
5
suggested approved edit on Thin categories: up to isomorphism Vs up to equivalance
May
5
revised Left adjoint in a functor category
corrected grammar
May
5
suggested approved edit on Left adjoint in a functor category
Apr
30
comment Category theory $\subset$ Set theory or vice versa?
@Bento What book is it?
Apr
11
comment Category theory - what's the intuition behind diagrams?
actually I would slightly generalize and say that a commutative diagram is a functor on a preorder.
Mar
27
revised Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$?
Edited some small typos
Mar
27
answered Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$?
Mar
26
comment Distinguishing equality and isomorphism as relations
@alancalvitti The article on Equality is indeed a little bit confusing. I have already made some minor editing to it today, but I will extensively edit it in the next few days. You can look at its talk page in the mean time. Basically the problem lays in the fact that it uses the same symbol "=" to represent 2 different concepts: the equality, which is a logic constant and the identity relation, which is just another (albeit important) relation in set theory. I will also emphasize/clarify this point in my answer.
Mar
26
revised Distinguishing equality and isomorphism as relations
Edited some small typos
Mar
26
suggested rejected edit on Objects whose morphisms are all injective
Mar
26
comment Colimit of $\frac{1}{n} \mathbb{Z}$
Just a clarification: what is $\frac{1}{n}\mathbb{Z}$ ?
Mar
26
comment Distinguishing equality and isomorphism as relations
@alancalvitti By the way, in the upcoming release of WildCats, it will be possible to define isomorphisms within a category. Unfortunately i am experiencing a technical problem with my PC , so I cannot finalize the documentation.
Mar
26
comment Distinguishing equality and isomorphism as relations
@alancalvitti 1-anti-transitive? what do you mean? 2-Please write down how you would like to change the formula(s), so that we understand each other exactly.
Mar
25
comment Distinguishing equality and isomorphism as relations
@alancalvitti aha! Now I understand exactly what is the crux of your question. The isomorphism relation is reflexive, symmetric and transitive, but it is definitely not antisymmetric! So it is not a partial order. So it does not compete with the unique position held by equality, which is instead antisymmetric (besides being symmetric). Conclusion: the Wikipedia article is correct in saying that equality is the only reflexive, symmetric, antisymmetric and transitive relation.
Mar
25
comment Distinguishing equality and isomorphism as relations
Continued. Cat theory is nothing special. It is a theory with its axioms written in first order logic with equality. The equality sign appears directly in the axioms and is necessary in order to define composition and the dom and cod functions. You then define isomorphisms and it just happens that all important properties are conserved by isomorphisms, like all important topological properties are conserved by homeomorpisms (which are - not surprisingly - isomorphisms in $\bf Top$).
Mar
25
comment Distinguishing equality and isomorphism as relations
I knew @alancalvitti that most of what I wrote was probably known to you, but perhaps it will be useful for others. I have reread you question and I am not sure I understand what exactly is your issue here. I covered the linguistic/logic part, I think. So I just want to stress that in mathematics, also in set theory and also in cat theory, it is very clear and distinguishable when two "things" are equal, when they are not equal but they are "isomorphic" and when they are not equal and not isomorphic. Continued....