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 Jan 10 comment What is the definition of a commutative diagram? @drhab In Wikipedia (en.wikipedia.org/wiki/Commutative_diagram) commutative diagrams are defined exactly as you say with posets categories as schemes/indices. This is basically the same definition given by Martin for commutative diagrams. I do not see it as more restrictive than the one given with paths by Martin. Indeed also nlab (nlab.mathforge.org/nlab/show/commutative+diagram) describes a commutative diagram as a quiver which factors via a poset Jan 10 comment In category theory, does $X \leq Y$ have a nice characterization in terms of the existence of a morphism $X \rightarrow Y$? So, this means that this order relation is NOT a good generalization of the subset relation. Right? Jan 10 comment How sensitive is the Yoneda lemma to set-theoretic subtleties? Martin: could you perhaps indicate what you mean by "U-category"? I am familiar with "U-small/moderate category". Jan 10 comment In category theory, does $X \leq Y$ have a nice characterization in terms of the existence of a morphism $X \rightarrow Y$? I knew about ordering relation for subobjects , but I was not aware of this more general definition. Does it have a standard name? where can I read more about it? Jan 8 comment Question about the category $\textbf{2}$ As you will soon discover (see @Aaron answer below, for example), "the" category $\mathbf 2$ has different meanings for different authors. You mean the category with no arrows between different objects while, for example, MacLane means the category with 2 objects and exactly one arrow joining the first to the second object. Jan 8 comment Question about the category $\textbf{2}$ your category $\mathbf 2$ is different from the category mentioned by the OP. The category described by the OP is simply the discrete category with 2 objects, while yours is the linear order with three objects (corresponding to the ordinal number 3). Jan 4 revised How to prove hom functor preserves pullbacks edited some typos Jan 4 suggested rejected edit on Hom-functor preserves pullbacks Jan 4 suggested approved edit on How to prove hom functor preserves pullbacks Dec 30 comment prove that two linear maps over a finite dimensional vector space are conjugate isn't the category-theory tag a bit far-fetched? Dec 30 comment Most important aspects of differential geometry for general relativity Then you will need to learn differential...diagnostic and integral... nutrition :-) Dec 29 answered Most important aspects of differential geometry for general relativity Dec 29 asked Zorn's lemma in categorical language Dec 29 comment How do morphisms in a comma category single out commuting squares? This question has been posed before and fully answered by David Moews and me: math.stackexchange.com/a/98358/19609 . Unfortunately the title of that question was very vague, so it is not obvious what it is all about, by just browsing the site index. Basically most/all textbooks use a sloppy notation for a morphism in a comma category. The correct one should be a quadruple, as described in the cited reference or in Berci's answer below. Dec 27 revised What is the benefit of the theory of categories? edited some typos Dec 27 suggested approved edit on What is the benefit of the theory of categories? Dec 27 revised Understanding the Category of open subsets of a top. space X $Op_X$ edited some typos Dec 27 suggested approved edit on Understanding the Category of open subsets of a top. space X $Op_X$ Dec 24 revised How to get used to commutative diagrams? (the case of products). edited small typos Dec 24 suggested approved edit on How to get used to commutative diagrams? (the case of products).