Timeline for Union of a object (a set) in the Elementary Theory of the Category of Sets
Current License: CC BY-SA 3.0
8 events
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| Nov 22, 2011 at 1:09 | comment | added | David Roberts♦ | @Buschi - you cannot take the intersection of arbitrary sets defined as elements of a category of sets as we do not have a global membership relation. And intersections are not colimits, but limits, when it is meaningful to talk about them (e.g. two subsets of a given set). | |
| Nov 22, 2011 at 0:54 | comment | added | Todd Trimble | You misunderstood me entirely. I was considering $A = 1$, the terminal object in $Set$. In other words, let $J$ be the subpreorder of $I$ given by full subcategory of $Set/X \times Y$ whose objects are $(m, n): 1 \to X \times Y$ where $m: 1 \to X$ and $n: 1 \to Y$ are monic. Well, any morphism out of $1$ is monic! So, this $J$ is equivalent to the discrete category whose objects are indexed by elements of $X \times Y$. The colimit of the restriction of $F$ to $J$ is the union of all elements of $X \times Y$, i.e., is $X \times Y$. It follows easily that the colimit of $F$ is also $X \times Y$. | |
| Nov 21, 2011 at 23:19 | comment | added | Buschi Sergio | @Trimble. sorry, I realized too late get a mistake. But the colimit above define the intersection $X\cap Y$ (is roughtly the maximal common subobject), then the union is the pushout of $X\leftarrow X\cap Y\to Y$. A object of $I$ is simply a span $X\leftarrow A\to Y$ where the two arrow are monomorphism isnt true that if $(m, n)$ is mono then $m$ and $n$ are mono (this is true about $m\times n$) viceversa is enough that just one of these is Mono, consider the graph of a function. The terminal object $(X\times Y, \pi_1, \pi_2)$ isnt in $I$ just because $\pi_1$ and $\pi_2$ aren't mono. | |
| Nov 21, 2011 at 20:47 | comment | added | Todd Trimble | @Buschi Sergio: I don't understand your last comment. Your preorder $I$ contains all such instances where $A$ is restricted to the terminal object (where it is automatic that $m$ and $n$ are monic for any $m$, $n$), and the colimit over just that subpreorder is all of $X \times Y$, not "$X \cup Y$". In fact, the only categorical meaning of union $X \cup Y$ that is stable under categorical equivalence is the disjoint union. | |
| Nov 21, 2011 at 20:28 | comment | added | Buschi Sergio | If we admit colimits the union of two abject $X,\ Y$ the union $X\cup Y$ is the colimit $F: I\to Set$ where $I$ is the (small) category of alla monomorphism of type $(m,n): A\to X\times Y$ with $m$ and $n$ monomorphisms, (see as a subcategory of the comma $Set\downarrow X\times Y$ and $F: (A, m, n)\mapsto A$ is the forgetfull functor. | |
| Nov 21, 2011 at 12:18 | comment | added | Buschi Sergio | thank you very much for your reply. I seems that that naive set was a natural very large place for make mathematic bulding in past, now (axiomatic set theory) become no so natural (when two set $X$ and $Y$ are equal? Or it is a "ontological" property or because have equal elements, but how two elements of these two sets could be equal? If they have equal elements.... ecc. ). | |
| Nov 20, 2011 at 22:36 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
added 87 characters in body
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| Nov 20, 2011 at 22:00 | history | answered | David Roberts♦ | CC BY-SA 3.0 |