Category Theory - 2020-05-08

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Vinothkumar Raman

01:17

Actually the definition is like this $A$ is a subcategory of $B$ and $f: X \rightarrow Y$ and $X$ is in B and $Y$ is in $A$ and $f$ factors through some $Z$ in $A$ for every $X$ and which is unique in $A$.

But if I take your definition It doesnt say anything about subcategory at all.

My question was $id_X$ is definitely considered as B-morphism hence should also have a reflector in $A$, since $X \in A$ too. And the reflector could infact be just $id_X$ and $X$ itself.

But the proposition goes over another iff condition which expects $A$ to be full sub category, which i dont find necessary.

So I am definitely missing something here

And yea its in Abstract and Concrete categories

1 hour later…

Malice Vidrine

02:42

@VinothkumarRaman - I know what the definition of a reflective subcategory is and I know the theorem you're referring to in ACC. Above I was referring to what the definition says regarding two objects that are in the subcategory in question.

@VinothkumarRaman - It is NOT the case in general that $id_X$ will be a reflector, though.

If $id_X$ is a reflector, then $X$ is already in $A$, and any morphism $f:X\to Y$, with $Y$ also in $A$, will factor uniquely as $f=g\circ id_X$ for some $g$ in $A$.

but $g=f$ so $f$ was already in $A$. I.e. any morphism from $X$ to $Y$ is in $A$, for any $Y$ in $A$.

I.e. $A$ is full.

Conversely, if $A$ is full and $X$ is in $A$, then $id_X$ it's obvious that $id_X$ satisfies the property required of a reflector.

So the contrapositive of the first direction above is that if there is any morphism $X\to Y$ that is not in $A$, then $id_X$ is not a reflector.

Even though it is in $A$ if $X$ is.

Another way to put this is if $X\in A$ and $\eta:X\to X$ is the reflector, it is true that there is a unique $h:X\to X$ with $h\eta=id_X$. But all that means is that $\eta$ is a split monomorphism. It does not let you conclude that $\eta$ is in $A$, or that it's an isomorphism (which are, in fact, equivalent)

6 hours later…

Vinothkumar Raman

08:55

Yea, I kind of understood that after I posted it.

There is an example about $Rel$ and $Sym$ categories

Its definied as $\rho \cup \rho^{-1}$ in the reflector example and $\rho \cap \rho^{-1}$ in co-reflector. Not sure what it really is

But my confusion is $id$ arrows are automatically symmetric and hence should be there in the subcategory it is defined to be reflector too and it will have all objects too, since they all have to have $id$ arrows which makes the subcategory have all the objects

And full criterion makes it have all the arrows too, but all of them are not symmetric

That example is super confusing

Malice Vidrine

09:59

@VinothkumarRaman - In that example, $id_X$ is not the identity on $(X,\rho)$, as you can tell by $(X,\rho\cup\rho^-1)$ being different objects in Rel. It's just using the name of the underlying map in the category of sets.

And arrows aren't "symmetric" in Rel.

That doesn't even mean anything.

Recall that these authors' version of Rel is not the category whose objects are sets and morphisms are relations.

It's the category whose objects are sets equipped with binary relations, and the morphisms are functions between those sets that preserve the relation.

So the subcategory of sets equipped with symmetric relations in no way has all of the objects of Rel.

Malice Vidrine

10:17

Or maybe I misunderstand the nature of your confusion.

It sounds like you're confusing the morphisms with the relations that help constitute the objects.

6 hours later…

Vinothkumar Raman

16:42

I think you are right. I was thinking about $Rel$ as a category of where morphisms are relations

Now this clears things up

Thanks a lot :D

Malice Vidrine