Category Theory - 2018-07-04

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user131753

10:07 AM

The conceptual hurdle you have to pass here is to think of a category as really just being a collection of abstract dots called "objects" and abstract arrows between them called "morphisms" satisfying certain axioms. Just as an abstract group isn't defined with reference to a specific set it acts...

user131753

2:04 PM

@LeakyNun:

user131753

in Homotopy Theory, 8 mins ago, by user 170039

Let $\mathbf{A}$ be a category and $\mathbf{A}^{\text{op}}$ be the dual category of $\mathbf{A}$. Let $A,B$ be any two $\mathbf{A}$-object. In this book it is written that $\operatorname{Hom}_{\mathbf{A}^{\text{op}}}(A,B)=\operatorname{Hom}_{\mathbf{A}‌}(B,A)$.

user131753

in Homotopy Theory, 4 mins ago, by user 170039

In particular if we consider $\mathbf{A}=\mathbf{Set}$ then how can the same $f$ belong to the both $\text{Hom}$-sets? Isn't a function $f\in \operatorname{Hom}_{\mathbf{A}}(A,B)$ a subset of $A\times B$ and $f\in \operatorname{Hom}_{\mathbf{A}^{\text{op}}}(B,A)$ a subset of $B\times A$?

Martin Sleziak

Isn't this just the definition of the dual category. This is related to the answer you linked to above.

Here is a related post on the main: What is the opposite category of $Set$?

user131753

@MartinSleziak The answer overcomes this by naming the morphisms in $\mathbf{Set}^{\text{op}}$ by $f^{\text{op}}$ whereas in the book that I linked they do not. I am wondering the rationale behind this choice.

Martin Sleziak

liked, linked, or both? :-)

To be honest, I do not see why it should be named differently - since it is the same object.

user131753

2:11 PM

@MartinSleziak Edited.

Martin Sleziak

A morphism from $A$ to $B$ in $\mathbf{Set}^{op}$ is simply a function from $B$ to $A$.

user131753

@MartinSleziak That's the problem I asked here.

Martin Sleziak

But why should it be subset of $B\times A$?