When does adding inverses of morphisms preserve commutativity of a diagram?

2010-04-16T16:56:32Z

Here is the essence of a problem I have run in to: I have a finite poset D with a terminal object. If I formally invert all of the morphisms, and add these into my diagram, does the new diagram D' still commute?

I think that the resulting diagram will still commute basically because I have done a lot of examples. Working out a few examples you can see that it basically follows by doing it for the "commutative triangle", and applying this finitely many times. It feels like I should be able to do some kind of messy induction, but I do not really want such a proof cluttering up my work.

Is there a reference I could quote for a result like this? It seems like if it is true it should be a "folk lemma".

Of course, if you have more relaxed criteria for when the result will hold, that would be helpful too.

Also if you know of a conceptual proof which does not fall back on some messy induction, that would be wonderful!

EDIT: An example might help to clarify my question. (How do you draw diagrams on MO?)

 a-->b
 ^   ^
 |   |
 c-->d

is my poset. b is the terminal object. Now say someone told you that this was actually a subcategory of a larger category, and in that larger category all of the arrows were invertible. Now consider the larger diagram consisting of the 4 original arrows and their inverses. Is this diagram also commutative? Yes! It is just one or two lines of formal manipulation.

Answer by Reid Barton for When does adding inverses of morphisms preserve commutativity of a diagram?

2010-04-16T18:01:43Z

There is an easy conceptual proof using the fact that the category obtained by formally inverting all the arrows in a category C is equivalent to the fundamental groupoid of the nerve NC of C, and that the nerve of a category with a final object is contractible. Without the assumption of a final object your assertion is false in general, e.g., reverse the arrows from c in your example.

But it should also be easy to prove by induction: for any zigzag of arrows between a and b, the corresponding map in the category with all arrows inverted, when composed with the map from b to the original terminal object, is equal to the map from a to the original terminal object (this is by induction); and so any two maps from a to b in the category with all arrows inverted are equal. In symbols: let me write $t_x$ for the unique morphism in C from $x$ to the terminal object and $[f]$ for the image of $f$ in the category with all arrows inverted. Suppose $[f_1]^{\pm 1} \cdots [f_n]^{\pm 1}$ is a typical map in the category with all arrows inverted with domain $a$ and target $b$. Then the inductive claim is that $[t_b] [f_1]^{\pm 1} \cdots [f_n]^{\pm 1} = [t_a]$, and so $[f_1]^{\pm 1} \cdots [f_n]^{\pm 1} = [t_b]^{-1} [t_a]$.

When does adding inverses of morphisms preserve commutativity of a diagram? - MathOverflow

When does adding inverses of morphisms preserve commutativity of a diagram?

Answer by Reid Barton for When does adding inverses of morphisms preserve commutativity of a diagram?