- #1

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(Here X,Y are topological spaces)

colim(X-->Y<--X) where the first arrow is a map f, the second is a map g.

colim(X==>Y), where there are 2 maps f,g from X to Y (indicated by double lines, but couldn't draw 2 arrow heads)

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- Thread starter galoiauss
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- #1

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(Here X,Y are topological spaces)

colim(X-->Y<--X) where the first arrow is a map f, the second is a map g.

colim(X==>Y), where there are 2 maps f,g from X to Y (indicated by double lines, but couldn't draw 2 arrow heads)

- #2

matt grime

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The answer will be a (co)fibration in this kind of example.

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- #4

Hurkyl

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- #5

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No, the original colim notation is accurate. I would appreciate knowing what the relationship is. Thank you.

- #6

Hurkyl

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X --> Y <-- X

is very easy: it's Y.

What do you know about colimits?

- #7

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X --> Y <-- X

is very easy: it's Y.

What do you know about colimits?

On a very informal level. I haven't learned category theory, but one of my teachers introduced some basic terminology and ideas. He mentioned something about if there's a commutative diagram, with a lot of things D going into the colimit, and for any other object Z that the D's map to, there is a unique map from the colimit to that object Z. I'm not really sure I'm getting this, and I'm confused about the 2 examples I listed because I don't see how the 2 are different (it seems like both cases have 2 maps from X to Y, and only the placement of the X is different). Can you explain why the colimits are what you say they are? Thanks.

- #8

Hurkyl

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- #9

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I'm sorry, I'm confused because the example said there were 2 maps in X==>Y, not 1.

- #10

Hurkyl

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Code:

```
X ===> Y
\ |
\ |
\ |
\ |
_| \/
V
```

I was referring to the fact there is only one map from X to V, as opposed to a cone from the other diagram in which X appears twice, so each X gets its own map to V.

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- #12

Hurkyl

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- #13

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If * represents a single point, then colim(X<--*-->Y)=XvY, where v represents the wedge product.

Where is XvY supposed to fit into the diagram X<--*-->Y? Is it just placed in there and we make arrows from every vertex pointing to it? How do we know it's XvY? I'm just lost as to how I'm even supposed to approach this kind of problem. Sorry for my lack of understanding.

- #14

Hurkyl

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That's how you said the colimit is defined, right? Actually, it's often drawn like this:Is it just placed in there and we make arrows from every vertex pointing to it?

Code:

```
* ---> X
| |
| |
V V
Y --> XvY
```

and we don't explicitly draw the arrow from *, because it can be inferred from the diagram.

Grind through the definition!How do we know it's XvY? I'm just lost as to how I'm even supposed to approach this kind of problem.

I think maybe you're in the wrong mindset at the moment? Maybe you're wondering "How would I calculate that?" -- but for now you might be better off thinking "How can I prove that's the answer?" (Once you get some practice, and learn some tricks, then you are much better equipped to try and answer "How would I calculate that?")

But, either way you look at it, your only real approach at this point is to simply grind through the definition of a colimit.

Code:

```
* ---> X
| |\
| | \
V V \
Y --> XvY \
\ |
\ V
\-----> Z
```

What if you had a diagram like this? Can you find one map from XvY to Z that makes the diagram commute? Can you show there aren't two?

(Again, I didn't draw the arrows from *. Can you see why I can ignore them?)

- #15

matt grime

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