View source for Concatenation of paths and loops (homotopy)

{{Stub page|grade=A|msg=Flesh out, link to other parts, make sure [[loop (topology)]] and [[path (topology)]] both link back here}}
__TOC__
==Definition==
Let {{M|p,q:[0,1]\rightarrow X}} be {{link|path|topology|s}} (possibly {{link|loop|topology|s}}) in a [[topological space]] {{Top.|X|J}} such that {{M|1=p(1)=q(0)}} - the terminal point of {{M|p}} is the initial point of {{M|q}}<ref group="Note">Or, if they're both loops, we could just say "both loops have the same basepoint"</ref> - then we define their [[concatenation]] ({{AKA}}: composition, product){{rITTMJML}} as follows:
* {{M|f*g:[0,1]\rightarrow X}} given by {{M|f*g:t\mapsto\left\{\begin{array}{lr}f(2t)& \text{if }t\in[0,\frac{1}{2}]\\g(2t-1)& \text{if }t\in[\frac{1}{2},1]\end{array}\right. }} - we claim this is a path. 
** in words this is the path that goes first around {{M|f}} (at double the speed of {{M|f}}) and then around {{M|g}} (again at double the speed of {{M|g}})
** Note that {{M|1=t=\frac{1}{2} }} is in both parts of the [[piecewise function]], this is to emphasise that {{M|(f*g)(\frac{1}{2})}} is the same in either case.
'''Note: ''' that if {{M|f}} and {{M|g}} are loops based at {{M|x_0}} then so is {{M|f*g}}, and also that if {{M|1=f(0)=g(1)}} (in addition to the {{M|1=f(1)=g(0)}} required for concatenation) then {{M|f*g}} is a loop.
==Caution==
Don't be over-eager and think "I see the [[group]] structure!" the [[constant loop]] is the identity and for a path {{M|p}} it done backwards is the inverse! 

Not quite. Mainly because if you do {{M|f*\text{backwards}(f)}} you do not end up with the [[constant loop based at|constant loop based at {{m|f(0)}}]], you end up with a loop that goes around {{M|f}} then back again!

See the "see next" section below.
==Proof of claims==
{{Requires proof|grade=A|msg=Elementary from [[pasting lemma]]}}
==See next==
* [[First homotopy group]]
** [[Homotopic paths]]
** [[Homotopy (overview)]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Homotopy Theory}}