I'm doing this exercise:

Prove that $\mathcal{S}[\beta]$ is a bijection.

Here, $\beta:M\rightarrow N$ is a bijection of finite sets, $\mathcal{S}$ is a species, and $\mathcal{S}[\beta]:\mathcal{S}[M]\rightarrow\mathcal{S}[N]$ is "the induced map" by $\mathcal{S}$.

My problem is that I am having trouble writing what a general $\sigma\in\mathcal{S}[M]$ looks like. I understand how I would prove what is to be shown when $\mathcal{S}$ is the species of finite graphs, the power set, the permutations, or any of the other specific cases of species that I've seen. I understand completely how these work. However, these species all have their particular notation where I can go in and apply $\beta$ to elements of $M$ to show how $\beta$ acts on $\sigma\in\mathcal{S}[M]$, and thus describe the induced map $\mathcal{S}[\beta]$. But how do I do this when $\mathcal{S}$ is kept general? How would I notate $\mathcal{S}[\beta]$ well enough to prove it's bijective?

  • $\begingroup$ Should it be $\mathcal{S}[\beta]: \mathcal{S}[M] \to \mathcal{S}[N]$ is the "induced map" which we are trying to prove is a bijection? $\endgroup$ – John Machacek Jun 15 '14 at 20:58
  • $\begingroup$ Use the definition. Functors take isomorphisms to isomorphisms. $\endgroup$ – Qiaochu Yuan Jun 15 '14 at 21:49
  • $\begingroup$ The definition given in my book is "A species is defined to be a mapping $\mathcal{S}$ that associates to each finite set $M$ a finite set $\mathcal{S}$ consisting of elements $\sigma\in\mathcal{S}[M]$ that can be expressed in terms of the labels $m$ of the elements of $M$ only." I don't know what to do with this definition, @QiaochuYuan, which is why I'm asking the question. $\endgroup$ – user157261 Jun 15 '14 at 22:49
  • $\begingroup$ That seems ambiguous. The correct definition is that a species is a functor from the category of finite sets and bijections to, say, the category of sets. $\endgroup$ – Qiaochu Yuan Jun 15 '14 at 23:05
  • $\begingroup$ @Qiaochu I feel like I should prove that that's true before I can use it. (which I'd like to.) But in doing so I think I have the same problem as mentioned in my post - I don't know how to describe an element in the general case. I mean, I could do it given a specific functor / species, like for graphs, but if I'm not given one what can I do? $\endgroup$ – user157261 Jun 16 '14 at 1:15

Your book certainly requires that $\mathcal{S}$ carries composite functions to composites, in the sense that if $M,M',M''$ are finite sets and $M\xrightarrow{f}M'\xrightarrow{g}M''$ are maps between them, then $$\mathcal{S}[g\circ f]=\mathcal{S}[g]\circ\mathcal{S}[f]$$ and furthermore it should carry identity maps to identity maps, in the sense that for any finite set $M$ $$\mathcal{S}[\mathrm{id}_M]=\mathrm{id}_{\mathcal{S}[M]}$$

Now suppose $\beta:M\to N$ is a bijection of finite sets. Apply the two points above to $\beta$ and its inverse map $\alpha$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.