How to define 0-sphere in a category with zero object?

Question

The 0-sphere $S^0$ is the disjoint union of two points, $$S^0 \simeq \ast \coprod \ast$$

How to define 0-sphere in a category with zero object?

Answer 1

The notion of "the $0$-sphere" has no reasonable interpretation in an arbitrary pointed category $C$ in the absence of any more information. If $C$ is equipped with a forgetful functor $U$ to $\mathrm{Set}$, then $S^0$ can be characterized as an object representing $U$. Thus if $U$ admits a left adjoint $F$, then $S^0=F(*)$ is the "free" $C$-object on a point. This works to describe a $0$-sphere analogue in many model categories.

However, this isn't really connected to the question of spherical objects, because even the $0$-sphere in pointed spaces is not a homotopy cogroup, since there's no natural group structure on $\pi_0$. The most natural model category in which to consider spherical objects is that of pointed connected spaces. Non-connected pointed spaces are a bit, well, pointless.

Answer 2

I think I agree with Kevin Carlson that you probably need a bit more structure than just a pointed category in order to have a reasonable analog of $S^0$. But here are a few more ways you might do it:

• $S^0$ is the unit for a symmetric monoidal closed structure on the category $Top_\ast$ of pointed spaces (namely, the smash product). So if $(\mathcal C, \otimes, I)$ is a symmetric monoidal closed pointed category, it's reasonable to think of the unit $I$ as an analog of $S^0$.

  For instance, if $\mathcal C$ is the category of chain complexes or the category of spectra or the category of pointed spaces, this gives the expected answer.

  (This is really an instance of Kevin Carlson's context, since if $\mathcal C$ is a symmetric monoidal close category, the functor $\mathcal C(I,-): \mathcal C \to Set$ is a canonical forgetful functor which is corepresented by $I$.)

• There is a canonical adjunction $(-)_+: Top^\to_\leftarrow Top_\ast: U$ and $S^0 = pt_+$. More generally, if $\mathcal D$ is a category with terminal object and $\mathcal C$ is the category of pointed objects in $\mathcal D$, then there is a canonical adjunction $(-)_+ : \mathcal D^\to_\leftarrow \mathcal C: U$, and it's reasonable to think of $pt_+$ as an analog of $S^0$ where $pt$ is the terminal object in $\mathcal D$.

  Note that for any pointed category $\mathcal C$ there is a category $\mathcal D$ such that $\mathcal C$ is the the category of pointed objects in $\mathcal D$, namely $\mathcal C$ itself -- although in this case the above adjunction is just the identity.

  (Again, this is an instance of Kevin Carlson's context, since $S^0$ is then the representing object for the functor $\mathcal D(pt, U-): \mathcal C \to Set$.)

• Generalizing the previous case, if we have any favorite adjunction $F: \mathcal D^\to_\leftarrow \mathcal C$ where $\mathcal D$ is some other category, and if we have a favorite object $D \in \mathcal D$, then $FD \in \mathcal C$ corepresents $\mathcal D(D,U-)$, and so it should be our favorite object of $\mathcal C$.

  For instance, if $\mathcal C$ is is the category of algebras for some algebraic theory / operad / whatnot on $\mathcal D$, then there is a canonical adjunction $F: \mathcal D^\to_\leftarrow \mathcal C: U$ to use here.

Here's a bit more perspective. The question seems to be

For some class $\mathbb C$ of categories, is there a "good" way to pick out some "canonical" nonzero object from each of these categories?

To make this a bit more precise, let's stipulate that "good" means "functorial with respect to the most important functors between categories in $\mathbb C$".

Now Kevin Carlson's suggestion makes sense when we realize that most important categories $\mathcal C$ come equipped with a right adjoint forgetful functor $U: \mathcal C \to Set$, and most of our favorite right adjoint functors between these categories commute with these forgetful functors. Dually, most important categories come equipped with a left adjoint functor $F: Set \to \mathcal C$ and our favorite left adjoint functors commute with these left adjoints. So it makes sense to take $F(pt)$ as our favorite object of $\mathcal C$, where $pt$ is the terminal set.