I came across the concept of a category with suspension $(\mathcal{C},\Sigma)$ which is defined as a category $\mathcal{C}$ together with an endofunctor $\Sigma:\mathcal{C}\rightarrow \mathcal{C}$. We call $(\mathcal{C},\Sigma)$ (strictly) stable if $\Sigma$ is an isomorphism of categories.
The "correct" morphism $(F,\varphi):(\mathcal{C},\Sigma)\rightarrow(\mathcal{D},T)$ between categories with suspension will be called weakly stable functor, which is a functor $F:\mathcal{C}\rightarrow\mathcal{D}$ with a natural isomorphism $\varphi:F\Sigma\xrightarrow{\sim} TF$. And $(F,\varphi)$ will be further called stable if $F\Sigma=TF$ and $\varphi=\operatorname{id}$.
There are two universal ways to make a category with suspension stable, which are dual to each other. One is called costabilization (the other stabilization of course), which enjoys the following universal property:
- The costabilization of a category with suspension $(\mathcal{C},\Sigma)$ is a stable category with suspension $(\mathcal{RC},\hat\Sigma)$ together with a weakly stable functor $(R,\gamma):(\mathcal{RC},\hat\Sigma)\rightarrow(\mathcal{C},\Sigma)$ s.t.
- For any stable category with suspension $(\mathcal{D},T)$ and weakly stable functor $(F,\varphi):(\mathcal{D},T)\rightarrow(\mathcal{C},\Sigma)$, there exists a unique stable functor $\tilde{F}:(\mathcal{D},T)\rightarrow(\mathcal{RC},\hat\Sigma)$ s.t. $R\tilde{F}=F, \gamma\tilde{F}=\varphi$ strictly.
These concepts makes think of the action of a group (or monoid) on a set and to take the invariants/coinvariants. But one can also think of an action of a monoid $M$ on a set as a presheaf on $M$ considered as a category with one object, and the process of taking (co)invariants as taking (co)limits.
I'm having some trouble understanding the construction of (co)stabilization so want to make some analogies. I think a category with suspension will be equivalently a $\mathbf{Cat}$-valued presheave over some 2-categorical flavored structure, and the (co)stabilization will be simply taking some 2-(co)limits, but I can't be sure what the framework is. Can somebody help me?
The references of (co)stabilizations would be Remark 4 in Co-Gorenstein algebras by Sondre Kvamme, Rene Marczinzik or Stable homotopy categories by Alex Heller
Yes, the intuition you have has been formalized for the first time in 1969 by Miles Tierney (but I think it was "known" to algebraic topologists from before?) in https://link.springer.com/book/10.1007/BFb0101425
Consider in particular its chapter 2, which can be summarized as follows, when monoids are considered as categories with a single object.
A category with endomorphism can be regarded as a functor $\mathbb{N}\to Cat$ (more generally, an object with endomorphism is a functor $\mathbb{N}\to \mathcal C$, i.e. a representation of the monoid $(\mathbb N, +,0)$ of natural numbers into $\mathcal C$).
A category with automorphism is, similarly, a functor $(\mathbb Z,+,0)\to Cat$ (a representation of the additive group of integers).
Clearly, there is a forgetful functor $J : Cat^{\mathbb Z} \to Cat^{\mathbb N}$, and the two universal stabilization procedures that you have in mind are nothing but the left and the right adjoint to $J$. If you note that $J$ is just the precomposition with the obvious inclusion $j : \mathbb N \hookrightarrow \mathbb Z$, the left and right adjoints to $J$ can also be characterized as left and right Kan extensions, respectively, along $j$.
What Tierney does in chapter 2 of that small book is to describe the left adjoint explicitly.