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Jonathan Beardsley
Jonathan Beardsley
3:56 AM
@DylanWilson yeah I think it's E_n again as well, where nilpotent completion is just the limit of the Adams tower
 
user131753
Let $\mathbf{C}$ be a category. Denote it's object class by $\text{ob}(\mathbf{C})$ and define a relation $\le$ on $\text{ob}(\mathbf{C})$ as follows, for all $A,B\in \text{ob}(\mathbf{C})$, $A\le B$ iff there exists a $\mathbf{C}$-morphism $f$ such that $f:A\to B$.
 
user131753
Now note that, (1) for all $A\in \text{ob}(\mathbf{C})$, $A\le A$ due to the existence of identity morphism for all $A\in \text{ob}(\mathbf{C})$ and (2) if $A\le B$ and $B\le C$ for some $A,B,C\in \text{ob}(\mathbf{C})$ then $A\le B$ since the composition of any two $\mathbf{C}$-morphisms is defined.
 
user131753
So $(\text{ob}(\mathbf{C}),\le)$ is a quasi-ordered set (henceforth, qoset) and naturally we may expect to investigate this qoset in order to have some further insight into category theory. Does anyone here knows of any particular study of category theory along this lines?
 
Denis Nardin
Denis Nardin
8:31 AM
@user170039 You seem to be describing the left adjoint of the inclusion of posets into categories (sometimes called "(-1)-truncation"). Note that if you consider categories only up to equivalence (as you should) what you get is a poset (you cannot distinguish between elements $x,y$ that are such that $x\le y$ and $y\le x$)
Of course posets are hugely important in category theory, both as examples and as "approximations" to other categories (see e.g. the recent paper on exodromy which uses ∞-cats such that the map to its -1-truncation is conservative as model for stratified spaces), but maybe you were thinking of something different?
 
 
15 hours later…
skd
skd
11:56 PM
@JonathanBeardsley dylan gave a reference for how to do this in general. if k is just a finite field, then you can define it as follows: the morphism Z_p -> W(k) is 'etale, and goerss-hopkins obstruction theory allows you to uniquely lift this to an E_oo-ring morphism from S_p to some other E_oo-ring; this target is S_{W(k)}
 

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