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Stream: learning: questions

Topic: Localising at a set of fake arrows

fosco (Oct 31 2020 at 16:19):

Let $p : X \rightsquigarrow Y$ be a profunctor between small categories. Consider the collage $X\uplus_p Y$, also called cograph, of $p$. One among many possible intuition for $X\uplus_p Y$ is that $p(x,y)$ is a set of "fake arrows" $x \searrow y$, that in $X\uplus_p Y$ become real arrows. The fact that each $p(x,y)$ can be seen as such a set of arrows is embodied in the request that there is a cowedge $X(x,x')\times p(x',y)\to p(x,y)$ for each $x,y$.

Now, I would like to know what is the category that results from formally inverting the arrows in $X\uplus_p Y$ contained in the various $p(x,y)$, or in other words, I want to consider the category of fractions $X\uplus_p Y[\Sigma_p^{-1}]$, where $\Sigma_p = \coprod_{x,y\in X\times Y} p(x,y)$.

fosco (Oct 31 2020 at 16:25):

I suspect this is something already well-known; yet, I don't see any relevant property that might present $X\uplus_p Y[\Sigma_p^{-1}]$ as another guy in disguise. An interesting observation is that this construction always yields a nontrivial result (as soon as $p$ is nonempty: this because an inverse to $f\in p(x,y)$ is an arrow $y\to x$, i.e. an element of a set $X\uplus_p Y(y,x)$ that previously was empty.

sarahzrf (Oct 31 2020 at 16:31):

random guess: if the cograph is a lax colimit, i don't suppose what you're doing here might possibly be a pseudo colimit? image.png

fosco (Oct 31 2020 at 16:31):

A motivation, which is also the example I care about now:

Let $X,Y$ be two posets, and let $R$ be a profunctor between them regarded as categories; $R$ is a relation between $P$ and $Q$ with the property of being downward closed on one side, and upward closed on the other side. For reasons irrelevant to the discussion, I want to look at functors $F : P \to {\sf Set}$ with the property that, for a certain relation $R$ on $P$ compatible with the order, $(a,b)\in R$ implies $Fa \cong Fb$.

I have never seen such a thing before, and this broader picture seems the right one into which this can be framed.

fosco (Oct 31 2020 at 16:33):

sarahzrf said:

what you're doing here might possibly be a pseudo colimit? image.png

Sure; in the end, a category of fractions is a particular instance of a coinverter!

Amar Hadzihasanovic (Nov 01 2020 at 14:14):

In the case of a representable profunctor $\mathrm{Hom}(-,F-)$ for a functor $F: X \to Y$, its collage can be built as a kind of “directed mapping cylinder” for the functor $F$, with the walking arrow category $I$ playing the role of the interval: it's isomorphic to the quotient $((X \times I) \amalg Y)/\sim$, where $\sim$ identifies $X \times \{1\}$ and $Y$ along $F$.

Presumably your “localised” version can be constructed in the same way, except now we replace $I$ with the “walking isomorphism” category $I_\simeq$?

Amar Hadzihasanovic (Nov 01 2020 at 14:16):

For the general (non-representable) case, I think it should be possible to build the collage in a similar way. First we replace a profunctor $p$ with the corresponding two-sided discrete fibration $X \leftarrow E_p \to Y$. Then the collage is a quotient of $X \amalg (E_p \times I) \amalg Y$ where $E_p \times \{0\}$ is identified with $X$ and $E_p \times \{1\}$ with $Y$ along the two legs of the fibration.

Amar Hadzihasanovic (Nov 01 2020 at 14:16):

And then again, the localised version should be the same thing with $I_\simeq$ replacing $I$.

Amar Hadzihasanovic (Nov 01 2020 at 14:22):

(I've only tested that this is equivalent to what I said above in the representable case, so I may be wrong -- but I think it feels right...)

fosco (Nov 01 2020 at 20:43):

I was hoping for something like this, yes. I'll check tomorrow!