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There is an obvious similarity between the four adjoints describing cohesion for the Sierpinski $(\infty, 1)$-topos in Example 6.1.2 of dcct and four of the seven adjoints (third to sixth) for arrow categories of pointed categories having pullbacks and pushouts MO answer+comment. So
$\Pi([P \to X]) = X$ ; $Disc(X) = [X \to X]$; $\Gamma([P \to X]) = P$ ; $coDisc(Q) = [Q \to \ast]$.
$[f: A \to B] \mapsto coker(f)$; $A \mapsto [0 \to A]$; $[A \to B] \mapsto B$; $A \mapsto [A \to A]$; $[A \to B] \mapsto A$; $A \mapsto [A \to 0]$; $[f: A \to B] \mapsto ker(f)$.
It seems there’s a linearization happening, and we might want to consider, as we did a while ago, whether there is a Freyd-like result for AT-$(\infty, 1)$-categories between stable ones and toposes.
Forming the (co)monads from the adjunction strings, the Sierpinski case gives us the expected three for cohesion, ʃ $\dashv\; \flat \;\dashv\; \sharp$. The other case gives us six. Now, are there traces of these extra three in the nonlinear case, trying to show themselves?
Consulting the cohesion - table, ʃ does have a left adjoint, that is, if one lifts it up to infinitesimal shape $\Im$. It’s the reduction modality, $\Re$. And by the same move, its lift to $\rightsquigarrow$ has a further left adjoint $\rightrightarrows$.
So is there a connection between $\Re$ and the modality $[A \to B] \to [0 \to B]$? Maybe we’re not so far from the tangent (infinity,1)-category construction as a halfway house, with an element above a space $X$, a parameterized spectrum, as an infinitesimal there.
And what of the other two modalities, $[A \to B] \mapsto [0 \to coker f]$ and $[A \to B] \mapsto [ker f \to 0]$?
To a certain extent at least, linearization is characterized by the existence of extra adjoints.
Above I was wondering about the monad $[A \to B] \mapsto [0 \to coker f]$. Something similar but for a nonlinear base occurs in the Thom space construction:
$\array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.$So the nonlinear version is the cofiber construction, presumably. Do these get viewed as monads in an arrow category?
Presumeably this is what Mike was meaning to allude to with his pointer in #2: to define $[A \overset{f}{\to} B] \mapsto [ker f \to \ast]$ and its dual, all one needs is basepoints (and existence of (co-)fibers).
Passing from plain objects to pointed objects is like a 0th step in “linearizing”, in that it makes a zero-object appear (the point) and gives rise (if starting from good enough objects) to a non-Cartesian (hence “non-non-linear”) symmetric monoidal product, the smash product.
Thanks. I’m left with a couple of matters:
(1) Is it ever the case that one would be better off when there are lots of commutative squares about, trying to express everything as maps in an arrow category? So, e.g., in differential cohomology diagram, along with plenty of squares which are (co)units of modalities applied to arrows, one forms cofibers and fibers, but might it be better to remember the fourth corners (even when just $0$ or $\ast$), or take the appearing there to be in full the monad $Spectra^I &[A \to B] \mapsto [0 \to B]$, etc.?
And the initial motivation for the thread:
(2) Behind the string of 7 adjoints in the linear case, is there a nonlinear version struggling to emerge (requiring Aufhebung to express itself)? The connection between $\Re$ and the modality $[A \to B] \to [0 \to B]$ is quite plausible, no? Or maybe better the other way around, from the table and the ’defect’ that the negative of the $\emptyset$ modality is the maybe monad, it’s not surprising that a linear (or non-non-linear) version of the table appears.
Again the idea of an adjoint 7-tuple appears, here for pointed derivators, in Mike’s
and this is infinitely extendable in the stable case with a periodicity 6.
Coming back to this thread and the string of adjoints
$[f: A \to B] \mapsto coker(f)$; $A \mapsto [0 \to A]$; $[A \to B] \mapsto B$; $A \mapsto [A \to A]$; $[A \to B] \mapsto A$; $A \mapsto [A \to 0]$; $[f: A \to B] \mapsto ker(f)$,
I hadn’t seen above that $[A \to B] \mapsto A$ is a dependent sum map, so we see the positive and negative polarities of the dependent sum type via the adjunctions on each side.
And, as suggested in #1, we did observe that a form of reduction modality appears.
Re #7, with cartesian product as a (nondependent) dependent sum, we might expect the third adjoint, $[A \to B] \mapsto B$, to have something to do with coproduct.
I guess it does, since we can see
$\mathcal{C}^{\,2} \stackrel{\stackrel{+}{\longrightarrow}}{\stackrel{\stackrel{\Delta}{\longleftarrow}}{\stackrel{\times}{\longrightarrow}}} \mathcal{C}$as the composite
$\mathcal{C}^{\,2} \stackrel{\stackrel{p_1 \to p_1 + p_2}{\longrightarrow}}{\stackrel{\stackrel{(dom, cod)}{\longleftarrow}}{\stackrel{p_1 \times p_2 \to p_2}{\longrightarrow}}} \mathcal{C}^{\,\to} \stackrel{\stackrel{cod}{\longrightarrow}} {\stackrel{\stackrel{Id}{\longleftarrow}}{\stackrel{dom}{\longrightarrow}}} \mathcal{C}.$1 to 8 of 8