Consider a category $\mathsf C$ admitting a quadruple adjunction as below.


$$(\Pi\_0 \dashv \text{disc} \dashv \Gamma \dashv \text{codisc}) : 
 \mathsf{C}
 \stackrel{\stackrel{\longrightarrow}{\longleftarrow}}{\stackrel{{\longrightarrow}}{{\longleftarrow}}}
 \mathsf{B}
 \;$$
A good example is the category of locally connected spaces with $\mathsf{B}=\mathsf{Set}$. In this case the connected-components functor $\Pi\_0$ implies $\mathsf C$ is extensive.


Suppose now $\mathsf C$ is extensive and admits the exponentials below. In *Axiomatic Cohesion*, Lawvere defines an adjoint quadruple as above to be *cohesive* if:


1. $\Pi\_0$ preserves finite products and $\text{codisc}$ is fully faithful.
2. $\Pi\_0$ preserves $\mathsf{B}$-parametrized powers in the sense of a natural isomorphism $$\Pi\_0(X^{\text{disc}I})\cong \Pi\_0(X)^{I}.$$
3. The canonical arrow $\Gamma\Rightarrow \Pi\_0$ is epimorphic.


Lawvere later defines such an adjunction to be *sufficiently cohesive* if each object has a "contractible envelope":


4. For every $X$ there exists a monic map $X\to Y$ with $Y$ contractible in the sense that $\Pi\_0(Y^A)=\mathbf 1$ for all $A$.


Then it is proved that a topos of cohesion ($\mathsf C$ is a topos) is sufficiently cohesive iff the truth-value object is connected, and also iff all injective objects are connected. Here, connectedness of $X$ means $\Pi\_0(X)\cong \mathbf 1$.


For locally connected spaces it seems the injective objects are the inhabited indiscrete spaces, whose connectedness reflects their cohesion. This is opposed to case of sets where $\text{codisc}$ is the identity and sets are seldom singletons.



> 
> Can a sufficiently cohesive category (quadruple adjunction) have a conservative $\Gamma$?
> 
> 
> 


I ask for the following reason: For locally connected spaces $\Gamma$ is the "points functor" and I think of the fact it is faithful but *not* conservative as reflecting that "discontinuous" arrows (i.e breaking cohesion) may be underlain by "isomorphisms on stuff"; I want to know if this is "correct" intuition or not.



Sufficient cohesion and conservativity of "underlying stuff"
Let ${\phi : \mathrm{disc} \rightarrow \mathrm{codisc}}$ be the canonical map.


The map ${\Gamma \phi }$ is an iso in the base so, if $\Gamma$ is conservative, $\phi$ is an iso.


In other words, if $\Gamma$ is conservative, the original string of adjoints is a quality type. So, if the original string is sufficiently cohesive then the whole thing is degenerate.


