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)^{\text{disc}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.