Completely connected topos

Contents

1. Definition§

A Grothendieck topos $\mathcal E$ is completely connected if its global-sections functor $\Gamma:\mathcal E\to\mathbf{Set}$ belongs to an adjoint chain

$$C\dashv\pi_0\dashv\Delta\dashv\Gamma.$$

Here $\pi_0\dashv\Delta\dashv\Gamma$ is the chain of a locally connected topos, and $C$ is a further left adjoint. See Definition 3.1.

A container object is an object $K$ such that $\mathcal E(K,-)$ is left adjoint to $\Delta$. Its existence is equivalent to complete connectedness. See Definition 3.2 and Proposition 3.3.

2. Properties§

Theorem 2.1 (Site characterization)§★

A Grothendieck topos is completely connected exactly when it has a presentation by a small site whose underlying category has an initial object for which the only covering sieve is the maximal sieve.

See Theorem 5.4.

Proposition 2.2 (Presheaf topoi)§★

For a small category $\mathcal C$, its presheaf topos is completely connected exactly when the Cauchy completion of $\mathcal C$ has an initial object.

See Proposition 6.1.

Theorem 2.3 (Family construction)§★

For any Grothendieck topos $\mathcal E$, the category $\operatorname{Fam}(\mathcal E)$ of set-indexed families of its objects is a completely connected Grothendieck topos. Moreover, $\mathcal E$ is a closed subtopos of it.

See Theorem 8.2 and Corollary 8.3.

3. Examples§

Sets, the Sierpiński topos, augmented simplicial sets, the object classifier and the topos of trees are completely connected. See §6.

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5. References§

Ryuya Hora, Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor, Proc. Amer. Math. Soc. 154 (2026), no. 2, 567–584.