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Prequotient

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Prequotients describe closure under finite limits and small colimits without requiring a right adjoint to the inclusion.

1. Definition

A prequotient of a cocomplete topos $\mathcal E$ is a full subcategory closed under isomorphisms, finite limits and small colimits computed in $\mathcal E$.

For a class $A$ of objects, write $\langle A\rangle$ for the smallest prequotient containing $A$. For an object $X$, abbreviate $\langle\{X\}\rangle$ to $\langle X\rangle$.

The generative order is the preorder on objects defined by

$$X\trianglelefteq Y\quad\Longleftrightarrow\quad\langle X\rangle\subseteq\langle Y\rangle.$$

See Definition 4.2, Lemma 4.6 and Definition 4.8.

2. Properties

Proposition 2.1 (Quotients and prequotients)★

Every quotient of a cocomplete topos is a prequotient. Under Vopěnka's principle, every prequotient of a Grothendieck topos is a quotient. For discrete dynamical systems, the converse holds without this additional assumption.

See Lemma 4.4, Remark 4.5 and Corollary 5.24.

Proposition 2.2 (Smallness)★

The poset of prequotients is small if and only if the poset obtained from the generative preorder by identifying objects related in both directions is small. When small, these two posets are isomorphic via $X\mapsto\langle X\rangle$.

See Proposition 4.14 and Theorem 4.17.

3. Examples

For the topos of discrete dynamical systems, all prequotients are classified by ideals of a product of two posets of natural numbers. See the classification on that page and Theorem 5.23.

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5. References