Topos Institute Colloquium, 9th of May 2024.
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In 1998 Manes introduced the notion of collection monad on the category of sets as a suitable semantics for collection types. The canonical example of collection monad is the finite powerset monad.

In order to account for the algorithmic aspects the category of sets should be replaced with other categories, whose arrows are maps computable by "low complexity" algorithms.

We extends Manes' definition of collection monad to models for weak versions of Algebraic Set Theory (AST). AST was proposed by Joyal and Moerdijk in the '90 as a category-theoretic counterpart of Bernays' set theory based on classes.

We give a systematic way to construct such models, which include
categories whose arrows are "low complexity" functions between countable sets.