We give suucient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability  and the other on Sheaf Models in Algebraic Set Theory . Sheaf theory and realisability have been effective methods for constructing models of various constructive and intuitionistic… (More)
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We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure , and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style… (More)
1 Introduction This paper is the second in a series on the relation between algebraic set theory  and predicative formal systems. The purpose of the present paper is to show how realizability models of constructive set theories fit into the framework of algebraic set theory. It can be read independently from the first part ; however, we recommend… (More)