Generalized Enrichments of Categories for Operads

• 来源期刊: 代数集刊（英文版）
• 关键词: monoidal categories enrichments operads PROPs
• 年，卷（期）: 2007,（1）
• 页码范围: 61-78
• 页数: 18页
• 语种: 英文

摘要

Most enriched categories also have an ordinary category structure which is compatible with the enrichment on them. In this paper, enrichments in a monoidal category are generalized to arbitrary categories. These specialize to the classical enrichments when sets are regraded as discrete categories. We also generalize the definitions of PROs and PROPs as some generalized enrichments of categories. Then an operad in some monoidal category corresponds to a generalized PROP. Algebras of operads induce some special kind of monoidal functors. In the category of small categories, we construct several operads to define lax monoids and lax commutative monoids which are formal descriptions of natural associativity and commutativity. Using this identification, operads and their algebras can be studied by lax commutative monoids and morphisms between them.