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I was reading 'Category for Scientists' by David Spivak and I'd like some references on operads with that kind of approach, using only "basic category theory", nothing too advanced.

I appreciate it :]

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Most textbooks have a portion of the front-matter devoted to telling the reader what mathematical prerequisites are expected. Tom Leinster's excellent book has already been mentioned in an answer (and probably only downvoted because initially the answer was just a link to an unusual place to download the book). In the front matter, he writes:

We will build on various ‘classical’ notions. Those traditionally considered the domain of category theorists are in Chapter 1: ordinary categories, bicategories, strict n-categories, and enrichment. Classical operads and multicategories have Chapter 2 to themselves. They should be viewed as categorical structures too, although, anomalously, operads are best known to homotopy theorists and multicategories to categorical logicians.

When discussing Part II, on operads, he writes:

This introduces the central idea of the text: that of generalized (‘higher’) operad and multicategory. The definitions—of generalized operad and multicategory, and of algebra for a generalized operad or multicategory—are stated and explained in Chapter 4.

For a beginner, I recommend Donald Yau's book Colored Operads. In the preface, he writes:

The intended audience of this book includes students and researchers in mathematics, physics, computer science, and other sciences where operads and colored operads are used. Since this book is intended for a broad audience, the mathematical prerequisite is kept to a minimum. Specifically:

(1) The reader is assumed to be familiar with basic concepts of sets and functions.

(2) The reader is assumed to be comfortable with basic proof techniques, including mathematical induction. Such concepts are covered in most books about the introduction to advanced undergraduate level mathematics.

Some knowledge of permutations and categories is certainly useful but not required. These concepts and many others will be recalled in this book. In a few instances, we mention some objects—such as topological spaces—that are neither defined nor discussed at length in this book. In those cases, we provide an appropriate reference for the reader to consult.

So, this seems to fit the bill in terms of being accessible to someone whose background is basic category theory. In an answer to the other question about books about operads, I recommended Fresse's book, but that tends to have higher prerequisites, including some familiarity with model categories and homological algebra.

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Higher Operads, Higher Categories by Tom Leinster might suit your purposes: http://www.maths.ed.ac.uk/~tl/hohc/

And you can read the book online.

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    $\begingroup$ I think it is better to post answer which is readable even without having to click on an external link. (After all, the link might eventually die.) Is this Higher Operads, Higher Categories by Tom Leinster? I guess it would be better to link to the authors' website and to the freely available version on arXiv. $\endgroup$ Aug 26, 2017 at 11:13

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