Lurie model operads

Remark 12.4.4. Our definition of \(\infty \)-operad is a priori different from that of Lurie (2017), who instead works with functors \(p_{\Oo }\) whose target is the category \(\Fin _*\) of finite pointed sets. To relate the two approaches, recall from Lemma 5.3.8 that \(\Fin _*\) is equivalent to the subcategory \(\Span (\Fin ,\inj ,\all )\) of \(\Span (\Fin )\) in which the left-pointing morphisms are required to be injective. We will show in Section 17.4 below that pullback along the resulting inclusion \(\Fin _* \hookrightarrow \Span (\Fin )\) induces an equivalence between our \(\infty \)-category of \(\infty \)-operads and Lurie’s one.

One advantage of working over \(\Fin _*\) is that it is a strict 1-category. In Lurie’s model, the category \(\Oo ^{\otimes }\) from Construction 12.4.1 also becomes a strict 1-category, and can therefore be directly imported into the world of \(\infty \)-categories, showing that every colored operad gives rise to an \(\infty \)-operad. While the same can in principle be done at the level of \((2,1)\)-categories, we will not develop the theory of \((2,1)\)-categories in this book, and instead import classical operads via Lurie’s model; see Proposition 17.4.9.

On the other hand, there are a variety of reasons to work over \(\Span (\Fin )\) rather than \(\Fin _*\). First, the span approach makes clear the orthogonal roles played by the backwards and forwards maps; this separation is less visible in \(\Fin _*\). Second, allowing only injective backwards maps breaks the property that concatenation of unordered tuples is a categorical product, so conditions on \(\Oo ^{\otimes }\) need to be formulated in a slightly more complicated way. And third, the span perspective generalizes more easily to other settings, making it easier to connect the theory developed here to concepts like Mackey functors, normed \(\infty \)-categories and six-functor formalisms. For example, it allows us to treat the theory of (co)cartesian monoidal structures as an instance of Barwick’s more general unfurling construction in Chapter 15.