By Cheng E., Lauda A.
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Extra info for Higher-dimensional categories: an illustrated guide book
1 Introduction The opetopic approach was first proposed by Baez and Dolan in . The most striking feature of this definition is the underlying shapes of its cells. These shapes are called “opetopic” and look like ooooyyy yyy jjj oo 6j y6 yy && ooo UUoo && o 66 y×& UU&& 6××× & o o UUoo UU jjjj&& & ×× × × This is not just a whimsical artistic foible or even an arbitrary ideological decision — opetopic cells are this shape in order to express composition. That is, in this definition cells do not just play the role of “being cells”; they also directly give composition.
We can also do this to unlabelled cells by applying T to the terminal globular set 1, which has precisely one cell of each dimension. 1 Intuitions 1 0-cells 31 T1 • 1-cells 2-cells • • G• • & id on • id on all 1-cells h • G • • & • G • h • • G • & • & h• G • h • G • f •t ( f •t • 3-cells BR 08 xÐ • & id on all 2-cells h• & BR 08 xÐ • h• & BR 08 xÐ • G & • h G • & BR 08 xÐ h• ... • ( • G • h• BR 19 w BR 2@ v~ • G #q• G %i • G • So to make labelled composites as above, we want a pullback of the following form in GSet: · c ccc cc cc cc 1 Wc TA cc cc cc cc 1 T1 A priori, W could be any globular set of “weak configurations” that we choose.
5 Which operad do we want? ” we now have to ask: Question: What makes a particular operad a sensible choice for defining n-categories? Recall that we want to use operads to express weak composition of globular cells. We must be careful not to confuse the composition of cells in an ncategory with the composition in an operad. The former will be obtained by putting some extra structure on our operad. • Composition in an n-category will be given by requiring certain kinds of cells to be present in our operad.