Bivariant operadic categories

Abstract

We develop a self-dual, bivariant extension of the concept of an operadic category, its associated operads and their algebras. Our new theory covers, besides all classical subjects, also generalized traces and bivariant versions of Kapranov's charades. It is, moreover, combinatorially rich and aesthetically pleasing.

Figures (5)

• Figure 1: Dioperadic duad (left) and bioperadic triad (right).
• Figure 2: The standard form of $S \xrightarrow h T \xrightarrow g Z$.
• Figure 3: The structure of a bimodule. The rectangles represent the corresponding pieces of the bimodule, the triangles the corresponding pieces of the operad and the cooperad. The equalities $a = a_1 + \cdots + a_n$ and $b = b_1 + \cdots + b_m$ following from the axioms of the operadic categories ${\tt Q}_L$ and ${\tt Q}_R$ are assumed.
• Figure 4: An attempt to rewrite bimodule action \ref{['Byly Dusicky.']}.
• Figure 5: A $\frac{1}{2}$PROPic version of Figure \ref{['Porad mne pali ty\n ruce.']}. The rectangles represent the corresponding pieces of ${\EuScript M}$, the triangles the pieces of ${\EuScript P}'$ and ${\EuScript P}"$, respectively. The equalities $a = a_1 + \cdots + a_n$ and $b = b_1 + \cdots + b$ are assumed.

Theorems & Definitions (59)

• Definition 1
• Definition 2
• Proposition 3
• Proposition 4
• proof
• Example 5
• Example 6
• Proposition 7
• Example 8
• Definition 9