The Operator Dictionary
Chapter 1 ยท Reference
Three vocabularies for the same structure. This table is the Rosetta Stone.
The three-column crosswalk
Video codec engineers, temporal graph theorists, and algebraic topologists all work on directed dependency graphs with time-ordered composition. Each field names the parts differently. The sheaf column is this series' proposed reading โ the other two columns use each field's own published terminology. Where a cell is blank, that field lacks the concept โ and the blank is where the open problems live.
| Video codecs | Temporal graph theory | Sheaf cohomology | Shared meaning |
|---|---|---|---|
| prediction residual | โ | coboundary ฮดโฐ | Per-edge error: current minus predicted |
| irreducible error | โ | Hยน (cohomology) | Error no encoder/activation choice can remove |
| I-frame (checkpoint) | Snapshot graph | Clamped sub-complex | Self-contained state, resets dependencies |
| P-frame (delta) | Edge delta | 1-cochain | Forward diff from a reference |
| chain break | journey break | exactness failure | Sequential dependency severed |
| prediction DAG | event graph (TEG) | directed cell complex | Time-ordered dependency structure |
| R-D optimization | โ | โ | Bits vs. reconstruction quality |
| โ | foremost / fastest / shortest | โ | Three distinct path optimality criteria |
| โ | connectivity class | โ | 13 classes with strict inclusion |
| โ | temporal spanner | โ | Sparse subgraph preserving journeys up to stretch ฮฑ |
| โ | โ | Hodge decomposition | Split error into removable + irreducible |
| โ | โ | max-flow/min-cut | Capacity bounds via sheaf cohomology |
| scene-change detection | temporal disconnection | โ | Forward prediction has failed |
| DPB (buffering) | bounded waiting | โ | Deferred usability under temporal constraints |
| B-frame | โ | monodromy ฮฆ | Bidirectional / cyclic dependency |
Reading the table
The row with terms in all three columns โ chain break / journey break / exactness failure โ is the strongest parallel. That's the next chapter.
Rows with blanks mark where one field has concepts the others lack. These aren't missing vocabulary โ they're missing operations. Codecs have lossy approximation and error budgets. TVGs have impossibility proofs and formal journey types. Sheaves have removable/irreducible decomposition. The third chapter inventories who has what. The blanks are the open problems.
Sources
- Casteigts et al. (2012) โ TVG framework, 13 connectivity classes
- Seely (2025) โ sheaf cohomology of predictive coding networks
- Krishnan (2014) โ flow-cut dualities for sheaves on digraphs
- Ghrist & Hiraoka โ sheaf-cohomological max-flow min-cut
- Temporal state machines โ tropical invariance constraint
- Bilรฒ et al. (2022) โ temporal spanners
- Sullivan & Wiegand (1998) โ R-D optimization for video
- Chou & Miao (2006) โ packet dependency DAG
- Mellor (2018), Kivelรค et al. (2018) โ event graphs
Neighbors
Related sections
- ๐ฑ Category Theory โ composition, functors
- ๐ก Information Theory โ entropy, compression
- โ๏ธ Algorithms โ graph algorithms