12.20.4 Tensor Composition Notation

Tensor Composition Notation provides a structured way to combine tensors, defining how their components interact in algebraic operations.

Tensor Composition Notation is the specific symbolic convention used to write the chaining of two or more tensor operations — most commonly maps such as pullbacks, pushforwards, and linear endomorphisms — into a single combined operation, encompassing the small-circle symbol ∘, the ordering convention determining which operation is understood to act first, and the star-placement rules that track how composition interacts with the reversed direction of the pullback as opposed to the preserved direction of the pushforward. Because composition order carries substantive mathematical meaning for tensor operations — reversed for the pullback, preserved for the pushforward — its notation must communicate that order unambiguously rather than leaving it to be inferred from context.

The Composition Symbol and Reading Order

Right-to-Left Application Convention

(g \circ f) (v) = g (f (v))

The symbol ∘ denotes composition, and by standard convention g ∘ f means "apply f first, then apply g," matching the order in which the two maps would be applied if written as nested function calls. This right-to-left reading is fixed by mathematical convention and is essential to parse correctly, since reversing it would silently swap which map acts first, changing the meaning of the entire expression.

Explicit Parenthesization for Longer Chains

h \circ g \circ f

For chains of three or more maps, h ∘ g ∘ f is understood, by associativity of composition, to mean the same map regardless of how it is parenthesized — (h∘g)∘f or h∘(g∘f) — so explicit parentheses are typically omitted once associativity has been established, though they may still be added for clarity when a chain is being evaluated in a specific order for pedagogical or computational purposes.

Composition Notation for the Pullback

Reversed Order in the Induced Operation

(g \circ f)^{*} = f^{*} \circ g^{*}

The pullback's composition notation places the star outside the composed maps on the left side, (g∘f)*, and swaps the order of the individual pullbacks on the right side, f*∘g*. This visible reversal in the notation is not a stylistic choice but a direct encoding of the contravariant composition law: the notation makes it immediately visible, without needing to consult the underlying formula, that pullback reverses composition order.

Reading a Chain of Pullbacks

(h \circ g \circ f)^{*} = f^{*} \circ g^{*} \circ h^{*}

For longer chains, every pullback star notation reverses the entire order of the original composition, so f, applied first among the original maps, ends up applied last among the pullback stars, and h, applied last among the original maps, ends up applied first among the pullback stars — a complete order reversal that the notation communicates directly through the left-to-right order in which the starred symbols are written.

Composition Notation for the Pushforward

Preserved Order in the Induced Operation

(g \circ f)_{*} = g_{*} \circ f_{*}

By contrast, the pushforward's composition notation places the star as a subscript and preserves the same order on both sides of the equation: (g∘f)_* equals g_*∘f_*, with f still applied first and g still applied second. The visual similarity between this notation and the pullback's — differing only in whether the star is a subscript or superscript — is deliberate, since it lets a reader immediately distinguish the two composition laws by the single detail of star placement alone.

Diagram Contrasting the Two Composition Notations

Consistency Requirements for Composition Notation

Distinguishing Composition from the Tensor Product Symbol

The composition symbol ∘ must not be confused with the tensor product symbol ⊗; the two denote fundamentally different operations — combining maps versus combining tensors — and using them interchangeably, even where context might seem to disambiguate, undermines the precision composition notation is meant to provide.

Marking Which Star Convention Is in Force

Because star-as-superscript and star-as-subscript carry opposite composition-order behavior, any document mixing pullback and pushforward expressions must maintain the superscript-versus-subscript distinction consistently throughout; conflating the two notations, even once, reintroduces exactly the ambiguity about composition order that the notation exists to eliminate.

Practical Value of Explicit Composition Notation

Verifying a Chained Computation by Inspection

Given a long chain of pullback or pushforward operations written in standard composition notation, the order in which the underlying source maps must actually be applied numerically can be read directly off the notation without re-deriving the composition law from first principles each time, since the star placement already encodes which composition law is in effect.

Supporting Correct Simplification of Composite Expressions

When simplifying an expression involving several composed tensor operations, correctly manipulating composition notation — applying associativity, substituting a composite map for a single symbol, or expanding a single symbol back into its constituent composed maps — depends on faithfully preserving the order dictated by whichever composition law (contravariant for pullback, covariant for pushforward) the notation in use signals.