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A duality principle variously described as  "...a metamathematical principle that corresponding to a theorem there is a dual theorem (each of these dual theorems being proved separately)"  [[#References|[a4]]],  "...a guiding principle to the homotopical foundations of algebraic topology..."  [[#References|[a1]]],  "...a principle or yoga rather than a theorem"  [[#References|[a5]]], and  "...a commonplace of experience among topologists, accepted as obvious"  [[#References|[a3]]]. The duality provides a categorical point of view for clarifying and unifying various aspects of pointed homotopy theory, but is often heuristic rather than strictly categorical.

Any notion (definition, theorem, etc.) in a [[Category|category]] $C$ which can be expressed purely category-theoretically admits a formal dual in the opposite or [[dual category]] $C ^ { * }$, which can then be re-interpreted as a notion in the original category $C$; this latter notion is the (Eckmann–Hilton) dual of the original notion. As examples, the notions of monomorphism in $C$ and epimorphism in $C$ are dual, as are the notions of product of objects in $C$ and co-product of objects in $C$. Pursuing the second example, an object $X$ in $C$ (assumed to have zero-mappings) is group-like if there is a [[Morphism|morphism]] $X \times X \rightarrow X$ in $C$, $X \times X$ denoting the product of $X$ with itself, satisfying the group axioms, expressed arrow-theoretically; dually, $X$ in $C$ is co-group-like if there is a morphism $X \rightarrow X \vee X$ in $C$, $X \vee X$ denoting the co-product of $X$ with itself, satisfying the group axioms with arrows reversed. If $X$ is group-like (respectively, co-group-like), the morphism set $\operatorname{mor}( W , X )$ (respectively, $\operatorname{mor}( X , W )$) inherits a natural [[Group|group]] structure.

In the category ${\cal H}_*$ of pointed CW-complexes (cf. also [[CW-complex|CW-complex]]; [[Pointed space|Pointed space]]) and pointed homotopy classes of mappings,  "product"  is Cartesian product and  "co-product"  is one-point union. The most familiar group-like (respectively, co-group-like) objects are loop spaces (respectively, suspensions) (cf. also [[Suspension|Suspension]]; [[Loop space|Loop space]]). If the requirements of associativity and existence of inverses are dropped from the group axioms, the resulting objects are $H$-spaces (respectively, co-$H$-spaces; cf. also [[H-space|$H$-space]]; [[Co-H-space|Co-$H$-space]]). An important generalization of co-$H$-spaces is obtained by considering the notion of Lyusternik–Shnirel'man category (cf. also [[Category (in the sense of Lyusternik–Shnirel'man)|Category (in the sense of Lyusternik–Shnirel'man)]]). Originally conceived as a geometric invariant of a space $X$ ($\operatorname { cat } ( X ) = - 1 +$ the minimum cardinality of an open covering of $X$, each of whose members is contractible in $X$), the definition can be recast (G.W. Whitehead, T. Ganea) in ways that are susceptible to dualization (see [[#References|[a3]]] for a useful bibliography, including references to sources for $\operatorname{cat}\,( X )$ and various candidates for a dual, $\operatorname{cocat}( X )$). $X$ is a co-$H$-space if and only if $\operatorname{cat} ( X ) \leq 1$.

At the beginning of their work on duality in 1955, B. Eckmann and P. Hilton studied the category $\mathcal{M}$ of modules (over some ring) and developed two dual notions of module homotopy (cf. also [[Homotopy|Homotopy]]): injective homotopy, where, for a module $M$, an injective module containing $M$ plays the role of the  "cone"  over $M$; and projective homotopy, where a projective module mapping onto $M$ plays the role of the  "path space"  over $M$. These two versions of homotopy in $\mathcal{M}$ are different but their analogues in ${\cal T}_{*}$, the category of pointed CW-complexes and pointed mappings, are identical owing to the adjunction equivalence

\begin{equation*} \operatorname { map }_{ *}( X \bigwedge Z , Y ) \approx \operatorname { map }_{ *} ( X , \operatorname { map } _ { * } ( Z , Y ) ), \end{equation*}

applied with $Z = [ 0,1 ]$; here $\operatorname{map}_{ *} ( ., . )$ is the morphism set in ${\cal T}_{*}$, suitably topologized, $\wedge$ is smash product and $\approx$ is homeomorphism [[#References|[a1]]]. In other words, the duality between projective and injective modules in $\mathcal{M}$ becomes an internal duality in the topological context.

The adjunction equivalence $\operatorname { map } _ { * } ( X \wedge S ^ { 1 } , Y ) \approx \operatorname { map } _ { * } ( X , \operatorname { map } _ { * } ( S ^ { 1 } , Y ) )$, with $S ^ { 1 }$ the circle, induces an adjunction equivalence

\begin{equation*} [ \Sigma X , Y ] \cong [ X , \Omega Y ] \end{equation*}

in ${\cal H}_*$; here $\Sigma$, $\Omega$ and $[ \cdot  \ , \ \cdot ]$ denote suspension, loop space and morphism set in ${\cal H}_*$. Using iterated suspensions and loop spaces, one introduces

\begin{equation*} \pi _ { n } ( X , Y ) = [ \Sigma ^ { n } X , Y ] \cong [ X , \Omega ^ { n } Y ], \end{equation*}

simultaneously generalizing the cohomology groups of $X$ (when $Y$ is an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]) and the homotopy groups of $Y$ (when $X$ is the $0$-sphere and $n \geq 1$). (When $X$ is a [[Moore space|Moore space]] of type $( A , 2 )$, with $A$ an Abelian group, one obtains homotopy groups with coefficients.) The $\pi _ { n } ( X , Y )$ may be generalized to $\pi _ { n } ( \alpha , \beta )$, where $\alpha$, $\beta$ are mappings; namely, $\pi _ { n } ( \alpha , \beta )$ is set equal to the homotopy classes of commutative diagrams $g \circ \alpha = \beta \circ f$. Various relative groups are special cases of this general construction and the standard exact sequences of [[Algebraic topology|algebraic topology]] ensue, in dual pairs. Also, the Postnikov decomposition of a path-connected space $Y$ (where the basic building blocks are Eilenberg–MacLane spaces) and the Moore decomposition of a $1$-connected space $X$ (where the basic building blocks are Moore spaces) are dual to one another; these two decompositions appear as special cases of what Eckmann and Hilton describe as the homotopy decomposition, respectively the homology decomposition of a map $\varphi : X \rightarrow Y$.

Again appealing to the adjunction equivalence in ${\cal T}_{*}$, one observes that the topological notion of [[Fibration|fibration]] (homotopy lifting property) dualizes to the notion of [[Cofibration|cofibration]] (homotopy extension property). Similarly, HELP (homotopy extension and lifting property) dualizes to co–HELP; [[#References|[a1]]], Thms. 4; 4$\square ^ { \color{blue} * }$. But HELP leads to the theorem of J.H.C. Whitehead that a mapping of path-connected CW-complexes inducing isomorphisms on homotopy groups is a homotopy equivalence [[#References|[a1]]], Thm. A, while co–HELP leads to another Whitehead theorem that a mapping of path-connected nilpotent CW-complexes inducing isomorphisms on homology groups is a homotopy equivalence [[#References|[a1]]], Thm. B. Thus, the two illustrious Whitehead theorems are Eckmann–Hilton dual, with dual proofs. (Cf. also [[Homotopy group|Homotopy group]].)

It is not true, however, that dual theorems necessarily admit dual proofs. An example is afforded by theorems of I.M. James and T. Ganea characterizing path-connected $H$-spaces and path-connected co-$H$-spaces, respectively. Thus, the path-connected space $Y$ is an $H$-space if and only if the canonical mapping $Y \rightarrow \Omega \Sigma Y$, adjoint to the identity mapping of $\sum Y$, admits a left homotopy inverse and the path-connected space $X$ is a co-$H$-space if and only if the canonical mapping $\Sigma \Omega X \rightarrow X$, adjoint to the identity mapping of $\Omega X$, admits a right homotopy inverse. No known proof of either theorem dualizes to a proof of the other.

It is also possible that the dual of a theorem is false. As an example, consider the well-known result that the suspension of the loop space of a sphere is homotopy equivalent to a co-product of spheres. The dual would assert that the loop space of the suspension of an Eilenberg–MacLane space is homotopy equivalent to a product of Eilenberg–MacLane spaces. However, this assertion fails already in the case that the Eilenberg–MacLane space is the circle $S ^ { 1 }$.

Sometimes the strict dual of a result turns out to admit a surprisingly interesting variant. A theorem of Hilton [[#References|[a2]]], Ref. H82, asserts that co-product cancellation fails for finite CW-complexes; there exist $1$-connected $2$-cell CW-complexes $V$, $W$ and a sphere $S$ such that $V \vee S \simeq W \vee S$ but $V \ncong W$. This theorem dualizes straightforwardly to an example of the failure of product cancellation; there exist $1$-connected, 2-stage Postnikov systems $Y , Z$ and an Eilenberg–MacLane space $K$ such that $Y \times K \simeq Z \times K$ but $Y \ncong Z$. The much more delicate question of failure of product cancellation for $1$-connected, finite CW-complexes was studied by P. Hilton and J. Roitberg [[#References|[a2]]], Ref. H98. One of their examples leads to the existence of a finite CW-complex which is an $H$-space (indeed, a loop space) not homotopy equivalent to any of the  "classical"  $H$-spaces. This example, along with other examples of A. Zabrodsky [[#References|[a6]]] helped usher in a new subdiscipline of homotopy theory, that of  "finite H-spaces"  and  "finite loop spaces" .

A conscientious Eckmann–Hilton dualist might enquire about the existence of duals of finite $H$-spaces; these would be finite Postnikov spaces which are co-$H$-spaces. Since a non-contractible, $1$-connected, finite Postnikov space cannot be a co-$H$-space (the Lyusternik–Shnirel'man category of such a space is infinite according to Y. Félix, S. Halperin, J.-M. Lemaire and J.-C. Thomas [[#References|[a7]]]), it follows that $S ^ { 1 } \vee \ldots \vee S ^ { 1 }$, the co-product of finitely many circles, is the only path-connected finite Postnikov space admitting a co-$H$-space structure. Thus the dual of a spicy piece of homotopy theory can be rather bland.

The space $S ^ { 1 } \vee \ldots \vee S ^ { 1 }$ calls to mind another example, discussed in [[#References|[a2]]], and given here. If $Y$ is a path-connected $H$-space of finite homotopical type (all homotopy groups are finitely generated), then there exists an $H$-space $Y ^ { 1 }$ with $H ^ { 1 } ( Y ^ { 1 } ; \mathbf{Z} ) = 0$ and a homotopy equivalence from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120020/e120020124.png"/> to $Y$. Dually, if $X$ is a path-connected co-$H$-space of finite homological type (all homology groups are finitely generated), there exists a $1$-connected co-$H$-space $X ^ { 1 }$ and a mapping from $X$ to $X ^ { 1 } \vee S ^ { 1 } \vee \ldots \vee S ^ { 1 }$ inducing homology isomorphisms. In 1971, Ganea posed the question of whether $X$ is homotopy equivalent to such a co-product. Recently (1997), N. Iwase has announced a negative answer to this question.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  J.P. May,   "The dual Whitehead theorems"  I.M. James (ed.) , ''Topological Topics. Articles on Algebra and Topology presented to Prof. P.J. Hilton in celebration of his sixtieth birthday'' , ''London Math. Soc. Lecture Notes'' , '''86''' , Cambridge Univ. Press  (1983)  pp. 46–54</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Mislin,   "Essay on Hilton's work in topology"  I.M. James (ed.) , ''Topological Topics. Articles on Algebra and Topology presented to Prof. P.J. Hilton in celebration of his sixtieth birthday'' , ''London Math. Soc. Lecture Notes'' , '''86''' , Cambridge Univ. Press  (1983)  pp. 15–30</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P. Hilton,   "Duality in homotopy theory: a restrospective essay"  ''J. Pure Appl. Algebra'' , '''19'''  (1980)  pp. 159–169</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  E.H. Spanier,   "Review of: B. Eckmann, Homotopie et dualité, Colloque de topologie algebrique, Louvain 1956, Masson, 1957, 41–53"  ''Math. Reviews'' , '''19'''  (1958)  pp. 570c</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Stasheff,   "Hilton–Eckmann duality revisited"  ''Contemp. Math.'' , '''37'''  (1985)  pp. 149–152</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Zabrodsky,   "Homotopy associativity and finite CW-complexes"  ''Topology'' , '''9'''  (1970)  pp. 121–128</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  Y. Felix,   S. Halperin,   J.-M. Lemaire,   J.-C. Thomas,   "Mod $p$ loop space homology"  ''Invent. Math.'' , '''95'''  (1989)  pp. 247–262</td></tr></table>