We prove the recognition principle for relative
–loop pairs of
spaces for
.
If
,
this states that a pair of spaces homotopy equivalent to CW–complexes
is homotopy equivalent
to
for a functorially
determined relative space
if and only if
is a
grouplike
–space,
where
is any cofibrant resolution of the Swiss-cheese relative operad
.
The relative recognition principle for relative
–loop pairs of spaces states
that a pair of spaces
homotopy equivalent to CW–complexes is homotopy equivalent to
for a functorially determined
relative spectrum
if and only if
is a
grouplike
–algebra,
where
is a
contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal
relative operad
of continuous homomorphisms of commutative monoids. These principles are proved
as equivalences of homotopy categories.
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