Editing Enriched category (section)

==Definition==
Let {{math|('''M''', ⊗, ''I'', ''α'', ''λ'', ''ρ'')}} be a [[monoidal category]]. Then an ''enriched category'' '''C''' (alternatively, in situations where the choice of monoidal category needs to be explicit, a ''category enriched over'' '''M''', or '''M'''-''category''), consists of
* a [[class (set theory)|class]] ''ob''('''C''') of ''objects'' of '''C''',
* an object {{math|''C''(''a'', ''b'')}} of '''M''' for every pair of objects ''a'', ''b'' in '''C''', used to define an arrow <math>f: a \rightarrow b</math> in '''C''' as an arrow <math>f:I\rightarrow C(a,b)</math> in '''M''',
* an arrow {{math|id<sub>''a''</sub> : ''I'' → ''C''(''a'', ''a'')}} in '''M''' designating an ''identity'' for every object ''a'' in '''C''', and
* an arrow {{math|°<sub>''abc''</sub> : ''C''(''b'', ''c'') ⊗ ''C''(''a'', ''b'') → ''C''(''a'', ''c'')}} in '''M''' designating a ''composition'' for each triple of objects ''a'', ''b'', ''c'' in '''C''', used to define the composition of <math>f:a\rightarrow b</math> and <math>g:b\rightarrow c</math> in '''C''' as <math>g \circ_{\textbf{C}} f = {^\circ}_{abc}(g\otimes f)</math> together with three commuting diagrams, discussed below.

The first diagram expresses the associativity of composition:

:[[Image:Math-enriched category associativity.svg]]

That is, the associativity requirement is now taken over by the [[associator]] of the monoidal category '''M'''.

For the case that '''M''' is the [[category of sets]] and {{math|(⊗, ''I'', ''α'', ''λ'', ''ρ'')}} is the monoidal structure {{math|(×, {•}, ...)}} given by the [[cartesian product]], the terminal single-point set, and the canonical isomorphisms they induce, then each {{math|''C''(''a'', ''b'')}} is a set whose elements may be thought of as "individual morphisms" of '''C''', while °, now a function, defines how consecutive morphisms compose.  In this case, each path leading to {{math|''C''(''a'', ''d'')}} in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms {{math|''a'' → ''b'' → ''c'' → ''d''}}, i.e. elements from {{math|''C''(''a'', ''b'')}}, {{math|''C''(''b'', ''c'')}} and {{math|''C''(''c'', ''d'')}}. Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories.

What is new here is that the above expresses the requirement for associativity without any explicit reference to individual morphisms in the enriched category '''C''' &mdash; again, these diagrams are for morphisms in monoidal category '''M''', and not in '''C''' &mdash; thus making the concept of associativity of composition meaningful in the general case where the hom-objects {{math|''C''(''a'', ''b'')}} are abstract, and '''C''' itself need not even ''have'' any notion of individual morphism.

The notion that an ordinary category must have identity morphisms is replaced by the second and third diagrams, which express identity in terms of left and right [[unitor]]s:

:[[File:Math-enriched category identity1.svg]]
and
:[[File:Math-enriched category identity2.svg]]

Returning to the case where '''M''' is the category of sets with cartesian product, the morphisms {{math|id<sub>''a''</sub>: ''I'' → ''C''(''a'', ''a'')}} become functions from the one-point set ''I'' and must then, for any given object ''a'', identify a particular element of each set {{math|''C''(''a'', ''a'')}}, something we can then think of as the "identity morphism for ''a'' in '''C'''". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in '''C'''" behave exactly as per the identity rules for ordinary categories.

Note that there are several distinct notions of "identity" being referenced here:
* the ''monoidal identity object'' {{mvar|I}} of '''M''', being an identity for ⊗ only in the [[monoid]]-theoretic sense, and even then only up to canonical isomorphism {{math|(''λ'', ''ρ'')}}.
* the ''identity morphism'' {{math|1<sub>''C''(''a'', ''b'')</sub> : ''C''(''a'', ''b'') → ''C''(''a'', ''b'')}} that '''M''' has for each of its objects by virtue of it being (at least) an ordinary category.
* the ''enriched category identity'' {{math|id<sub>''a''</sub> : ''I'' → ''C''(''a'', ''a'')}} for each object ''a'' in '''C''', which is again a morphism of '''M''' which, even in the case where '''C''' ''is'' deemed to have individual morphisms of its own, is not necessarily identifying a specific one.