Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 4 / c

Table of contents
- Solution c

Solution

By part (b),

i : P \to H_{B}

is the equaliser of

i p

and

1_{H_{B}}

. Full faithfulness of the Yoneda embedding writes

i p = H_{b}

for a unique endomorphism

b : B \to B

. Choose an equaliser

e : A \to B

of

b

and

1_{B}

in

C

.

For each

c

, the maps

c \to A

are precisely the maps

u : c \to B

with

b u = u

. Thus

H_{e} : H_{A} \to H_{B}

is the pointwise equaliser of

H_{b}

and

1_{H_{B}}

, hence their equaliser in the presheaf category. Both

i

and

H_{e}

have the same universal property, so their vertices are isomorphic. This proves the representable retract criterion:

P ≅ H_{A} .

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