Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 119 / 2 / ii

Table of contents
- Solution ii

Solution

Apply condition

(ii)

with the singleton

S = 1

. It says that the category of elements

E = (1 ↓ F)

has all finite limits, including a terminal object. The opposite category

E^{op}

therefore has all finite colimits. In particular it is nonempty; two objects map to their binary coproduct in a category; and any parallel arrows are equalized by their coequalizer. These are precisely the conditions for a filtered category. Equivalently, every finite diagram has a cocone, obtained by reversing a categorical limit in

E

. Thus

(ii) \Rightarrow (iii) : (1 ↓ F)^{op} is filtered .

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