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added pointer to:
In this paper, Mitchell says that representable presheaves are tiny when the base category C has finite products, but the nlab page says that this is true with no assumptions about the base category. I don’t see any dependence on finite products in the construction, am I missing something? Maybe he is just confusing this with the fact that powering by a representable has a simpler formulation when the base category has cartesian products.
Let’s see. I suppose this is because Mitchell requires the internal hom out of a tiny object to preserve colimits (p. 1).
A sufficient condition for reducing this requirement, in the case of representable presheaves, to the argument given on the nLab page for the external hom is that the site has finite products.
Oh and I see now that this is mentioned already on the page (2.4). Carry on!
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