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# $\mathcal{U}$-small

A set $S$ is said to be *$\mathcal{U}$-small* if it is isomorphic to an element of $\mathcal{U}$ (i.e., there is a bijection between $S$ and some element of $\mathcal{U}$).

A category $C$ is *$\mathcal{U}$-small* (or just *small*, if no confusion is likely to arise) if the set of objects of $C$ is isomorphic to a set in $\mathcal{U}$, and is a *$\mathcal{U}$-category* if for every pair of objects $A$, $B$ in $C$, $\Hom(A,B)$ is isomorphic to a set in $\mathcal{U}$.

These definitions amount to restrictions on the cardinality of the objects involved, and are intended to provide a condition that will allow operations such as extracting the category of functors or taking the direct limit to give results that are reasonable, that is, either isomorphic to an object of $\mathcal{U}$ or made up of objects of $\mathcal{U}$.

Observe that the category of subsets of $\mathcal{U}$ is a $\mathcal{U}$-category but is not $\mathcal{U}$-small.

# References

- SGA4
Grothendieck et al.,
*Séminaires en Gèometrie Algèbrique 4*, tomes 1, 2, and 3. - Mur68
Murphy, O.
*Some modern methods in the theory of lion hunting*, American Mathematical Monthly 75 (2), Feb., 1968, 185–187.

## Mathematics Subject Classification

18A15*no label found*03E30

*no label found*

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## Corrections

emphasis on defined terms by Mathprof ✓

synonym by CWoo ✓

few minor issues by CWoo ✓

minor issues by CWoo ✓

## Comments

## title too general

I initially named this entry "small" because that is the technical term used to mean this. But a correction was filed asking me to change it (to $U$-small, which is occasionally also used). Noticing that about fifty entries pointed here, three of which meant the right thing, I did, but filed corrections against those three. Which brings me to my point:

Can I do a search for "what would point here if it were titled X?" This might be useful for entries that define common words in unusual ways. The entry "fix", which I usually have to linkescape, is maybe not such a good candidate as there are many places that want to link to it. But surely there are others...

## Re: title too general

as far as I know

there IS a list

of words

that won't be linked by default

precisely because they are frequent words with common snese meanings (opposed to math meanings), I guess fix and small should be added

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f