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Homemonomorphic set

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# monomorphic set

Let $\mathcal{C}$ be a category and $M:=\{f_{i}:A\to B_{i}\mid i\in I\}$ a set (indexed by a set $I$) of morphisms with common domain $A$ in $\mathcal{C}$. Then $M$ is said to be a *monomorphic set* if for any pair of morphisms $g,h:C\to A$, $f_{i}\circ g=f_{i}\circ h$ for all $i\in I$ imply that $g=h$. A *monomorphic pair* is a monomorphic set $M$ such that the cardinality of $M$ is 2.

Monomorphic sets are generalizations of monomorphisms. Indeed, for if $\{f:A\to B\}$ is a monomorphic set, then $f$ is a monomorphic.

For example, in Set, the category of sets, let $R$ be an $n$-ary relation on a set $A$. For each $i=1,\ldots,n$, let $p_{i}$ be the projection of the $i$-th coordinate of $R$ into $A$. Then

$\{p_{i}\mid i=1,\ldots,n\}$ |

is a monomorphic set in Set. To see this, observe first that, since $R$ is a subset of $A^{n}$, any function $f:B\to R$ has $n$ components, $f_{i}:B\to A$, given by $f_{i}=p_{i}\circ f$. Now, suppose $g,h:B\to R$ are functions, such that $p_{i}\circ g=p_{i}\circ h$. Then $g_{i}=h_{i}$ for all $i$. In other words, all components of $g$ and $h$ match. Therefore $g=h$.

More generally, a relation $R$ between sets $A_{1},\ldots,A_{n}$ is a subset of the cartesian product $A_{1}\times\cdots\times A_{n}$. The set of projections $\{p_{i}:R\to A_{i}\mid i=1,\ldots,n\}$ is also a monomorphic set in Set. Using this concept, one may generalize the notion of a relation on sets to a relation on objects in a category.

Remark. One can dually define *epimorphic sets* and *epimorphic pairs*.

# References

- 1
F. Borceux
*Basic Category Theory, Handbook of Categorical Algebra I*, Cambridge University Press, Cambridge (1994)

## Mathematics Subject Classification

18A20*no label found*

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