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Are the following definitions are equivalent?

An embedding of topological spaces is just a monic in Top, i.e. it is a continuous $e: A \rightarrow X$ s.t. for any space $Z$ the function $f \mapsto e \circ f:$ Top $\left(Z,A\right)\rightarrow$ Top $\left(Z,X\right)$ is injective i.e. $e_\circ f_1=e\circ f_2 \implies f_1 =f_2$.

and

$e:A \rightarrow X$ is an embedding iff

  • $U\left(e\right):U\left(A\right)\rightarrow U\left(X\right)$ is injective in Set, where $U$ is the underling set functor.
  • for any $Z \in$ Top a set map $U\left(f\right):U\left(Z\right) \rightarrow U\left(A\right)$, $f$ is continues iff $e \circ f : Z \rightarrow X$ is continuous.

Is the analogue definitions for epic and quotient space are also equivalent? Many thanks.

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No. An embedding of topological spaces is - by definition - something stronger: A continuous map which is injective and induces a homeomorphism onto its image. –  Martin Brandenburg May 29 '13 at 19:58

3 Answers 3

The first definition actually characterises injective continuous maps, but the second definition does indeed characterise topological embeddings. More generally there is a notion of initial topology and final topology for a family of maps, and an embedding is precisely an injective map that induces the initial topology on its domain, and a quotient is precisely a surjective map that induces the final topology on its codomain. The universal property you have written down is a translation of that into the language of category theory.

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Let Top be the category of topological spaces (and continuous maps) and Haus be the subcategory of Hausdorff topological spaces.

  • monomorphisms in Top and Haus are precisely injective continuous maps;
  • epimorphisms in Top are precisely surjective continuous maps;
  • epimorphisms in Haus are precisely dense continuous maps (i.e., maps $f\colon X\to Y$ such that $\overline{f[X]}=Y$); see this question.

Note that not every monomorphism is an embedding. Not every epimorphisms is a quotient map.

Your second description says that $A$ has the initial topology w.r.t. to the map $e$. This is the same as saying that $e$ is an embedding. Similarly, final topology corresponds to a quotient map.

It might be also useful to mention that in Top the extremal monomophisms and regular monomorphisms coincide with embeddings. In Haus we get closed embeddings. Regular and extremal epimorphisms are quotient maps in both Top and Haus.

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There is also the notion of an extremal monic in a category. This is a monic $m$ such that for each factorization $m=f\circ e$ where $e$ is an epic, $e$ is also an isomorphism.

Using this notion, one can characterize the topological embeddings categorically: They are just the extremal monics in Top.

Dually, an extremal epic is an extremal monic in the opposite category. Specifically, it is an epic $e$ such that each monic $m$ in a factorization $e=m\circ f$ is an isomorphism.

The extremal epics in Top are precisely the quotient maps.

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