Is the category of elements induced by a topological functor topological? Let $F: \mathcal{A} \to {\bf Set}$ be a topological functor (see Adamek - The joy of cats) and consider the category of elements $el(F)$ as defined in
http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/category+of+elements
Does there exist a topological functor from $el(F)$ to ${\bf Set}$?
 A: There are no topological functors $G\colon A\to\mathbf{Set}$ for which $el(G)\to\mathbf{Set}$ is also topological. This is because $G\colon A\to\mathbf{Set}$ being topological implies both that $A$ has a strict initial object and that $el(G)$ has an initial object that is not strict.

For the first claim, recall that

*

*the empty set $\emptyset$ is an initial object, i.e. for every set $S$ there is a unique morphism $\emptyset\to S$, and

*that for a topological functor $G\colon A\to\mathbf{Set}$ the empty set $\emptyset$ has a reflection whose unit is an isomorphism, i.e. there is an object $\mathbf0\in A$ and an isomorphism $\emptyset\cong\mathbf0$ such that the unique morphism $\emptyset\to GY$ that has to factor as $\emptyset\cong G\mathbf0\to FY$ for a unique morphism $\mathbf 0\to Y$.

It follows from the definitions that properties 1. and  2. are equivalent to  $A$ having an initial object $\mathbf0$ preserved by $G$. Moreover, in that case any morphism $X\to\mathbf0\in A$ is a retraction of the unique morphism $\mathbf0\to X$, i.e. such that $\mathbf0\to X\to\mathbf0$ is the identity.
Next, recall that


*the empty set is a strict initial object, i.e. any morphism $S\to\emptyset$ is an isomorphism, and moreover its inverse is the unique morphism $\emptyset\to S$;

*$G\colon A\to\mathbf{Set}$ is faithful on morphisms, i.e. $Gf=Gg$ implies $f=g$.

It follows from 3. that $GX\to G\mathbf 0\cong\emptyset\to GX$, which is the image of $X\to\mathbf0\to X$, is the identity. Then 4. implies that $X\to\mathbf0\to X$ must also be the identity, and so $\mathbf 0$ is a strict initial object of $A$.
More generally, if a faithful functor preserves an initial object and sends it to a strict initial object, then the initial object in the domain is also strict. In particular, this is the case with topological functors to $\mathbf{Set}$.

For the second claim, recall that if $G\colon A\to\mathbf{Set}$ is topological, then every set $S$ has a reflection across $G$ whose unit $S\to GFS$ is monic, i.e. $G$ has a faithful left adjoint $F\colon\mathbf{Set}\to A$. Explcitily, for every set $S$ there is an object $FS\in A$ and a monomorphism $\eta_S\colon S\hookrightarrow GFS$ so that each morphism $S\to GY$ factors as $S\to GFS\to GY$ for a unique morphism $FS\to Y$.
In the case where $S$ is a singleton $\{*\}$, we have that a morphism $\{*\}\to GY$, a so-called element of $G$, factors as $\{*\}\hookrightarrow G1\to GY$ for a unique morphism $1\to Y$, where $1=F\{*\}$. Thus, the category of elements of $G$ is the undercategory $1/A$. In particular, the category $el(G)$ of elements of $G$ has an initial object corresponding to $\mathrm{id}\colon1\to1$.
A morphism to this initial object is a factorization $1\to Y\to 1$ of the identity, and the initial object would fail to be strict if for some such factorization, the morphism $Y\to1$ is not an isomorphism. Since $Y\to1$ would have a section $1\to Y$, a suficient condition for it to not be an isomorphism is for it to have a second distinct section $1\to Y$ (each section, i.e. left inverse, of an isomorphism is its unique inverse).
But a set $S$ has more than one element precisely when $S\to\{*\}$ has two distinct sections, and $F$ being faithful implies $FS\to1$ has two distinct secitons. In particular, the initial object in the category of elements of $F$ is not strict.
