According to the nLab page of locally connected spaces (Theorem 4.1), the category of locally connected spaces is a coreflective subcategory of the category of topological spaces. The coreflector (right adjoint to the canonical inclusion) assigns to a topological space the coarsest topology finer than the original one.
I assume this holds analogously for connected, path-connected and locally path-connected spaces. So far, I sketched the following:
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Prove, that the intersection of two topologies (which is a topology again), one of which has one of the upper properties, inherits that respective property. The coreflector can then be defined as the intersection of all topologies finer and with the property desired.
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Prove, that this is a functor. Since both codomain and domain get a finer topology, the preservation of continuous maps depends on the respective property.
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Given topological spaces $X$ and $Y$ as well as a continuous map $f\colon X\rightarrow Y$, where $X$ has one of the above properties, $f$ factors above $Y$ with the respective coreflector of that property applied. (This is a more general case than the second step, which could therefore be omitted.) For example, let $\operatorname{Conn}\colon\mathbf{Top}\rightarrow\mathbf{Conn}$ be the coreflector into the category of connected topological spaces, then if $X$ is connected, $f\colon X\rightarrow Y$ factors as $f\colon X\xrightarrow{f}\operatorname{Conn}(Y)\xrightarrow{\operatorname{id}_Y}Y$, where the identity is continuous as the topology of $\operatorname{Conn}(Y)$ is per construction finer than that of $Y$. This makes the inclusion $\subseteq$ in:
$$\mathbf{Conn}(X,\operatorname{Conn}(Y))\cong\mathbf{Top}(X,Y),$$
trivial, while the inclusion $\supseteq$ is given by that lemma. Naturality follows from the fact, that a continuous map stays the same on the underlying sets, when $\operatorname{Conn}$ is applied.
I have not proven all the lemmas for all four properties yet, but am sure it will work out just fine. I still think, that there is a simpler way to prove the existence of respective coreflectors than just by construction, for example by a formulation of Vopěnka's principle found on the nLab page of coreflective subcategories (3. Properties): For a locally presentable category, every full subcategory which is closed under colimits is a coreflective subcategory.
Unfornutly, I am not familiar with locally presentable categories. Are there any remarks to my construction and what could maybe be shortened or proven directly with category theory? Can Vopěnka's principle be applied and how?
Best Answer
The subcategory of (path-)connected spaces is not coreflective. To show that the subcategory of locally (path-)connected spaces is coreflective, it suffices to check that the discrete topology is locally (path-)connected, and that the intersection of locally (path-)connected topologies is still locally (path-)connected.
A full subcategory $\mathcal C$ of topological spaces is coreflective if for each topological space $X$ has a coreflector, that is, a continuous map $CX\to X$ from $CX$ in $\mathcal C$ such that every continuous map $Y\to X$ for $Y$ in $\mathcal C$ factors as $Y\to CX\to X$ for a unique continuous map $Y\to CX$.
Note that if $X$ is in $\mathcal C$ then for $CX=X$ with the identity morphism is such a coreflector.
Note also that given two coreflectors $CX\to X$ and $CY\to Y$ and a continous map $X\to Y$, the composite $CX\to X\to Y$ must factor through $CY\to Y$ via a unique continuous map $CX\to CY$. It turns out that a choice of coreflectors for each topological space, together with the corresponding association of morphisms, is exactly the data of a right adjoint to the inclusion functor.
If $\mathcal C$ is the category of (path-)connected spaces and $CX\to X$ is a coreflector, then on the one hand the image of $CX\to X$ is contained in one (path-)component of $X$, and on the other hand the inclusion $C\hookrightarrow X$ of every (path)-component $C$ of $X$ into $X$ ha sto factor through $CX\to X$, whence must lie in the (path-)component of $X$ containing the image of $CX$. Thus, $X$ has only one (path-)component, i.e. is (path)-connected.
Thus the subcategories of (path-)connected space are not coreflective.
In the case of locally (path-)connected topological spaces, this amounts to verifying that for every topological space $X$ has a coarsest locally (path)- connected topology finer than it.
Given all continuous functions $Y\to X$ for $Y$ in $\mathcal C$, there is a coarsest topology on $X$ for which they are continuous. This topology is at least as fine as that of $X$, so we have a continuous map $CX\to X$ (the identity set-function changing to a possibly coarster topology) through which every continuous function $Y\to X$ for $Y$ in $\mathcal C$ factors uniquely.
It follows that a sufficienct (but not necessary) condition for the subcategory to be coreflective is that the above-defined $CX$ be in $\mathcal C$ for each topological space $X$. In particular, it suffices for the discrete topologies to be in $\mathcal C$ and for the intersection of topologies in $\mathcal C$ on the same set to also be a topology in $\mathcal C$.