Let me begin with a reference: Marta Bunge and Jonathon Funk, Singular coverings of toposes. What follows is a very concise digest of very few of the features of that book.

Seems like the correct way of dualizing the correspondence between sheaves and local homeomorphisms is the following.

A continuous map $f:E\to X$ is a local homeomorphism iff the canonical map $$ \operatorname{stalk}_x(f):=\varinjlim_{U\ni x}\Gamma\left(\left.f\right|_U\right)\to f^{-1}(x) $$ given by $$ (\text{germ at $x$ of a section $\sigma$})\mapsto\sigma(x) $$ is bijective for any $x\in X$, where $\Gamma\left(\left.f\right|_U\right)$ denotes the set of all global sections of the map $\left.f\right|_U:f^{-1}(U)\to U$.

Now considering that $\Gamma$ is right adjoint to the inverse image, it is most natural to try the left adjoint to the inverse image. It does not always exist but in good cases (related to local connectedness) is given by connected components. So in case there are appropriate adjunctions $\pi_0\dashv f^*\dashv\Gamma$, it makes sense to consider the following condition on $f$ as a first approximation to the dual notion: it is a "co-local homeomorphism" if the canonical maps $$ f^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(f^{-1}(U))=:\operatorname{costalk}_x(f) $$ given by $$ f^{-1}(x)\ni e\mapsto\left(\text{connected component of $e$ in $f^{-1}(U)$}\right)_{U\ni x} $$ are bijective. In this context then, it is natural to call matching families of connected components of inverse images of neighborhoods of a point cogerms (of cosections?) at that point.

This approach shows that cosheaves are "more special" than sheaves: to begin with, the $\pi_0$'s involved might be quite nasty. Even if they are nice, usually they are non-discrete. And even if they are discrete, still in general the inverse limit carries a natural nondiscrete inverse limit topology, so it is natural to require of the above map to be not just a bijection but a homeomorphism. In this way one more or less gets complete spreads introduced by R. H. Fox in 1957 and studied by several people ever since. Their total spaces behave much better than those of local homeomorphisms (which might be non-Hausdorff even over very nice spaces). More or less, complete spreads are branched coverings.

There are several other subtleties to take into account, but, at any rate, Bunge and Funk build a theory dual to that of sheaves based on the notion of complete spread, and obtain a duality with several nice properties. In particular they deepen further the insight of of Lawvere that sheaves are "like functions" while cosheaves are "like distributions".

Let me begin with a reference: Marta Bunge and Jonathon Funk, Singular coverings of toposes. What follows is a very concise digest of very few of the features of that book.

Seems like the correct way of dualizing the correspondence between sheaves and local homeomorphisms is the following.

A continuous map $f:E\to X$ is a local homeomorphism iff the canonical map $$ \operatorname{stalk}_x(f):=\varinjlim_{U\ni x}\Gamma\left(\left.f\right|_U\right)\to f^{-1}(x) $$ given by $$ (\text{germ at $x$ of a section $\sigma$})\mapsto\sigma(x) $$ is bijective for any $x\in X$, where $\Gamma\left(\left.f\right|_U\right)$ denotes the set of all global sections of the map $\left.f\right|_U:f^{-1}(U)\to U$.

Now considering that $\Gamma$ is right adjoint to the inverse image, it is most natural to try the left adjoint to the inverse image. It does not always exist but in good cases (related to local connectedness) is given by connected components. So in case there are appropriate adjunctions $\pi_0\dashv f^*\dashv\Gamma$, it makes sense to consider the following condition on $f$ as a first approximation to the dual notion: it is a "co-local homeomorphism" if the canonical maps $$ f^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(f^{-1}(U))=:\operatorname{costalk}_x(f) $$ given by $$ f^{-1}(x)\ni e\mapsto\left(\text{connected component of $e$ in $f^{-1}(U)$}\right)_{U\ni x} $$ are bijective. In this context then, it is natural to call matching families of connected components of inverse images of neighborhoods of a point cogerms (of cosections?) at that point. So every point in the inverse image of $x$ determines a cogerm at $x$, and the condition is that every cogerm is of this kind for a unique point in the inverse image.

This approach shows that cosheaves are "more special" than sheaves: to begin with, the $\pi_0$'s involved might be quite nasty. Even if they are nice, usually they are non-discrete. And even if they are discrete, still in general the inverse limit carries a natural nondiscrete inverse limit topology, so it is natural to require of the above map to be not just a bijection but a homeomorphism. In this way one more or less gets complete spreads introduced by R. H. Fox in 1957 and studied by several people ever since. Their total spaces behave much better than those of local homeomorphisms (which might be non-Hausdorff even over very nice spaces). More or less, complete spreads are branched coverings.

There are several other subtleties to take into account, but, at any rate, Bunge and Funk build a theory dual to that of sheaves based on the notion of complete spread, and obtain a duality with several nice properties. In particular they deepen further the insight of of Lawvere that sheaves are "like functions" while cosheaves are "like distributions".

Let me begin with a reference: Marta Bunge and Jonathon Funk, Singular coverings of toposes. What follows is a very concise digest of very few of the features of that book.

Seems like the correct way of dualizing the correspondence between sheaves and local homeomorphisms is the following.

A continuous map $f:E\to X$ is a local homeomorphism iff the canonical map $$ \operatorname{stalk}_x(f):=\varinjlim_{U\ni x}\Gamma\left(\left.f\right|_U\right)\to f^{-1}(x) $$ given by $$ (\text{germ at $x$ of a section $\sigma$})\mapsto\sigma(x) $$ is bijective for any $x\in X$, where $\Gamma\left(\left.f\right|_U\right)$ denotes the set of all global sections of the map $\left.f\right|_U:f^{-1}(U)\to U$.

Now considering that $\Gamma$ is right adjoint to the inverse image, it is most natural to try the left adjoint to the inverse image. It does not always exist but in good cases (related to local connectedness) is given by connected components. So in case there are appropriate adjunctions $\pi_0\dashv f^*\dashv\Gamma$, it makes sense to consider the following condition on $f$ as a first approximation to the dual notion: it is a "co-local homeomorphism" if the canonical maps $$ f^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(f^{-1}(U))=:\operatorname{costalk}_x(f) $$ given by $$ f^{-1}(x)\ni e\mapsto\left(\text{connected component of $e$ in $f^{-1}(U)$}\right)_{U\ni x} $$ are bijective.

This approach shows that cosheaves are "more special" than sheaves: to begin with, the $\pi_0$'s involved might be quite nasty. Even if they are nice, usually they are non-discrete. And even if they are discrete, still in general the inverse limit carries a natural nondiscrete inverse limit topology, so it is natural to require of the above map to be not just a bijection but a homeomorphism. In this way one more or less gets complete spreads introduced by R. H. Fox in 1957 and studied by several people ever since. Their total spaces behave much better than those of local homeomorphisms (which might be non-Hausdorff even over very nice spaces). More or less, complete spreads are branched coverings.

There are several other subtleties to take into account, but, at any rate, Bunge and Funk build a theory dual to that of sheaves based on the notion of complete spread, and obtain a duality with several nice properties. In particular they deepen further the insight of of Lawvere that sheaves are "like functions" while cosheaves are "like distributions".

Let me begin with a reference: Marta Bunge and Jonathon Funk, Singular coverings of toposes. What follows is a very concise digest of very few of the features of that book.

Seems like the correct way of dualizing the correspondence between sheaves and local homeomorphisms is the following.

A continuous map $f:E\to X$ is a local homeomorphism iff the canonical map $$ \operatorname{stalk}_x(f):=\varinjlim_{U\ni x}\Gamma\left(\left.f\right|_U\right)\to f^{-1}(x) $$ given by $$ (\text{germ at $x$ of a section $\sigma$})\mapsto\sigma(x) $$ is bijective for any $x\in X$, where $\Gamma\left(\left.f\right|_U\right)$ denotes the set of all global sections of the map $\left.f\right|_U:f^{-1}(U)\to U$.

Now considering that $\Gamma$ is right adjoint to the inverse image, it is most natural to try the left adjoint to the inverse image. It does not always exist but in good cases (related to local connectedness) is given by connected components. So in case there are appropriate adjunctions $\pi_0\dashv f^*\dashv\Gamma$, it makes sense to consider the following condition on $f$ as a first approximation to the dual notion: it is a "co-local homeomorphism" if the canonical maps $$ f^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(f^{-1}(U))=:\operatorname{costalk}_x(f) $$ given by $$ f^{-1}(x)\ni e\mapsto\left(\text{connected component of $e$ in $f^{-1}(U)$}\right)_{U\ni x} $$ are bijective. In this context then, it is natural to call matching families of connected components of inverse images of neighborhoods of a point cogerms (of cosections?) at that point.

This approach shows that cosheaves are "more special" than sheaves: to begin with, the $\pi_0$'s involved might be quite nasty. Even if they are nice, usually they are non-discrete. And even if they are discrete, still in general the inverse limit carries a natural nondiscrete inverse limit topology, so it is natural to require of the above map to be not just a bijection but a homeomorphism. In this way one more or less gets complete spreads introduced by R. H. Fox in 1957 and studied by several people ever since. Their total spaces behave much better than those of local homeomorphisms (which might be non-Hausdorff even over very nice spaces). More or less, complete spreads are branched coverings.

There are several other subtleties to take into account, but, at any rate, Bunge and Funk build a theory dual to that of sheaves based on the notion of complete spread, and obtain a duality with several nice properties. In particular they deepen further the insight of of Lawvere that sheaves are "like functions" while cosheaves are "like distributions".