I want to formulate something using the language of (possibly higher) category theory, but my knowledge in category theory is what most graduate students have learned in a first course in algebraic topology. So hopefully someone can help me out.
Let me provide some background to my question. A paper I am reading says "Let Cat be an $(\infty, 1)$-category of (small) categories, obtained by localizing at categorical equivalences. Define the $\infty$-functor
$$\textrm{Vect}_\nabla:\textrm{Man}^{\textrm{op}}\to\textrm{Cat}$$
where $\textrm{Man}$ is the category of smooth manifolds, so that for each manifold $M$, $\textrm{Vect}_\nabla(M)$ is the category of vector bundles over $M$ with connections." Okay, I understand only the bold part.
My question is, let say I want to define a "functor" $T$ so that for each manifold $M$, $T(M):\textrm{Vect}_\nabla(M)\to\Omega(M)$ is a functor from the category of vector bundles over $M$ with connections to the functor $\Omega^*:\textrm{Man}^{\textrm{op}}\to\textrm{Set}$ from the opposite category of $\textrm{Man}$ to the category of sets, which assigns to each manifold $M$ the set $\Omega(M)$ of differential forms on $M$ (probably with more algebraic structures, but anyway) such that for each $(E, \nabla)$, $T(M)(E, \nabla)\in\Omega(M)$. What the object should $T$ be? I expect it has something to do with $\infty$-functor?
Of course one can think of $T$ as Chern character form, where the Chern character is a natural transformation between the $K$-theory to ordinary cohomology as functors. But I cannot use K-theory here for some reason. Moreover, the "functor" $T$ must take a manifold $M$ first, and then take a vector bundle over $M$ with a connection.
Google shows that there are several books, notes and papers about higher category theory and related areas, but I am not certain where to start and it seems that most of them are too abstract for me, while I just need to learn all the necessary mathematics to formulate my question. You are very welcome to recommend any good source with concrete examples.
Thank you.
Edit: Thanks to AT0 for correcting a mistake.
Best Answer
$T$ can be formalized as a natural transformation $\def\Vect{{\rm Vect}} \def\Vectc{\Vect_\nabla} \Vectc→Ω^n$ of functors $\def\Man{{\sf Man}} \def\op{{\sf op}} \def\Grpd{{\sf Grpd}} \Man^\op → \Grpd$.
The functor $\Vectc$ sends a smooth manifold $M$ to the groupoid $\Vectc(M)$ of vector bundles with connection over $M$ and connection-preserving isomorphisms. It also sends a smooth map $f\colon M→M'$ of smooth manifolds to the corresponding pullback functor $$\Vectc(f)\colon \Vectc(M')→\Vectc(M).$$ There are many ways to make $\Vectc$ preserve composition (as required by the definition of a functor). Some of the easiest approaches are (1) strictify $\Vectc(M)$ by adding formal pullbacks; (2) make minor adjustments to the definition of vector bundles and pullbacks, ensuring the pullback preserves compositions on the nose; or (3) use the site of cartesian manifolds (diffeomorphic to $\def\R{{\bf R}} \R^n$), which yields an equivalent category of ∞-sheaves.
The last approach is the one used most often in practice. Observe that on cartesian manifolds, $\Vectc$ can be defined as the groupoid of connection 1-forms (every bundle on $\R^n$ is trivial), and differential forms pull back strictly.
The functor $Ω^n$ sends a smooth manifold $M$ to the set of differential $n$-forms on $M$, which is turned into a discrete groupoid by adding identity morphisms. It sends a smooth map $f\colon M→M'$ to the pullback map $$Ω(f)=f^*\colon Ω(M')→Ω(M).$$
Now $T\colon \Vectc→Ω^n$ is a natural transformation, whose components are given by functors $$T(M)\colon \Vectc(M)→Ω^n$$ that send a vector bundle with connection over $M$ to the corresponding differential $n$-form (given by the Chern–Weil homomorphism, for example) and a connection-preserving isomorphism gets mapped to an equality of differential forms.