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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.

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    $\begingroup$ I think 2-categories would suffice here, but I may be wrong $\endgroup$ Commented Aug 27, 2021 at 11:10
  • $\begingroup$ Thanks, I will check if 2-categories suffice $\endgroup$
    – Ho Man-Ho
    Commented Aug 27, 2021 at 12:35
  • $\begingroup$ I dont understand what is 'the category of differential forms on M', what category structure are you considering? $\endgroup$
    – AT0
    Commented Aug 27, 2021 at 13:29
  • $\begingroup$ I am probably wrong. In principle $\Omega^*:\textrm{Man}^{\textrm{op}}\to\textrm{Set}$ should be a functor to the category of sets. I will revise my question. Thank you. $\endgroup$
    – Ho Man-Ho
    Commented Aug 27, 2021 at 15:31
  • $\begingroup$ Superficially, it looks like you want a natural transformation between functors $\operatorname{Man} \to \operatorname{Cat}$. However, for that to make sense, you need to make sense of $\Omega(M)$ as a category, and pointwise $T(M): \operatorname{Vect}_\Delta(M) \to \Omega(M)$ as a functor of categories. $\endgroup$ Commented Aug 27, 2021 at 15:55

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$T$ 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.

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  • $\begingroup$ I would argue that the most natural approach to defining $\operatorname{Vect}_\nabla$ is to construct the cartesian fibration it classifies. I understand it might not feel as elementary, but if we're using ∞-categories anyway... $\endgroup$ Commented Aug 27, 2021 at 16:10
  • $\begingroup$ @DenisNardin: The OP only mentioned (∞,1)-categories, not ∞-categories, and in the context of a paper he was reading. Ordinary Grothendieck fibrations are sufficient for encoding vector bundles with connection. But in terms of actual efficiency and ease of use, in my experience the site of cartesian manifolds always wins. $\endgroup$ Commented Aug 27, 2021 at 16:18
  • $\begingroup$ @DmitriPavlov Thank you for solving my question. Probably you have written your answer in the language that I can understand. $\endgroup$
    – Ho Man-Ho
    Commented Aug 27, 2021 at 16:41

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