Schreiber
differential cohomology - with grouplike coefficients

This is a subentry of the entry differential cohomology.

Recall from the discussion there that there are two ways to characterize the obstructions to lifts of $A$-cocycles to flat differential $A$-cocycles through the morphism $A_{\mathrm{flat}}\to A$.

• One is exact, the case where this morphism is a homotopy fiber. This happens when $A$ is group-like in that its delooping $BA$ exists. This is the case discussed here.

• If $A$ is not grouplike, one may use a Chern character morphism $A\to {\prod }_i{B}^{n_i}R$ to approximate $A$ by an object that is. This leads to a nonabelian generalization of the classical definition of abelian differential cohomology in terms of homotopy fibers of the Chern character map, and is described at differential cohomology - with general coefficients.

Throughout this entry we assume that the object $A$ is a group object and hence once deloopable. In the case that it is not the discussion at differential cohomology - nonabelian case applies.

We discuss in the following homotopy pullbacks in the global model structure on simplicial presheaves. The same results for the local model structure are obtained

• either by $\mathrm{\infty }$-stackigication, which preserves homotopy pullbacks

• or by assuming that we are on a site with contractible objects, in which case $\Pi ⊣(-{)}_{\mathrm{flat}}$ extends to a and all the below reasoning applies directly in the local model structure.

Proposition

(differential fibration sequence) For once deloopable $A\in \mathrm{SPSh}(C)$ there is in $\mathrm{SPSh}(C)$ a fibration sequence

$$A_{\mathrm{flat}}\to A\to B{A}_{\mathrm{dR}},$$A_{flat} \to A \to \mathbf{B}A_{dR} \,,

where the morphism on the left is the canonical one described at path ∞-groupoid.

This realizes the flat differential refinement of $A$ as the homotopy fiber of a morphism from $A$ to the deRham differential refinement of $BA$.

Proposition

(characterization of differential cocycles)

For fibrant $A\in \mathrm{SPSh}(C)$ as above there is an object $A_{\mathrm{diff}}\in \mathrm{SPSh}(C)$ such that

• there is a span

  $$A\stackrel{\simeq }{\leftarrow }{A}_{\mathrm{diff}}\to B{A}_{\mathrm{dR}}$$A \stackrel{\simeq}{\leftarrow} A_{diff} \to \mathbf{B}A_{dR}

  in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ with the left leg a weak equivalence and the right morphism a fibration in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ (i.e. a global fibration)

• for $Y$ cofibrant the set of objects in $H(Y,A_{\mathrm{diff}})$ is the set of equivalence classes of pairs consisting of a commuting diagram

  $$\begin{array}{ccc}Y& \stackrel{g}{\to }& A\\ \downarrow & & {\downarrow }^i\\ \Pi (Y)& \stackrel{\nabla }{\to }& EA\end{array}$$\array{ Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow^i \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A }

• the projection $H(Y,A_{\mathrm{diff}})\to H(Y,A)$ sends such a diagram to the plain $A$ cocycle given by the top horizontal morphism $(g:Y\to A)\in H(X,A)$ of such a diagram;

• for $[P]\in H(Y,B{A}_{\mathrm{dR}})$ a curvature characteristic class the $P$-twisted flat differential cohomology $H_{\mathrm{diff}}^{[P]}(Y,A)$ is equivalent to the ordinary (not homotopy-) pullback

  $$\begin{array}{ccc}{H}_{\mathrm{diff}}^{[P]}(Y,A)& \to & *\\ \downarrow & & {\downarrow }^{*\mapsto P}\\ H(Y,A_{\mathrm{diff}})& \to & H(Y,B{A}_{\mathrm{dR}})\end{array}$$\array{ \mathbf{H}_{diff}^{[P]}(Y,A) &\to& {*} \\ \downarrow && \;\;\downarrow^{\mathrlap{*\mapsto P}} \\ \mathbf{H}(Y,A_{diff}) &\to& \mathbf{H}(Y,\mathbf{B}A_{dR}) }

• the set of objects of $H_{\mathrm{diff}}^{[P]}(X,A)$ in this realization is the set of commuting diagrams

  $$\begin{array}{ccc}Y& \stackrel{g}{\to }& A\\ \downarrow & & {\downarrow }^i\\ \Pi (Y)& \stackrel{\nabla }{\to }& EA\\ {\downarrow }^{\mathrm{Id}}& & \downarrow \\ \Pi (Y)& \stackrel{P(\nabla )}{\to }& BA\end{array}$$\array{ Y &\stackrel{g}{\to}& A \\ \downarrow && \downarrow^i \\ \Pi(Y) &\stackrel{\nabla}{\to}& \mathbf{E}A \\ \downarrow^{Id} && \downarrow \\ \Pi(Y) &\stackrel{P(\nabla)}{\to}& \mathbf{B}A }

• a 2-cell in $H_{\mathrm{diff}}^{[P]}(X,A)$ is a degreewise 2-cell in the top two rows of this diagram such that everything commutes on the nose

  $$\begin{array}{ccc}Y& \stackrel{\nearrow \searrow }{\to }& A\\ \downarrow & & {\downarrow }^i\\ \Pi (Y)& \stackrel{\nearrow \searrow }{\to }& EA\\ {\downarrow }^{\mathrm{Id}}& & \downarrow \\ \Pi (Y)& \stackrel{}{\to }& BA\end{array}$$\array{ Y &\stackrel{\nearrow \searrow}{\to}& A \\ \downarrow && \downarrow^i \\ \Pi(Y) &\stackrel{\nearrow \searrow}{\to}& \mathbf{E}A \\ \downarrow^{Id} && \downarrow \\ \Pi(Y) &\stackrel{}{\to}& \mathbf{B}A }

Definition

(connection) We call a lift (meaning: lift up to equivalence) of a cocycle $c\in H(Y,A)$ through the morphism $H(Y,A_{\mathrm{diff}})\to H(Y,A)$ a choice of connection on $c$. Specifically the morphism $\nabla :\Pi (Y)\to EA$ that is part of the data of such a lift according to the above coroally is the connection chosen on $c$.

Corollary

(existence of connections) Every $A$-cocycle $c$ admits a connection.

If we identify $A$-cocycles with the principal ∞-bundles that they classify, then this reads:

Every $A$-principal ∞-bundle admits has a connection.

Recall that so far the assumption on $A$ is that it is once deloopable. This is a “semi-abelian” case.

We now turn to the general nonabelian case where $A$ is not assumed to be deloopable.

Proofs

In the following the model structure on $\mathrm{SPSh}(C)$ is always $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ the local projective one.

To prove the proposition differential fibration sequence we use a few lemmas.

For notation, conventions and facts about the Reedy model structures on the arrow category $[I,\mathrm{SPSh}(C)]$ that is used in the following see the discussion at relative cohomology.

Cofibrant objects in $[I,\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}=[I,\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{]}_{\mathrm{proj}}$ are cofibrations $YhookrightaroX$ of cofibrant objects in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ and fibrant objects are any morphism $A\to B$ between fibrant objects in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

Lemma

For any $A\in \mathrm{SPSh}(C)$ we have the following isomorphic expressions for $A$, and for $A_{\mathrm{flat}}$

$$A:U\mapsto [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}A\\ \downarrow \\ *\end{array})$$A : U \mapsto [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,, \array{ A \\ \downarrow \\ {*} } \right)

and

$$A_{\mathrm{flat}}:U\mapsto [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}A\\ {\downarrow }^{\mathrm{Id}}\\ A\end{array})$$A_{flat} : U \mapsto [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,, \array{ A \\ \downarrow^{Id} \\ A } \right)

Lemma

For $A\in \mathrm{SPSh}(C)$ there is in the arrow category $[I_{+},\mathrm{SPSh}(C{)}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$ a fibration sequence of natural transformations

$$\left[\begin{array}{c}A\\ \downarrow \\ A\end{array}\right]\to \left[\begin{array}{c}A\\ \downarrow \\ *\end{array}\right]\to \left[\begin{array}{c}*\\ \downarrow \\ BA\end{array}\right].$$\left[ \array{ A \\ \downarrow \\ A } \right] \to \left[ \array{ A \\ \downarrow \\ {*} } \right] \to \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right] \,.

Proof

Recall that $A\in \mathrm{SPSh}(C{)}^{\mathrm{loc}}$ is assumed to be fibrant. Using the list of notation and constructions in categories of fibrant objects we write $BA$ for the delooping of $A$ and

$$A\stackrel{i}{\to }EA\stackrel{p}{\to }BA$$A \stackrel{i}{\to} \mathbf{E}A \stackrel{p}{\to} \mathbf{B}A

for the corresponding principal ∞-bundle fibration sequence in $\mathrm{SPSh}(C{)}^{\mathrm{loc}}$.

We define the morphism

$$\left[\begin{array}{c}A\\ \downarrow \\ *\end{array}\right]\to \left[\begin{array}{c}*\\ \downarrow \\ BA\end{array}\right]$$\left[ \array{ A \\ \downarrow \\ {*} } \right] \to \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right]

in the (∞,1)-category $H_{\mathrm{rel}}$ that describes relative cohomology by the span in $[I_{+},\mathrm{SPSh}(C{)}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$

$$\left[\begin{array}{c}A\\ \downarrow \\ *\end{array}\right]\stackrel{\simeq }{\leftarrow }\left[\begin{array}{c}A\\ \downarrow \\ EA\end{array}\right]\stackrel{f}{\to }\left[\begin{array}{c}*\\ \downarrow \\ BA\end{array}\right]$$\left[ \array{ A \\ \downarrow \\ {*} } \right] \; \stackrel{\simeq}{\leftarrow} \; \left[ \array{ A \\ \downarrow \\ \mathbf{E}A } \right] \;\stackrel{f}{\to}\; \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right]

that as a naturality diagram in $\mathrm{SPSh}(C)$ is given by

$$\begin{array}{ccccc}A& \stackrel{\mathrm{Id}}{\leftarrow }& A& \to & *\\ \downarrow & & {\downarrow }^i& & \downarrow \\ *& \leftarrow & EA& \stackrel{p}{\to }& BA\end{array}.$$\array{ A &\stackrel{Id}{\leftarrow}& A &\to& * \\ \downarrow && \downarrow^i && \downarrow \\ {*} &\leftarrow& \mathbf{E}A &\stackrel{p}{\to}& \mathbf{B}A } \,.

Since all objects and all horizontal morphisms in this naturality diagram are fibrations, the horizontal morphism denoted $f$ above is a fibration between fibrant objects in $[I_{+},\mathrm{SPSh}(C{)}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$.

This means (as discussed at homotopy limit) that the homotopy fiber of this morphism is computed already by the ordinary limit over

$$\left[\begin{array}{c}A\\ \downarrow \\ EA\end{array}\right]\stackrel{f}{\to }\left[\begin{array}{c}*\\ \downarrow \\ BA\end{array}\right]\leftarrow \left[\begin{array}{c}*\\ \downarrow \\ *\end{array}\right]$$\left[ \array{ A \\ \downarrow \\ \mathbf{E}A } \right] \;\stackrel{f}{\to}\; \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right] \;\leftarrow\; \left[ \array{ {*} \\ \downarrow \\ {*} } \right]

in $[I_{+},\mathrm{SPSh}(C{)}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$.

Being a limit over a functor category this is computed objectwise. The top object is

$$\lim (\cdots {)}_0=\lim (A\to *\leftarrow *)\simeq A$$lim(\cdots)_0 = lim( A \to {*} \leftarrow {*} ) \simeq A

trivially, and the bottom object is

$$\lim (\cdots {)}_1=\lim (EA\to BA\leftarrow *)\simeq A$$lim(\cdots)_1 = lim ( \mathbf{E}A \to \mathbf{B}A \leftarrow {*}) \simeq A

be definition of delooping. Drawing the full diagram shows that the morphism $\lim (\cdots {)}_0\to \lim (\cdots {)}_1$ is the universal morphism from the commuting diagram

$$\begin{array}{ccc}A& \to & *\\ \downarrow & & \downarrow \\ EA& \to & BA\end{array}$$\array{ A &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{E}A &\to& \mathbf{B}A }

into the pullback of

$$\begin{array}{ccc}& & *\\ & & \downarrow \\ EA& \to & BA\end{array}.$$\array{ && {*} \\ && \downarrow \\ \mathbf{E}A &\to& \mathbf{B}A } \,.

But since the former is this pullback, that morphism is the identity morphism. So the homotopy fiber in $[I_{+},\mathrm{SPSh}(C{)}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$ in question is indeed the morphism ${\mathrm{Id}}_A:A\to A$ in $\mathrm{SPSh}(C{)}^{\mathrm{loc}}$.

Proof (differential fibration sequence)

In $\mathrm{SPSh}(C)$ all representable objects are cofibrant. Moreover $\Pi (-)$ preserves cofibrations, as described at path ∞-groupoid.

Moreover, in $[I_{+},\mathrm{SPSh}(C)]$ the cofibrant objects are the cofibrations between cofibrant objects in $\mathrm{SPSh}(C)$ and therefore for all representable $U$ in $\mathrm{SPSh}(C)$ the object $\left[\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array}\right]$

is cofibrant in $[I_{+},\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$. This means that with the fibration sequence of the previous lemma also

$$[I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}A\\ \downarrow \\ A\end{array})\to [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}A\\ \downarrow \\ *\end{array})\to [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}*\\ \downarrow \\ BA\end{array})$$[I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ A \\ \downarrow \\ A } \right) \to [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ A \\ \downarrow \\ {*} } \right) \to [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right)

is a fibration sequence (in SSet now), because the objects are the right (infinity,1)-categorical hom-spaces and hence preserve homotopy limits.

But by the first lemma above this is the component over $U$ of the morphism $A_{\mathrm{flat}}\to A\to B{A}_{\mathrm{dR}}$ of simplicial presheaves.

Since (homotopy) limits of presheaf categories are computed objectwise, this finally means with the two lemmas above that

$$A_{\mathrm{flat}}\to A\to B{A}_{\mathrm{dR}}$$A_{flat} \to A \to \mathbf{B}A_{dR}

itself is a fibration sequence.

Proof (characterization of differential cocycle)

• That $A_{\mathrm{diff}}\to A$ is a weak equivalence in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ follows because it is objectwise (on $C$) an objectwise (on $I_{+}$) weak equivalence in $[I_{+},{\mathrm{SPSh}}_{\mathrm{proj}}^{\mathrm{loc}}]$ because $E_{\mathrm{pt}}A\to *$ is a weak equivalence in $\mathrm{SPSh}(C{)}_{\mathrm{loc}}$ and with $\Pi (U)$ cofibrant ${\mathrm{SPSh}}_C(\Pi (U),-)$ preserves weak equivalences between fibrant objects.

• That $A_{\mathrm{diff}}\to A_{\mathrm{dR}}$ is a fibration in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}$ follows because, as we saw above, $\begin{array}{ccc}A& \to & *\\ \downarrow & & \downarrow \\ E_{\mathrm{pt}}A& \to & BA\end{array}$ is an objectwise fibration in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ and hence a fibration in $[I_{+},\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$. Being a simplicially enriched model category the Hom-functor preserves fibrations in the second argument so that for each $U\in C$ the map

  $$[I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}A\\ \downarrow \\ E_{\mathrm{pt}}A\end{array})\to [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}*\\ \downarrow \\ BA\end{array})$$[I, SPSh(C)] \left( \array{U \\ \downarrow \\ \Pi(U)} \,,\; \array{A \\ \downarrow \\ \mathbf{E}_{pt}A} \right) \to [I, SPSh(C)] \left( \array{U \\ \downarrow \\ \Pi(U)} \,,\; \array{{*} \\ \downarrow \\ \mathbf{B}A} \right)

  is a fibration of simplicial sets.

  • Using the adjunction $\Pi (-)⊢A_{\mathrm{flat}}$ with the above description of $B{A}_{\mathrm{dR}}$ in $[I,\mathrm{SPSh}(C)]$ we have that for any $Y\in \mathrm{SPSh}(C)$ the morphism

    $${\mathrm{SPSh}}_C(Y,A_{\mathrm{diff}})\to {\mathrm{SPSh}}_C(Y,B{A}_{\mathrm{dR}})$$SPSh_C(Y,A_{diff}) \to SPSh_C(Y,\mathbf{B}A_{dR})

    is the same as the morphism

    $$[I,{\mathrm{SPSh}}_C](\begin{array}{c}Y\\ \downarrow \\ \Pi (Y)\end{array},\begin{array}{c}A\\ \downarrow \\ EA\end{array})\to [I,{\mathrm{SPSh}}_C](\begin{array}{c}Y\\ \downarrow \\ \Pi (Y)\end{array},\begin{array}{c}*\\ \downarrow \\ BA\end{array}).$$[I,SPSh_C](\array{Y \\ \downarrow \\ \Pi(Y)}, \array{A \\ \downarrow \\ \mathbf{E}A} ) \to [I,SPSh_C](\array{Y \\ \downarrow \\ \Pi(Y)}, \array{ {*} \\ \downarrow \\ \mathbf{B}A}) \,.

    But $Y\hookrightarrow \Pi (Y)$ is a cofibrant object $[I,\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{]}_{\mathrm{Reedy}}$ and $A_{\mathrm{diff}}\to B{A}_{\mathrm{dR}}$ is a fibration in there, so that this morphism is a fibration (of simplicial sets).

    This means that the homotopy pullback in question is already the ordinary pullback. This ordinary pullback is what the proposition above describes.

Proof (existence of connections)

Identify the object $A_{\mathrm{diff}}$ with the refinement of $A$ used in the above proof:

$$A_{\mathrm{diff}}:U\mapsto [I,\mathrm{SPSh}(C)]\left(\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array}\begin{array}{c}A\\ {\downarrow }^i\\ EA\end{array}\right).$$A_{diff} : U \mapsto [I,SPSh(C)]\left( \array{ U \\ \downarrow \\ \Pi(U) } \,\; \array{ A \\ \downarrow^i \\ \mathbf{E}A } \right) \,.

As discussed above, the morphism $A_{\mathrm{diff}}\to A$ (the obvious one using the first lemma above) is a weak equivalence.

This implies that $H(X,A_{\mathrm{diff}})\to H(X,A)$ is a weak equivalence of Kan complexes, hence that every object on the right has a lift up to equivalence.