I am making an attempt to get extra intiution about larger Okay-theory, Hochschild homology and the hint map between by excited about these objects from an off-the-cuff $infty$-categorical perspective, as a substitute of utilizing very exact and concrete definitions:

Let $A$ be a small steady $infty$-category. If my understanding is appropriate, the topological Hochschild homology of $A$ could be described because the co-end:

$$ THH(A) := int^A textual content{Hom}(a,a) $$

Informally, it implies that a map of spectrum $THH(A) rightarrow X$ defines an $X$-valued hint for arrows in $A$, i.e. maps of spectrum $textual content{Tr}_a : textual content{Hom}(a,a) to X $ which satisfies some coherence relation, the primary one akin to “$textual content{Tr}(ab)=textual content{Tr}(ba)$“, and better one being coherence circumstances for the hint of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the goal of the common hint map for $A$.

Then again, the (connective) $Okay$-theory of $A$ could be seen as (for e.g. from Waldhausen development) the goal of the common “Euler Attribute” map for $A$, within the sense {that a} map of spectrum $Okay(A) to X$ corresponds to an Euler attribute $chi(a) in X$ for every $a in A$,such that if $a to b to c$ is a fiber sequence then we’ve an equivalence $chi(b) simeq chi(a)+chi(c)$ additionally topic to some larger coherence circumstances.

It therefore sounds pure, that the Denis hint map $Okay(A) to THH(A)$ ought to, on this perspective, corresponds to the map informally outlined as:

$$ start{array}{ccc} Okay(A) &to & THH(A)

chi(a) & mapsto & textual content{Tr}(textual content{Id}_a)

finish{array}$$

To indicate that this defines maps, one wants to point out that if $a to b to c$ is a fiber sequence in $A$ then, for any hint perform as above, we will assemble an equivalence $textual content{Tr}(textual content{Id}_b) simeq textual content{Tr}(textual content{Id}_a) + textual content{Tr}(textual content{Id}_c)$

After all for a full proof that this map is properly outlined as a map of spectra (or quite $E_infty$-algebras by identyfing connective spectra with grouplike $E_infty$-algebra) we’d additionally must cope with the “larger coherence circumstances”, and present that this induce a map of groulike $E_infty$ algebra, by coping with extra larger coherence situation and so one. However I am specializing in the primary situation as that is the primary one I don’t perceive.

**Query:** *Can we give a proper/elementary proof {that a} hint map as above routinely comes with equivalences $textual content{Tr}(textual content{Id}_b) simeq textual content{Tr}(textual content{Id}_a) + textual content{Tr}(textual content{Id}_c)$ for every fiber sequence $a to b to c$.*

I believe I understand how to show this utilizing deeper theorem, for instance the additivity of $THH$, however I am actually keen on a direct elementary proof of this.

To present an concept of the kind of argument I settle for as a solution, it’s straightforward to present a proper proof that if $b = a oplus c$ then $textual content{Tr}(textual content{Id}_b) = textual content{Tr}(textual content{Id}_a) + textual content{Tr}(textual content{Id}_c)$. Certainly, as $a$ and $c$ are retract of $b=a oplus c$, with idempotent $P_a,P_c:b to b$. The property of the hint exhibits that:

$$ textual content{Tr}(P_a) = textual content{Tr}(i_a p_a) = textual content{Tr}(p_a i_a) = textual content{Tr}(textual content{Id}_a) $$

and $textual content{Id}_b = P_a + P_c$ so:

$$textual content{Tr}(Id_b)= textual content{Tr}(P_a)+ textual content{Tr}(P_c) = textual content{Tr}(textual content{Id}_a) + textual content{Tr}(textual content{Id}_c)$$