I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end of spectra (where the Hom is the spectra enriched Hom of $A$):

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}(ab)=\text{Tr}(ba)$", and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Dennis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end of spectra (where the Hom is the spectra enriched Hom of $A$):

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}_b(fg)=\text{Tr}_a(gf)$" for $f:a \to b$ and $g: b \to a$, and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Dennis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end of spectra (where the Hom is the spectra enriched Hom of $A$):

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}(ab)=\text{Tr}(ba)$", and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Denis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end of spectra (where the Hom is the spectra enriched Hom of $A$):

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}(ab)=\text{Tr}(ba)$", and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Dennis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end:

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}(ab)=\text{Tr}(ba)$", and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Denis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of using very precise and concrete definitions:

Let $A$ be a small stable $\infty$-category. If my understanding is correct, the topological Hochschild homology of $A$ can be described as the co-end of spectra (where the Hom is the spectra enriched Hom of $A$):

$$ THH(A) := \int^A \text{Hom}(a,a) $$

Informally, it means that a map of spectrum $THH(A) \rightarrow X$ defines an $X$-valued trace for arrows in $A$, i.e. maps of spectrum $\text{Tr}_a : \text{Hom}(a,a) \to X $ which satisfies some coherence relation, the first one corresponding to "$\text{Tr}(ab)=\text{Tr}(ba)$", and higher one being coherence conditions for the trace of cyclic permutation of composite of an $n$-cycle of arrows in $A$. So $THH(A)$ is the target of the universal trace map for $A$.

On the other hand, the (connective) $K$-theory of $A$ can be seen as (for e.g. from Waldhausen construction) the target of the universal "Euler Characteristic" map for $A$, in the sense that a map of spectrum $K(A) \to X$ corresponds to an Euler characteristic $\chi(a) \in X$ for each $a \in A$,such that if $a \to b \to c$ is a fiber sequence then we have an equivalence $\chi(b) \simeq \chi(a)+\chi(c)$ also subject to some higher coherence conditions.

It hence sounds natural, that the Denis trace map $K(A) \to THH(A)$ should, in this perspective, corresponds to the map informally defined as:

$$ \begin{array}{ccc} K(A) &\to & THH(A) \\ \chi(a) & \mapsto & \text{Tr}(\text{Id}_a) \end{array}$$

To show that this defines maps, one needs to show that if $a \to b \to c$ is a fiber sequence in $A$ then, for any trace function as above, we can construct an equivalence $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$

Of course for a full proof that this map is well defined as a map of spectra (or rather $E_\infty$-algebras by identyfing connective spectra with grouplike $E_\infty$-algebra) we would also need to deal with the "higher coherence conditions", and show that this induce a map of groulike $E_\infty$ algebra, by dealing with more higher coherence condition and so one. But I'm focusing on the first condition as this is the first one I do not understand.

Question: Can we give a formal/elementary proof that a trace map as above automatically comes with equivalences $\text{Tr}(\text{Id}_b) \simeq \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$ for each fiber sequence $a \to b \to c$.

I think I know how to prove this using deeper theorem, for example the additivity of $THH$, but I'm really interested in a direct elementary proof of this.

To give an idea of the type of argument I accept as an answer, it is easy to give a formal proof that if $b = a \oplus c$ then $\text{Tr}(\text{Id}_b) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$. Indeed, as $a$ and $c$ are retract of $b=a \oplus c$, with idempotent $P_a,P_c:b \to b$. The property of the trace shows that:

$$ \text{Tr}(P_a) = \text{Tr}(i_a p_a) = \text{Tr}(p_a i_a) = \text{Tr}(\text{Id}_a) $$

and $\text{Id}_b = P_a + P_c$ so: $$\text{Tr}(Id_b)= \text{Tr}(P_a)+ \text{Tr}(P_c) = \text{Tr}(\text{Id}_a) + \text{Tr}(\text{Id}_c)$$