Here is a sketch of topological description of a Tamarkin-Tsygan precalculus.

Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $\mathbb{R}^2-\{(0,0)\}$, respectively. The collection $(C_n,D_{1,n})_n$ form a topological (colored) operad.

Claim: the collection $(H_{-\bullet}(C_n,\mathbb{Q}),H_{-\bullet}(D_{1,n},\mathbb{Q}))_n$ is the operad of Tamarkin-tsygan precalculi.

It is well-known that $(H_{-\bullet}(C_n,\mathbb{Q}))_n$ is the Gerstenhaber operad. Then the operations $L$ and $i$ are the two classes (of respective degrees $1$ and $0$) in $H_{-\bullet}(D_{1,1},\mathbb{Q})=H_{-\bullet}(S^1,\mathbb{Q})$.

The two compatibility conditions can be understood as identities in $H^1(D_{1,2},\mathbb{Q})$.

If you want to get the operator $d$ (i.e. get a calulus rather than a precalculus) you'll have to replace $D_{1,n}$ by its semi-direct product with $S^1$.

See also Section 11 (especially $\S 11.3$) of this paper by Kontsevich and Soibelman for a similar point-of-view and its relation to a generalization of Deligne's conjecture.

EDIT: there is also an algebraic motivation for the compatibility conditions, which is explained in the original paper of Tamarkin and Tsygan. Namely, if $A$ is a Gerstenhaber algebra then $A[\epsilon]$, with $deg(\epsilon)=1$, is a Gerstenhaber algebra with modified product $a*b=a\cdot b+(-1)^{|a|}\epsilon[a,b]$. And if $(A,M)$ is a precalculus then $M$ becomes a Gerstenhaber module over $A[\epsilon]$ with $$ (a+\epsilon b)*m=(-1)^{|a|}i_am\quad\textrm{and}\quad [a+\epsilon b,m]=L_am+i_bm $$

Here is a sketch of topological description of a Tamarkin-Tsygan precalculus.

Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $\mathbb{R}^2-\{(0,0)\}$, respectively. The collection $(C_n,D_{1,n})_n$ form a topological (colored) operad.

Claim: the collection $(H_{-\bullet}(C_n,\mathbb{Q}),H_{-\bullet}(D_{1,n},\mathbb{Q}))_n$ is the operad of Tamarkin-Tsygan precalculi.

It is well-known that $(H_{-\bullet}(C_n,\mathbb{Q}))_n$ is the Gerstenhaber operad. Then the operations $L$ and $i$ are the two classes (of respective degrees $1$ and $0$) in $H_{\bullet}(D_{1,1},\mathbb{Q})=H_{\bullet}(S^1,\mathbb{Q})$.

The two compatibility conditions can be understood as identities in $H_1(D_{1,2},\mathbb{Q})$.

If you want to get the operator $d$ (i.e. get a calulus rather than a precalculus) you'll have to replace $D_{1,n}$ by its semi-direct product with $S^1$.

See also Section 11 (especially $\S 11.3$) of this paper by Kontsevich and Soibelman for a similar point-of-view and its relation to a generalization of Deligne's conjecture.

EDIT: there is also an algebraic motivation for the compatibility conditions, which is explained in the original paper of Tamarkin and Tsygan. Namely, if $A$ is a Gerstenhaber algebra then $A[\epsilon]$, with $deg(\epsilon)=1$, is a Gerstenhaber algebra with modified product $a*b=a\cdot b+(-1)^{|a|}\epsilon[a,b]$. And if $(A,M)$ is a precalculus then $M$ becomes a Gerstenhaber module over $A[\epsilon]$ with $$ (a+\epsilon b)*m=(-1)^{|a|}i_am\quad\textrm{and}\quad [a+\epsilon b,m]=L_am+i_bm $$

Here is a sketch of topological description of a Tamarkin-Tsygan precalculus.

Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $R^2-\{(0,0)}$, respectively. The collection $(C_n,D_{1,n})_n$ form a topological (colored) operad.

Claim: the collection $(H_{-\bullet}(C_n,\mathbb{Q}),H_{-\bullet}(D_{1,n},\mathbb{Q}))_n$ is the operad of Tamarkin-tsygan precalculi.

It is well-known that $(H_{-\bullet}(C_n,\mathbb{Q}))_n$ is the Gerstenhaber operad. Then the operations $L$ and $i$ are the two classes (of respective degrees $1$ and $0$) in $H_{-\bullet}(D_{1,1},\mathbb{Q})=H_{-\bullet}(S^1,\mathbb{Q})$.

The two compatibility conditions can be understood as identities in $H^1(D_{1,2},\mathbb{Q})$.

If you want to get the operator $d$ (i.e. get a calulus rather than a precalculus) you'll have to replace $D_{1,n}$ by its semi-direct product with $S^1$.

See also Section 11 (especially $\S 11.3$) of this paper by Kontsevich and Soibelman for a similar point-of-view and its relation to a generalization of Deligne's conjecture.

Here is a sketch of topological description of a Tamarkin-Tsygan precalculus.

Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $\mathbb{R}^2-\{(0,0)\}$, respectively. The collection $(C_n,D_{1,n})_n$ form a topological (colored) operad.

Claim: the collection $(H_{-\bullet}(C_n,\mathbb{Q}),H_{-\bullet}(D_{1,n},\mathbb{Q}))_n$ is the operad of Tamarkin-tsygan precalculi.

It is well-known that $(H_{-\bullet}(C_n,\mathbb{Q}))_n$ is the Gerstenhaber operad. Then the operations $L$ and $i$ are the two classes (of respective degrees $1$ and $0$) in $H_{-\bullet}(D_{1,1},\mathbb{Q})=H_{-\bullet}(S^1,\mathbb{Q})$.

The two compatibility conditions can be understood as identities in $H^1(D_{1,2},\mathbb{Q})$.

If you want to get the operator $d$ (i.e. get a calulus rather than a precalculus) you'll have to replace $D_{1,n}$ by its semi-direct product with $S^1$.

See also Section 11 (especially $\S 11.3$) of this paper by Kontsevich and Soibelman for a similar point-of-view and its relation to a generalization of Deligne's conjecture.

EDIT: there is also an algebraic motivation for the compatibility conditions, which is explained in the original paper of Tamarkin and Tsygan. Namely, if $A$ is a Gerstenhaber algebra then $A[\epsilon]$, with $deg(\epsilon)=1$, is a Gerstenhaber algebra with modified product $a*b=a\cdot b+(-1)^{|a|}\epsilon[a,b]$. And if $(A,M)$ is a precalculus then $M$ becomes a Gerstenhaber module over $A[\epsilon]$ with $$ (a+\epsilon b)*m=(-1)^{|a|}i_am\quad\textrm{and}\quad [a+\epsilon b,m]=L_am+i_bm $$