Monoids in semiadditive ∞-categories

Proof. Let \(\simp ^{\leq 1}_{\mathrm {int}} \subseteq \simp \) be the subcategory with objects \([0]\) and \([1]\) and whose only non-identity morphisms are the two face maps \(d^0, d^1\colon [0] \to [1]\). We will show that restriction along the inclusions \[ j\colon \{[1]\} \hookrightarrow (\simp ^{\leq 1}_{\mathrm {int}})\catop \qquadtext {and} i\colon (\simp ^{\leq 1}_{\mathrm {int}})\catop \hookrightarrow \simp \catop \] define equivalences \[ \Mon (C) \iso \Fun ^{*}\big ((\simp ^{\leq 1}_{\mathrm {int}})\catop , C\big ) \iso C, \] where \(\Fun ^{*}\) denotes the full subcategory of those functors whose value at \([0]\) is terminal.

For the first equivalence, we use left Kan extension along \(i\). We apply the pointwise formula (Theorem 21.4.3) to compute the left Kan extension. For this, we need to understand the relative slice \(i_{/[n]}\) for each \([n] \in \simp \). Objects of this category are pairs \((x, \phi )\) where \(x \in \{[0], [1]\}\) and \(\phi \colon x \to [n]\) is a morphism in \(\simp \catop \), i.e. a morphism \([n] \to x\) in \(\simp \). There are \(n+2\) order-preserving maps \([n] \to [1]\), which we may label by the sequence of values they take: there are the two constant maps \(\underline {0}, \underline {1}\colon [n] \to [1]\), and for each \(1 \leq k \leq n\) a map \(\phi _k\) sending \(0, 1, \ldots , k-1\) to \(0\) and \(k, \ldots , n\) to \(1\). There is also a unique map \(\star \colon [n] \to [0]\). For morphisms in \(i_{/[n]}\), we observe that the only non-identity morphisms in \((\simp ^{\leq 1}_{\mathrm {int}})\catop \) are the maps \([1] \to [0]\) induced by the face maps \(d^0, d^1\colon [0] \to [1]\). By checking the commutative triangles, one sees that the only non-identity morphisms in \(i_{/[n]}\) are the maps \(\underline {0} \to \star \) and \(\underline {1} \to \star \). It follows that the category \(i_{/[n]}\) has precisely \(n+1\) connected components: there is a ‘span’ component \(\underline {0} \to \star \leftarrow \underline {1}\), and there are \(n\) isolated objects \(\phi _1, \ldots , \phi _n\).

Now let \(F\colon (\simp ^{\leq 1}_{\mathrm {int}})\catop \to C\) be a functor with \(F([0]) = *\), and write \(X := F([1])\). The left Kan extension \(i_!F\) evaluated at \([n]\) is the colimit over \(i_{/[n]}\) of the composite \(i_{/[n]} \to (\simp ^{\leq 1}_{\mathrm {int}})\catop \xrightarrow {F} C\). This colimit decomposes as a coproduct over the connected components. The span component contributes the colimit \(\colim (X \to * \leftarrow X) = *\), the terminal object, which is also initial in the pointed category \(C\). Each isolated object \(\phi _k\) contributes a copy of \(X\). Hence \((i_! F)([n]) \simeq X^{\sqcup n}\), the \(n\)-fold coproduct. Under these identifications, the Segal map \((i_!F)([n]) \to (i_!F)([1])^n\) is the canonical comparison from the \(n\)-fold coproduct of \(X\) to its \(n\)-fold product, hence is an isomorphism by semiadditivity. Thus \(i_!F\) is a monoid.

Conversely, let \(M\colon \simp \catop \to C\) be a monoid. There is a canonical comparison map from the left Kan extension of the restriction, \(i_! i^* M \to M\). To show this is an isomorphism, it suffices to check this at the underlying object \([1]\). There it is clear: as we just computed, the value at \([1]\) of both sides is simply \(M([1])\).

For the second equivalence, we use right Kan extension along \(j\). Since \(j\) is fully faithful, right Kan extension \(j_*\) is fully faithful whenever it exists. The relative slice \(j_{x/}\) over \(x \in \{[0], [1]\}\) is as follows: for \(x = [1]\), the identity \(\id _{[1]}\) is an initial object, so the limit evaluates to the original value. For \(x = [0]\), the relative slice is empty, so the limit is the terminal object. This shows that right Kan extension along \(j\) always exists and produces a functor in \(\Fun ^{*}((\simp ^{\leq 1}_{\mathrm {int}})\catop , C)\). Conversely, any functor with value \(*\) at \([0]\) is right Kan extended from \(\{[1]\}\).

For the claim about groups, note that if \(C\) is additive and \(M\) is a monoid in \(C\) with underlying object \(X\), then the shear map is given by \(\begin {psmallmatrix} \id _X & \id _X \\ 0 & \id _X \end {psmallmatrix}\colon X^2 \to X^2\). Its inverse is \(\begin {psmallmatrix} \id _X & -\id _X \\ 0 & \id _X \end {psmallmatrix}\), so every monoid is automatically a group. □