Variable-base cotensor corners and path objects

Cotensors in the variable base. For a simplicial set $K$ and a fixed-variable-base $A$-module $M$, define $M_n^K$ as the set of maps of simplicial sets $K\times \mathrm{\Delta }[n]\to M$ with the pointwise additive structure; the $A$-action is through the constant-precomposition map $A\to A^K$: an $n$-simplex $a\in A_n$ represents a map $\mathrm{\Delta }[n]\to A$ and multiplies a map $K\times \mathrm{\Delta }[n]\to M$ pointwise in the $K$-coordinate as well. For an $A$-algebra $C$ this gives $C^K$ its $A$-algebra structure through $A\to A^K\to C^K$, with pointwise multiplication in the nonunital case. All constructions commute with the face and degeneracy maps because these act on the $\mathrm{\Delta }[n]$-coordinate, so the underlying simplicial set of $X^K$ is exactly the simplicial exponent. For a fixed augmentation $C\to B$ one uses the relative cotensor $C^K{\times }_{B^K}B$, where $B\to B^K$ is the constant map in the $K$-variable; the fixed-base restriction is essential, since the unrestricted exponent would change the prescribed augmentation.

Corner lifting. Let $p\text{\hspace{0.17em}⁣}:X\to Y$ have underlying horn lifting and let $i\text{\hspace{0.17em}⁣}:K\to L$ be a monomorphism. By the exponent adjunction $\mathrm{Map}(Z,X^K)\cong \mathrm{Map}(Z\times K,X)$, a lifting problem for $p^i\text{\hspace{0.17em}⁣}:X^L\to X^K{\times }_{Y^K}{Y}^L$ against a horn inclusion is equivalent to a lifting problem for $p$ against the pushout product $i\mathrm{\square }({\mathrm{\Lambda }}^r[t]\subset \mathrm{\Delta }[t])$, which is anodyne by [F1]; hence $p^i$ has horn lifting. If $i$ is itself a horn inclusion, the corresponding boundary lifting problems for $p^i$ translate into pushout products of that horn with a boundary inclusion, again anodyne, so $p^i$ has boundary lifting. If instead $p$ has boundary lifting, then all corresponding pushout products of the monomorphism $i$ with a boundary inclusion are monomorphisms, and [F3] supplies lifting of $p$ against these monomorphisms, so $p^i$ has boundary lifting. These arguments apply verbatim to the variable-base additive and algebraic cotensors, because their underlying set corners are the exponent corners and all algebraic structure is fixed through constant precomposition.

Path objects in the unsliced case. Take $Y=0$ and $i\text{\hspace{0.17em}⁣}:\mathrm{\partial }\mathrm{\Delta }[1]\subset \mathrm{\Delta }[1]$. Every additive object $X$ is Kan by [F2], so the endpoint map $X^{\mathrm{\Delta }[1]}\to X\times X$ is a Kan fibration by step 2.1 with $p\text{\hspace{0.17em}⁣}:X\to 0$. The constant-path map $c\text{\hspace{0.17em}⁣}:X\to X^{\mathrm{\Delta }[1]}$ satisfies $e_{0}c=\mathrm{i}\mathrm{d}$ for the evaluation $e_0$ at $0$. Precomposing with the map $\mathrm{\Delta }[1]\times \mathrm{\Delta }[1]\to \mathrm{\Delta }[1]$ given on ordered vertices by the minimum gives a simplicial homotopy $c{e}_0\simeq \mathrm{i}\mathrm{d}$ on $X^{\mathrm{\Delta }[1]}$: at one endpoint it is the constant map at $0$ and at the other it is the identity. Pointwise operations and constant-$A$-precomposition make this homotopy compatible with the module and algebra structures at every simplicial stage, so the prism and free-additive argument of [F2] shows that $c$ is a weak equivalence on normalized additive homology. Hence the endpoint and constant-path assertions hold for simplicial modules and for unital or nonunital algebras over an arbitrary simplicial $A$.

Relative and sliced statements. For augmented $B$-algebras, every augmentation $D\to B$ has the section given by the structure map and is termwise surjective, so it is Kan by [F2]; applying steps 2.1 and 3.1 with $Y=B$ (or with the relative cotensor of step 1.1) proves the corresponding path facts in the slice. For an arbitrary slice ${\mathrm{A}\mathrm{l}\mathrm{g}}_A\mathrm{/}B$, the fibrant objects are exactly those whose augmentation is Kan; not every object of the slice is fibrant, and no such assertion is made. The Axiom of Choice is used exactly for the simultaneous lifting choices in step 2.1.