author also acknowledges the concurrent work of Eva Höning on topological Hochschild homology of $K(\mathbb{F}_q)$ using a different approach. Personal communication with Eva Höning resolved a differential that the author originally thought to be trivial.

THH-May spectral sequence {#section 2}
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The goal of this section will be to construct the THH-May spectral sequence in a minimal way. This section is a review of results from the author’s paper with A. Salch and more details about the constructions can be found in that paper [@thhmay]. To construct the spectral sequence, we first need to define what we mean by a filtered object in $\mathfrak{S}$. We first give a description that has the advantage of concreteness.

\[def:DFO1\] A decreasingly filtered commutative monoid in $\mathfrak{S}$ is a sequence of objects in $\mathfrak{S}$ $$\xymatrix{ ... \ar[r]^{f_3} & I_2 \ar[r]^{f_2} & I_1 \ar[r]^{f_1} & I_0 } ,$$ along with structure maps $$\rho_{i,j}: I_i \wedge I_j \lra I_{i+j}$$ satisfying commutativity, associativity, unitality, and compatibility [@thhmay]. We say that $I$ is a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$ if each $I_i$ is cofibrant and the maps $f_i$ are all cofibrations.

We avoid writing out all the commutativity, associativity, unitality, and compatibility diagrams by providing an 