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Consider a functor between small categories $i:\mathcal{A}\to \mathcal{B}$ which is faithful, not full, and bijective on objects. Consider a functor $F:\mathcal{A}\to \mathcal{M}$ where $\mathcal{M}$ is a combinatorial simplicial model category such that there is no functor $\overline{F}:\mathcal{B} \to \mathcal{M}$ such that $\overline{F}.i = F$. Let $\mathrm{Ho}(\mathcal{M})$ be the homotopy category of $\mathcal{M}$. The composite functor $F':\mathcal{A}\to \mathcal{M} \to \mathrm{Ho}(\mathcal{M})$ can be extended to a functor $\overline{F'}:\mathcal{B} \to \mathrm{Ho}(\mathcal{M})$, i.e. such that $\overline{F'}.i = F'$.

Are there similar situations in the mathematical literature which we could help me to understand what to do ?

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  • $\begingroup$ What's your goal here? $\endgroup$
    – Denis Nardin
    Commented Jun 4, 2018 at 14:33
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    $\begingroup$ My goal would be to find a Quillen equivalence $\mathcal{M}\to \mathcal{N}$ such that the composite functor $\mathcal{A}\to\mathcal{M}\to \mathcal{N}$ is extendable to a functor from $\mathcal{B}\to \mathcal{N}$. There is no canonical choice for $F'$. Maybe there is a way to encode all the possible choices. I would be interested in seeing how similar problems are treated in the literature. $\endgroup$
    – Philippe Gaucher
    Commented Jun 4, 2018 at 14:47

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