Is it possible that $\mathsf{L} = \mathsf{NP}$?
$L \subseteq NL \subseteq P \subseteq NP \subseteq PSPACE \subseteq EXP$

From space hierarchy and time hierarchy theorems we can prove that

$L \subsetneq PSPACE$

$NL \subsetneq NPSPACE$

Note that we know $PSPACE = NPSPACE$.

$P \subsetneq EXP$

$NP \subsetneq NEXP$

Note that we don't know if $EXP \subsetneq NEXP$.

Whether the other inclusions are strict, are still open problems. Even though there are no proofs, many, but not all, believe most of the inclusions to be strict.