In a Cox proportional hazards model, specifying strata allows for different baseline hazards between the strata, in your case for men versus women. If you specify the following model in R:
coxph(Surv(time,status) ~ treatment + strata(sex))
then you get a single hazard ratio for treatment, which is taken to be the same for both sexes. Differences between sexes in outcome with this model depend on the different baseline hazards, not on different hazard ratios.
If you are interested in different effects of treatment depending on sex then you need to include an interaction term (indicated by "
:" below, as in R). For example, you could specify:
coxph(Surv(time,status) ~ treatment + strata(sex) + treatment:strata(sex))
to get an interaction term representing how treatment-related hazard differs between sexes. This formulation allows both for different hazards between sexes and for different baseline hazards for the sexes.
Using strata for sex does not model a relationship between sex and outcome except via the different baseline hazards. If you specify:
coxph(Surv(time,status) ~ treatment + sex + treatment:sex)
you would get hazard ratios of treatment and for sex, plus an interaction term, but with the same baseline hazard assumed for both sexes.