Perhaps it makes sense to mention this example: The category of measurable spaces is equivalent to the category of hyperstonean topological spaces and hyperstonean maps between them.

To construct a measurable space (X,M,N) from a hyperstonean topological space (Y,T), set X=Y, let M be the set of all unions of open and meager sets, and let N be the set of all meager sets in (Y,T). (Here M is the set of all measurable sets and N is the set of all null sets, i.e., sets of measure 0. For more information see this answer: Is there an introduction to probability theory from a structuralist/categorical perspective?)

So in this particular case meager sets are precisely sets of measure 0.

Perhaps it makes sense to mention this example: The category of measurable spaces is equivalent to the category of hyperstonean topological spaces and hyperstonean maps between them.

To construct a measurable space (X,M,N) from a hyperstonean topological space (Y,T), set X=Y, let M be the set of all unions of open and meager sets, and let N be the set of all meager sets in (Y,T). (Here M is the set of all measurable sets and N is the set of all null sets, i.e., sets of measure 0. For more information see this answer: Is there an introduction to probability theory from a structuralist/categorical perspective?)

So in this particular case meager sets are precisely sets of measure 0.