I know 2 inch is a standard vent for a water closet with a 3 inch drain. I have seen an illustration using a 1 1/2 inch drain at the water closet. I need to run a 1 1/2 inch vent from a low heel 3 inch elbow to fit in the space I have. Can I run this for the first 4 feet horizontally until I can go 2 inch vertically?

I will probably get hammered here by the other answer experts. 1 1/2 is not to code in my area An it must be 2' vent but I see many of our old cast lines use 1 2/2 vents off the toilet soil line and work fine. The closer to the closet flange you can get the better. Code here is within 5' of the flange. Since you are up sizing it to a two that is where the real issues on all venting for air volume to get to the soil line. If you think about it the low flush toilets are a lot less to vent because of a short bust of liquid and waste. Not like the old gallons of tanks.
As long as you keep the 1 1/2 horizontal slanted as best as you can get and its such a short distance. Is this getting inspected? If not I would go for it Ive seen tons of 1 1/2 vents that work fine on the very old toilet soil lines

UPC say's 2". Keep in mind if the line is visible and not behind a wall when you go to sell your home and it is inspected the home inspector will probably make you change it.

Yes, Code says 2" and I also believe it should be 2". But 1 1/2" is certainly better than no vent at all. Question: How is it that you have room for 1 1/2" vent and don't have room for 2" vent ? Just curious...

Our code will allow a 1 1/2 vent but our inspectors push for 2" vents with one 3" vent. Usually off the kitchem sink. Like Milo sez." A 1 1/2" is certainly better than no vent at all." Good luck, Tom