For your time and money, just get a small PID controller. No need to reinvent the temperature controller. There are cheap knockoffs of the Omrons called "REX-C100". Kits sell for under $20 that include thermocouple and SSR.
It's not easy to program them but they can do ramps and multiple temperature processes etc.
Otherwise, I would get a NEST or an ESP8266 and make an IoT enabled one.

 

I'm gathering this is not for brewing beer but rather some top secret device for which we are not allowed to know the details.

If you want to keep something in a refrigerator just 3 or 4 degrees above the ambient set temperature in a refrigerator; first I must ask, why not set the temperature just 3 or 4 degrees higher? But if for some reason this is a part of your master plan, use a Peltier tile. One side heats while the other side cools. So if you're heating a surface you're not introducing much heat into the fridge because the other side is cooling. Over all there's going to be an increase in temperature, but I'm supposing not much.

Now, since we're not allowed to know details of what you're actually doing, at least tell us what kind of volume you're after. If you want to warm 100 cc air space then a Peltier tile will probably provide the temperature difference you've stated (particularly 3 to 4 degrees above ambient inside a fridge). If you want to warm 10,000 cc air space you'll need a bigger heat source.

I had an ice maker get stuck in the "Eject" mode. To eject it had to warm the ice tray so the ice could be liberated from the molds. Having gotten stuck in Eject it turned the freezer into an oven. Everything was thawed to slightly above room temperature. Heaters inside refrigerators isn't the brightest approach. The refrigerator is simply a heat pump. It will pump heat from inside the box to outside. Introducing heat, even low current draw, will make the refrigerator run a lot more. Possibly constantly. And your electric bill will shoot to the moon.

OK, Bond; James Bond, whatever it is you're building, it won't work with a 9V battery in a fridge. It just won't.

 

Or just buy a small used dorm refrigerator and set it to the temp you want. When living in hot Alabama I had a 50s GE full-sized refrigerator I bought cheap just for cold brewing. You can set their temps well up into the 40s, maybe even 50s F. Their insulation also keeps cold out as well as in not knowing exactly what ambient temps you are dealing with and for how long.

 

I like the idea of a separate unit but wouldn’t it eventually need heating because enough days at subzero temperatures such as we get in Chicago everything in that fridge would be frozen solid. When I hear shed, I picture an unheated space. Is this man cave heated? What are the ambient temps in the winter.

 

I'm gathering this is not for brewing beer but rather some top secret device for which we are not allowed to know the details.

It appears the OP is looking to grow a particular yeast strain.
Beer is generally brewed with a strain that is compatible with the type of ingredients and beer variety that gives it a certain characteristic taste,
Max.

 

@MaxHeadRoom

ok.... I have an idea of a kitchen gadget I’m thinking of building and want to protect my idea/ design I’m looking for something that will control a small heat pad at a very low temperature as this item will be in a refrigerator for long period of time. The idea is to raise the product temperature 3 or 4 degrees warmer than the fridge temp as this item will be the size of a mobile phone. Any suggestions on how to build such a small device would be greatly helpful to a non electric person.

Not sounding like a yeast culture to me. I may be wrong. Sounds like a secret project we're not supposed to know the true nature of the device.

 

Re post #12 !
Max.

 

@MaxHeadRoom, yeah, I read that already. But he started speaking about secrets and I concluded maybe he's not developing yeast. May be a biological weapon, might be yeast, might be some secret transmitter. Truth is - I have no idea for sure what he's up to. There have been plenty of inventors come here and tell us they were making one thing just to disguise his/her true intent. This could be one more of those who believe their secret is critical to some greater plan, thus, smoke screen us and bait us in a different direction with the hope we can help him/her engineer something entirely different from what we suspect.

Just why something in the fridge needs to be 3 to 4 degrees above ambient fridge temp - doesn't sound like a simple experiment to me. Sounds like more is going on.

Hey! I could be entirely wrong. Since that's a distinct possibility I will refrain from posting any further. No, I'm not insulted. Just don't want to go accusing someone over and over of having some other purpose than what has been disclosed.

 

The proof is in the taste test!
Max.