In 2010, a library contained a total of 11500 books in two categories - fiction and nonfiction. In 2015, the library contained a total of 12760 books in these two categories. During this period, there was 10% increase in the fiction category while there was 12% increase in the non-fiction category. How many fiction books were in the library in 2015?

Let the number of fiction and non-fiction books in 2010 be $F$ and $N$, respectively.

We are given:
$F+N=11500$  — (Equation 1)

In 2015, the fiction books increased by 10% and non-fiction books increased by 12%. The new total is 12760:
$1.1F+1.12N=12760$  — (Equation 2)

Multiply Equation 1 by 1.1:
$1.1F+1.1N=12650$  — (Equation 3)

Subtract Equation 3 from Equation 2:
$0.02N=110$
$N=\frac{110}{0.02}=5500$

Substitute $N=5500$ back into Equation 1:
$F=11500-5500=6000$

Thus, the number of fiction books in 2010 was 6000.

The number of fiction books in 2015 after a 10% increase is:
$F_{2015}=1.1\times 6000=6160$