For simplicial complexes, there is a combinatorial approach to defining the fundamental group $\pi_1$, involving maximal trees and generators with relations (see for instance Armstrong's book pg 133-135). This definition does not use the idea of "loops" explicitly.
How does induced homomorphism work in this context?
For instance, given a simplicial map $f:K\to L$, can we combinatorially define the induced homomorphism $f_*: \pi_1(K)\to\pi_1(L)$, without explicitly using "loops"?
What I mean is given a set of generators and relations in $\pi_1(K)$, can we determine what $f_*$ does to them? Ideally, without explicitly using loops, i.e. avoid defining $f_*$ by $f_*(\gamma)=[f\circ\gamma]$ for a loop $\gamma$.
I have been searching for this approach in books, but did not quite find it. Is there such a thing?
Thanks a lot.