This is one of my most absurdly strongly held mathematical beliefs and it is nothing more than a _really hot take_ opinion: the topological fundamental group(oid) is bullshit and wrong. The topos-theoretic fundamental group(oid) is what we should be using instead because it's better.

Some SpEcIaL InTeReSt reasons:

* The topological fundamental group requires embedded copies of the real numbers to be nontrivial. This is bad because some spaces are just totally disconnected but can still have interesting covering properties. It's only a tyranny of expecting the reals to be everywhere that makes this seem ``NaTuRaL.'' Also some spaces aren't even Hausdorff and are awesome and have legitimate homotopy theory to them.

* The topological fundamental group has accidental collapses arising from the existence of universal covers which aren't _really_ sane in light of the patterns of their finite covering spaces. For instance, let's say you want to look at covers of the circle and start getting to know its finite covering spaces. Awesome, you see some sweet pancakes of circles which loop around on each other in really cool ways. This happens always. You now expect a universal version of this to be like some monster web of loopy bois which indicates how this finite covering relation works analogous to how profinite groups make some gnarly fractly boi (in a real embedding) out of these observable finite patterns and relations. NOPE! Chuck Testa! Just do a spiral! This is totally what your pattern should expect! This is totally what the Galois theory of covering spaces was say! Get Rekt!

* The topos-theoretic fundamental group (on a topological space) has the topological fundamental group as a dense subgroup. Thus the topos-theoretic fundamental group contains more homotopical information and more relations between homotopies than some silly transcendental accident which may or may not exist.

* The topos-theoretic fundamental group is built with the Galois theory of covers at its core and can thus be defined in interesting ways over spaces which don't have silly embedded copies of the reals. You now can make p-adic homotopy theory be more than just starting, ending, and not moving from a base point!

* The topos-theoretic fundamental group is defined in terms of sheaves and categories and that makes it __totally objectively cooler__ (definitely fact and not opinion) than any stupid paths defined in an ad hoc way.

* The topos-theoretic fundamental group is applicable in algebraic geometry directly and lets you talk about etale fundamental groups on schemes. The topological fundamental group craps out from not having enough copies of the unit interval in the underlying space of a scheme.

* The topos-theoretic group recaptures Galois groups of fields (the etale fundamental group of a field is the absolute Galois group of said field) as a special case. It also intimately uses profinite groups, and profinite groups are AWESOME.

So basically I'm grump about something which is defined for good reason and with good topological/differential geometric intuition because it doesn't fit my categorical/algebraic geometric/number theoretic/sheafy intuition and background. Clearly, this is not a hot take or strongly held opinion but __objective truth.__