Two Topological spaces $X$ and $Y$ are homotopy equivalent if there exists Continuous functions $f: X \to Y$ and $g: Y \to X$ such that

$f \circ g \sim id_Y$ and $g \circ f \sim id_X$

where $\sim$ refers to the equivalence relation homotopic. $f$ here is then called a homotopy equivalence. We can also refer to the spaces as being homotopic

So I think that if we take the Category of Topological spaces, and mod out the set of arrows with the homotopic equivalence relation between functions, then homotopy equivalence between top spaces is just the categorical notion of Isomorphism

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A Homeomorphism is trivially a homotopy equivalence.

Spaces which are homotopy equivalent share the same Fundamental group. This can be seen from the fact that the existence of a Continuous function $f: X \to Y$ implies that for two paths homotopic in $X$, their images are homotopic in $Y$. Viceversa, if we also have cont. fun $g: Y \to X$. However, we can't guarantee that two non-homotopic paths in $X$ map to two non-homotopic paths in $Y$ unless we have the condition for homotopy equivalence. This condition guarantees that given path $p$ in $X$, $g(f(p))$ is homotopic to $p$, and this can be used to conclude (after thinking about basic implications, and the fact that homotopic is an equivalence relation) that {that two non-homotopic paths in $X$ map to two non-homotopic paths in $Y$}.

Some Criteria for Homotopy Equivalence

Two spaces related by a Retraction deformation are homotopy equivalent

Collapsing Subspaces

The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space. Here is a positive result in this direction: If (X, A) is a CW pair consisting of a CW complex X and a contractible subcomplex A , then the quotient map X → X/A is a homotopy equivalence.

generally, suppose X is any graph with finitely many vertices and edges. If the two endpoints of any edge of X are distinct, we can collapse this edge to a point, producing a homotopy equivalent graph with one fewer edge. This simplification can be repeated until all edges of X are loops, and then each component of X is either an isolated vertex or a Wedge sum of circles.

Attaching spaces

If (X 1 , A) is a CW pair and the two attaching maps f , g : A → X 0 are homotopic, then X 0 ⊔ f X 1 ≃ X 0 ⊔ g X 1 .