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Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd) Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

A personal comment. Nowadays, these kind of questions is completely out of fashion, because homotopy categories are not anymore the foundations of homotopical algebra. Not even their enhanced cousins derivators had much fortune after Joyal-Lurie's treatment of $\infty$-categories. After quite some juvenile reluctance, I myself stand on the side of $\infty$-categories, but I think it would be very interesting to investigate Freyd's conservative functor $\text{HoTop} \to \text{Set}$, as it indeed provides an interesting variant of homotopy groups that completely classifies spaces up to homotopy, and we really know nothing about its behavior.

Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd). Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

A personal comment. Nowadays, these kind of questions is completely out of fashion, because homotopy categories are not anymore the foundations of homotopical algebra. Not even their enhanced cousins derivators had much fortune after Joyal-Lurie's treatment of $\infty$-categories. After quite some juvenile reluctance, I myself stand on the side of $\infty$-categories, but I think it would be very interesting to investigate Freyd's conservative functor $\text{HoTop} \to \text{Set}$, as it indeed provides an interesting variant of homotopy groups that completely classifies spaces up to homotopy, and we really know nothing about its behavior.

Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd) Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

A personal comment. Nowadays, these kind of questions are completely out of fashion, because homotopy categories are not anymore the foundations of homotopical algebra. Not even their enhanced cousins derivators had much fortune after Joyal-Lurie's treatment of $\infty$-categories. After quite some juvenile reluctance, I myself stand on the side of $\infty$-categories, but I think it would be very interesting to investigate Freyd's conservative functor $\text{HoTop} \to \text{Set}$, as it indeed provides an interesting variant of homotopy groups that completely classifies spaces up to homotopy, and we really know nothing about its behavior.

Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd) Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

A personal comment. Nowadays, these kind of questions is completely out of fashion, because homotopy categories are not anymore the foundations of homotopical algebra. Not even their enhanced cousins derivators had much fortune after Joyal-Lurie's treatment of $\infty$-categories. After quite some juvenile reluctance, I myself stand on the side of $\infty$-categories, but I think it would be very interesting to investigate Freyd's conservative functor $\text{HoTop} \to \text{Set}$, as it indeed provides an interesting variant of homotopy groups that completely classifies spaces up to homotopy, and we really know nothing about its behavior.

Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd) Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

Following the Peter's suggestion, I'll turn my comment in an answer.

In general by algebraic category one means variety in the sense of universal algebra, or category of models of a (possibly multisorted) Lawvere theory. Yet, if we allow for infinitary operations, one could say that an algebraic category is the category of algebras for a Set-monad. Personally, I do not stand on any normative side, and I prefer to have an intuitive notion of what an algebraic category is.

Yet, for all the notions I can come up with, any algebraic category $\mathcal{A}$ has a faithful and conservative functor into $\text{Set}$, $$ \mathcal{A} \to \text{Set}.$$

Because $\text{Set}$ is itself a (trivial) algebraic category, one gets to the following conclusion.

Conclusion. A category $\mathcal{H}$ has a faithful/conservative functor into an algebraic category if and only if it has a faithful/conservative functor into $\text{Set}$.

Having a faithful functor into the category of sets is called concreteness in category theory. The main paper on the topic are due to Peter Freyd.

• Freyd, On the concreteness of certain categories. 1969.
• Freyd, Homotopy is not concrete. 1970.
• Freyd, Concreteness. 1973.

Freyd's main results (cfr. the papers) are the following.

Thm 1 (Freyd). Any locally small category has a conservative functor into Set.

Thm 2 (Freyd). A locally small category has a faithful functor into Set if and only if it verifies the Isbell condition.

It's a bit hard to phrase the Isbell condition, thus I will just state the lazy Isbell criterion which gives a sufficient condition that sets the correct flavour.

Thm 3 (Lazy Freyd) Let C be a category with finite limits. C is concrete if and only if it is regular well-powered.

In his paper, Freyd shows that the Homotopy category of spaces cannot be concrete.

In 2016 I was doing my master, and I was fascinated precisely by the very motivation of this question, i.e. the possibility of using algebraic gadgets to distinguish topological structures. While I found Freyd's answer absolutely beautiful, I was wondering whether variations of Ho(Top) could still be concrete.

In 2017, this line of thoughts led to a collaboration with Fosco Loregian.

DL and Loregian. Homotopical Algebra is not concrete. 2018.

In the paper we should that a vast majority of homotopy categories cannot be concrete, due to the existence of Eilenberg-Maclane functors. I gave a talk, of which this answer is a crude summary.

A personal comment. Nowadays, these kind of questions are completely out of fashion, because homotopy categories are not anymore the foundations of homotopical algebra. Not even their enhanced cousins derivators had much fortune after Joyal-Lurie's treatment of $\infty$-categories. After quite some juvenile reluctance, I myself stand on the side of $\infty$-categories, but I think it would be very interesting to investigate Freyd's conservative functor $\text{HoTop} \to \text{Set}$, as it indeed provides an interesting variant of homotopy groups that completely classifies spaces up to homotopy, and we really know nothing about its behavior.