#bijective

[arcsin27]

if youre bisexual youre also insexual and sursexual

[nontrivialproof]

I have three siblings and so a time-honored group activity is sorting ourselves into any other group of four. And since Thanksgiving I have been intermittently thinking about when we were doing this for The Beatles and my older brother said with zero hesitation, "Well, Raina is definitely Paul." What did he mean by that. It's not a compliment.

[nicholasandriani]

Mastering Isometric & Isomorphic Game Design: A Quickstart Guide to Unlocking 3D Potential in 2D Worlds

We say that: “oh I love that Baldur Gate, let’s crack out an isometric.” Or, “oh, digging this Final Fantasy Tactics stuff, let’s do that sideways angle thing.” But what does that mean? Let’s break it down mathematical. T:(X,∥⋅∥X)→(Y,∥⋅∥Y) is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is ∥T(x)∥Y=∥x∥X∀x∈X; T:(X,∥⋅∥X)→(Y,∥⋅∥Y) is a topological…

[unnonexistence]

well, it finally happened, i finally found the math version of "the average person probably only knows the formulas for olivine and one or two feldspars" in my writing

[cleverusername01]

pride month is getting to be a little much this year. I mean love is love and everything but what the FUCK is “bijective”

[despondent-beauty]

Tfw you believe you've somehow created an ordered list of every single real number even though such a thing contradicts its own existence and you back this up by raging about how everyone who disagrees is just spewing what they were taught at school because you can't deal with the fact that you've spent 15 years of your life failing to prove something in a field you don't understand.

God, I love bringing you topical content.

[trilobyteme]

countability

[fazedlight]

Awakening (Kara character study, small part supercorp softness)

“But why can’t I be matched to Tali, mom? She’s my best friend!”

Alura turned to Kara, her jaw tensed with frustration with the stubborn child. “Kara, you know better than this. Stable matching can only be achieved if there is a true bijection between disjoint sets-”

“We don’t even have a true bijection because the population is constantly changing, we don’t sort according to all possible preferences, we don’t even have-” “It is not in our nature, Kara,” Alura said, with a dangerous tone in her voice. “We are not Daxamites.”

“But-” 

“The answer is no.”

---

Kara is thirteen Earth years the first time she’s called “dyke.” She doesn’t know what it means. She had only been to school for a couple of weeks. Before that, the only substantial English she had spoken was the couple months with Eliza, Jeremiah, Alex, and Kal. 

Clark, not Kal. Saying “Kal” would put her baby cousin- her older cousin- her cousin in danger.

Alex’s face flushes, and her eyes almost burn with anger, as she shoves Jake Howell against a locker. Kara could do it easily herself, but showing her newfound strength to humans would put her in danger too, somehow. Kara doesn’t think asshole is a nice word, given how Alex growled it. But she suspects dyke isn’t a nice word either.

That night after dinner, Eliza sighs, and hugs Kara gently - and Kara resolves to never get called dyke again.

---

“She’s gorgeous, she’s smart, she smells nice. Hell, I want to date her.” Kara flushed with discomfort, as the words came unbidden from her lips. But Alex didn’t remark on the odd statement, and Kara shoved the thought away.

Just weeks later, awkwardness would turn to tension as Lucy growled. “You and Hank, why do you all lie?”

“When you are an alien,” Kara choked, “You’re willing to sacrifice anything, everything, betray your fundamental instincts - just to fit in.” Something tugged in Kara’s soul at that moment. That she had always tried to fit in, long before she became an alien. But there simply hadn’t been time to linger.

It was only weeks later, when Lucy was saying her goodbyes before leaving National City to rejoin the military, that Kara felt the uncomfortable spike again. “I do know what it’s like to hide,” Lucy confessed.

Kara tilted her head, questions like why? and what do you mean? floating through her mind. But she thought it would be kinder not to ask. “I hope someday, you can be all that you are.”

Lucy gave a small smile. “Me too.”

---

She hadn’t expected meeting Lena to feel like lightning in her veins. The younger Luthor was quick-witted, and beautiful, and playful. Kara felt herself flush with the gentle teasing during their first coffee, and found herself marveling at never quite having a friendship like this before. 

---

“So… so she’s gay?” Kara asked, the word heavy in her throat. “And are you saying, you’re gay too?”

Alex sighed and paced in front of Kara, her frustration just as apparent as her confusion. How can you not know if you’re gay?, Kara wondered, at the same time feeling strangely allergic to the conversation. Wouldn’t it be obvious? “What’s changed?” Kara asked.

---

Yeah, he was… immature. Irresponsible. But they connected - orphans of a lost planet, who spoke the same tongue, who had the same bewilderment in their first moments on an alien planet with newfound powers. And if being in his bed brought her pleasure, it was only proof of their connection, that a good relationship could come of it.

Sometimes there were those flashes - Mon-El had been confused by Alex’s coming out, not understanding the concept. The more the merrier would ring in Kara’s head, and she’d chase away the image of Lena’s face.

---

“I couldn’t have done it, Kara.”

Kara’s chest heaved as she gazed down at Lena, hearing Kal’s words flash through her mind. Lena clung to Kara’s arm as Kara hovered above the reservoir, and some corner of Kara’s mind knew that she should go land, that the danger was over. That Lena was safe. That the city was safe.

But she could only stare down at Lena, whose heart hammered in her chest, whose panting breaths from her climb had not yet slowed. I almost lost her, Kara thought, forlorn. I couldn’t lose her…

It was that moment that her world came crashing down, that realization made her feel like she was drowning. That romantic love wasn’t merely a combination of friendship and lust. That shared experience didn’t mean a shared connection. There was something that ran deeper.

She was in love with Lena, and she could no longer deny it.

---

It was a drunken movie night, after Lena’s breakup with James, when Kara heard I love you fall from Lena's lips.

“It was always you,” Lena confessed, her words slightly slurred from the alcohol as Kara finished pulling the covers over her. “I just wanted to be close to you.”

Kara stood back, feeling her heart pound as she watched Lena slip into slumber. I wish I had told you, Kara thought, her mind flashing to a moment long ago in a forest. I wish I had told you, before…

Kara spent a fitful night trying to sleep on her couch, and Lena’s eyes flashed with shame the next morning as she woke. But they left for Noonan’s, leaving the conversation behind.

---

It felt impossibly brief, that window of time after Kara had revealed her secret, where everything felt almost right with the world. Maybe someday, she and Lena could finish that conversation.

But she found herself in a kryptonite shell.

The universe ended soon after, and even magic couldn’t fix how they had broken. Until the day Kara finally found her hands in Lena’s, vowing together to take down her brother, and Kara felt again that hopeful wonder of what a future with Lena could hold.

And then she found herself in the Phantom Zone again, the words ringing in her head, I wish I had told you.

---

Sleep had eluded Kara in the weeks back from the Phantom Zone. So she was already wide awake at 2am, when she heard Lena’s heart begin to hammer. 

Kara tensed, rushing to her window and ears tuning in as she prepared to fight off an assassination attempt or catch Lena as she fell.

But as she shot into the sky, she nearly tumbled when she realized that Lena wasn’t in distress. The shaky breaths and small laughs caused Kara’s chest to tighten in anguish. She’s fine, Kara thought, feeling tears prick the edges of her vision. She’s fine. 

---

“Are you okay?” Lena said, when she finally found Kara in the Tower, sitting on a step. “Alex said she couldn’t find you - you were in the Fortress?”

Kara glanced up from the steps. “I just, um. I was reading in the Fortress, I fell asleep there.” It had the benefit of being true. The Fortress was far enough to drown the sound of Lena’s heart out.

Lena shuffled next to Kara, taking a seat. “I don’t remember seeing any beds there.”

“I float in my sleep,” Kara shrugged, staring at her hands as she let silence fall.

Lena shifted, uncomfortable with the quiet. “Are you okay?”

Yeah, Kara almost said, but something stopped her this time. Perhaps it was the poor sleep. Perhaps it was the litany of I wish I had told you that would replay in her mind.

How many more times am I going to do this?, Kara thought. How many more times am I going to carry that regret? “I love you,” Kara said finally, sensing Lena tense up next to her. “I know… I know that door is closed. But I love you. I should’ve told you so long ago.”

“You… you heard me last night,” Lena wondered softly. “So you went to the Fortress?”

Kara grimaced. “I stopped listening as soon as I realized,” Kara said, fighting a panic. Will she be angry? “I never meant to- to invade your privacy. I’ll be more careful.”

“The door isn’t closed,” Lena said. “If you don’t want it to be.”

Those words made Kara brave enough - or maybe just confused enough - to finally tilt her head up to meet Lena’s gaze. “But- last night-”

“I’ve been trying to get over you. Not very successfully,” Lena added, with a wry grin.

“Really?” Kara smiled.

“Really.” 

---

The matching laws had been long dismantled by the time a smiling Alura officiated their marital rites. Kryptonians didn’t have concepts like best man or matron of honor, but that didn’t stop the two women from inviting Alex and Kelly to stand at each of their sides as they said their vows.

Kara never imagined that it’d be a woman’s wrist she’d place her wedding bracelet on. Though she supposed she never imagined marrying on an asteroid of her father’s creation, or marrying for romantic love, or marrying someone her people would call Hero of Argo for the creation of a black rock.

She never imagined finally telling Lena her secret. She never imagined Lena’s forgiveness. She never imagined the feel of Lena’s lips pressed against her own, hands tugging at her robes, as she whispered zhao against Lena’s lips.

And she never imagined being the one to make Lena’s heart race.

[boycritter]

is this function bisexual i mean bijective

[lipshits-continuous]

A Short Intro to Category Theory

A common theme in mathematics is to study certain objects and the maps between that preserve the specific structure of said objects. For example, linear algebra is the study of vector spaces and linear maps. Often we have that the identity maps are structure preserving and the composition of maps is also structure preserving. In the case of vector spaces, the identity map is a linear map and the composition of linear maps is again a linear map. Category theory generalises and axiomatises this common way of studying mathematical objects.

I'll introduce the notion of a category as well as the notion of a functor, which is another very important and ubiquitous notion in category theory. And I will finish with a very powerful result involving functors and isomorphisms!

Definition 1:

We call the last property the associative property.

Here are some examples:

Examples 2:

Note that whilst all of these examples are built from sets and set functions, we can have other kinds of objects and morphisms. However the most common categories are those built from Set.

Functors:

In the spirit of category theory being the study of objects and their morphisms, we want to define some kind of map between categories. It turns out that these are very powerful and show up everywhere in pure maths. Naturally, we want a functor to map objects to objects and morphisms to morphisms in a way that respects identity morphisms and our associative property.

Definition 3:

Example 4:

For those familiar with a some topology, the fundamental group is another exmaple of a covariant functor from the category of based spaces and based maps to Grp.

We also have another kind of functor:

Defintion 5:

It may seem a bit odd to introduce at first since all we've done is swap the directions of the morphisms, but it turns out that contravariant functors show up a lot!

This example requires a little bit of knowledge of linear algebra.

Example 6:

In fact, this is somewhat related to example 5! We can produce a contravariant functor Mor(-,X) is a similar way. For V and W vector spaces over k, we have that Mor(V,W) is a vector space over k. In particular, V*=Mor(V,k). So really this -* functor is just Mor(-,k)!

Isomorphisms

Here we generalise the familiar notion of isomorphisms of any algebraic structure!

Definition 7:

For a category of algebraic objects like Vectₖ, isomorphism are exactly the same as isomorphisms defined the typical way. In Top the isomorphisms are homeomorphisms. In Set the isomorphisms are bijective maps.

Remark:

So if f is invertible, we call it's right (or left) inverse, g, the inverse of f.

Isomorphisms give us a way to say when two objects of a category are "the same". More formally, being isomorphic defines an equivalence relation.

Lemma 8:

A natural question one might as is how do functors interact with isomorphisms? The answer is the very important result I hinted at in the intro!

Theorem 9:

Remark: In general, the converse is not true. That is F(X) isomorphic to F(Y) does not imply X is isomorphic to Y. An example of this is the fundamental groups of both S² and ℝ² are isomorphic to the trivial group but these spaces are not homeomorphic.

Taking the negation of Theorem 9 gives us a very powerful result:

Corollary 10:

This means that if we can find a functor such that F(X) and F(Y) aren't isomorphic, we know that X and Y are not isomorphic. This is of particular importance in algebraic topology where we construct functors from Top or hTop to a category of a given algebraic structure. This gives us some very powerful topological invariants for telling when two spaces aren't homeomorphic or homotopy equivalent. (In fact, this is where category theory originated from!)

[mathhombre]

Are there more big dots or small dots?

Pluie, 1995

Yayoi Kusama

[leibnizstan]

I had a dream last night where I was really scared about voter fraud for the sole reason that if someone votes twice then it’s not a bijection anymore. Dream me was frantic. In shambles.

[bubbloquacious]

Okay so to get the additive group of integers we just take the free (abelian) group on one generator. Perfectly natural. But given this group, how do we get the multiplication operation that makes it into the ring of integers, without just defining it to be what we already know the answer should be? Actually, we can leverage the fact that the underlying group is free on one generator.

So if you have two abelian groups A,B, then the set of group homorphisms A -> B can be equipped with the structure of an abelian group. If the values of homorphisms f and g at a group element a are f(a) and g(a), then the value of f + g at a is f(a) + g(a). Note that for this sum function to be a homomorphism in general, you do need B to be abelian. This abelian group structure is natural in the sense that Hom(A ⊗ B,C) is isomorphic in a natural way to Hom(A,Hom(B,C)) for all abelian groups A,B,C, where ⊗ denotes the tensor product of abelian groups. In jargon, this says that these constructions make the category of abelian groups into a monoidal closed category.

In particular, the set End(A) = Hom(A,A) of endomorphisms of A is itself an abelian group. What's more, we get an entirely new operation on End(A) for free: function composition! For f,g: A -> A, define f ∘ g to map a onto f(g(a)). Because the elements of End(A) are group homorphisms, we can derive a few identities that relate its addition to composition. If f,g,h are endomorphisms, then for all a in A we have [f ∘ (g + h)](a) = f(g(a) + h(a)) = f(g(a)) + f(h(a)) = [(f ∘ g) + (f ∘ h)](a), so f ∘ (g + h) = (f ∘ g) + (f ∘ h). In other words, composition distributes over addition on the left. We can similarly show that it distributes on the right. Because composition is associative and the identity function A -> A is always a homomorphism, we find that we have equipped End(A) with the structure of a unital ring.

Here's the punchline: because ℤ is the free group on one generator, a group homomorphism out of ℤ is completely determined by where it maps the generator 1, and every choice of image of 1 gives you a homomorphism. This means that we can identify the elements of ℤ with those of End(ℤ) bijectively; a non-negative number n corresponds to the endomorphism [n]: ℤ -> ℤ that maps k onto k added to itself n times, and a negative number n gives the endomorphism [n] that maps k onto -k added together -n times. Going from endomorphisms to integers is even simpler: evaluate the endomorphism at 1. Note that because (f + g)(1) = f(1) + g(1), this bijection is actually an isomorphism of abelian groups

This means that we can transfer the multiplication (i.e. composition) on End(ℤ) to ℤ. What's this ring structure on ℤ? Well if you have the endomorphism that maps 1 onto 2, and you then compose it with the one that maps 1 onto 3, then the resulting endomorphism maps 1 onto 2 added together 3 times, which among other names is known as 6. The multiplication is exactly the standard multiplication on ℤ!

A lot of things had to line up for this to work. For instance, the pointwise sum of endomorphisms needs to be itself an endomorphism. This is why we can't play the same game again; the free commutative ring on one generator is the integer polynomial ring ℤ[X], and indeed the set of ring endomorphisms ℤ[X] -> ℤ[X] correspond naturally to elements of ℤ[X], but because the pointwise product of ring endomorphisms does not generally respect addition, the pointwise operations do not equip End(ℤ[X]) with a ring structure (and in fact, no ring structure on Hom(R,S) can make the category of commutative rings monoidal closed for the tensor product of rings (this is because the monoidal unit is initial)). We can relax the rules slightly, though.

Who says we need the multiplication (or addition, for that matter) on End(ℤ[X])? We still have the bijection ℤ[X] ↔ End(ℤ[X]), so we can just give ℤ[X] the composition operation by transfering along the correspondence anyway. If p and q are polynomials in ℤ[X], then p ∘ q is the polynomial you get by substituting q for every instance of X in p. By construction, this satisfies (p + q) ∘ r = (p ∘ r) + (q ∘ r) and (p × q) ∘ r = (p ∘ r) × (q ∘ r), but we no longer have left-distributivity. Furthermore, composition is associative and the monomial X serves as its unit element. The resulting structure is an example of a composition ring!

The composition rings, like the commutative unital rings, and the abelian groups, form an equational class of algebraic structures, so they too have free objects. For sanity's sake, let's restrict ourselves to composition rings whose multiplication is commutative and unital, and whose composition is unital as well. Let C be the free composition ring with these restrictions on one generator. The elements of this ring will look like polynomials with integers coefficients, but with expressions in terms of X and a new indeterminate g (thought of as an 'unexpandable' polynomial), with various possible arrangements of multiplication, summation, and composition. It's a weird complicated object!

But again, the set of composition ring endomorphisms C -> C (that is, ring endomorphisms which respect composition) will have a bijective correspondence with elements of C, and we can transfer the composition operation to C. This gets us a fourth operation on C, which is associative with unit element g, and which distributes on the right over addition, multiplication, and composition.

This continues: every time you have a new equational class of algebraic structures with two extra operations (one binary operation for the new composition and one constant, i.e. a nullary operation, for the new unit element), and a new distributivity identity for every previous operation, as well as a unit identity and an associativity identity. We thus have an increasing countably infinite tower of algebraic structures.

Actually, taking the union of all of these equational classes still gives you an equational class, with countably infinitely many operations. This too has a free object on one generator, which has an endomorphism algebra, which is an object of a larger equational class of algebras, and so on. In this way, starting from any equational class, we construct a transfinite tower of algebraic structures indexed by the ordinal numbers with a truly senseless amount of associative unital operations, each of which distributes on the right over every previous operation.

[svndaysaweek]

e^(iπ)+1=0 — {Feat. Minnie}

1.3k words

A/N: I've been studying math so hard that I came up with this... I don't know if all readers can understand this, so I recommend googling the inclined terms(which are mathematical ones). Also know that e^(iπ)+1=0 is the most beautiful formula in mathematics,,

p.s: Why is eˣ okay but e^(iπ)+1=0 not? Tumblr mysteries..

Tags: Math(?), Choking, Creampie, Loving Sex

******

Minnie is like the eˣ function to you. The way her consistency fills up your heart—eˣ differentiated is still e��—, her always positive character—its domain is always positive number—, her out-of-this-world beauty—e^(iπ)+1=0—, and–

The way her back forms an exponential arc in front of you.

"F-fuck,"

On her knees Minnie is taking your pounding, with her entire body. Her back arcs upward to meet you face to face, as she turns her head back to you and into a blissful kiss.

You draw the sin x graph with your tongue inside her mouth. Up and down, up and down, corresponding to her tongue that draws a cos x graph, intertwined with yours completely—but following the exact same period of 2π.

You push her down to the mattress, and now you can see the hyperbolic curve of her waist. You grab on the narrow valley of it and continue pounding.

What you are doing to Minnie is distorting the perfectness of her body, although it's your way of worshiping it. The circular dark orbs in her elliptical eyes disappear into her head.

You choke her from behind. The amount of air getting in her lungs converges to zero, while the pleasure diverges to infinity.

Her fingers dig into the mattress as you pace up. You feel her walls pulse like sin x. Instant grips and loosenings of her pussy indicate that she is now close. You contribute the last drop of your patience for her orgasm—the maximum, when the derivative of her pleasure hits the X axis.

"I'm cummi–"

Minnie cums with a scream silenced by your grip around her neck. You for a moment regret restraining her voice, but no. You can just make her cum again, again, and again.

You flip her around to face you and resume the race. The race of which finish line you all are aware of obviously.

You kiss her again, this time drawing infinity with your tongue. It of course doesn't mean that it'll last forever, but you feel like it lasts as long as eternity itself. Rather, it would mean the neverending high you two are sharing right now.

Integral. Integrity. You and Minnie in bed together can't be counted as two. Undivided pleasure travels your connected, shared body. No boundaries are found between you and Minnie as you are pulled into a tight hug by her. You can feel how smooth her skin is, how soft her breasts are like it's your body—forget about the breasts, maybe.

Actually, it sounds quite right because it is your body—it's been so long since you lost count how many times you two told each other "You're mine,". Minnie moaning beneath you is yours. Yours to savor, yours to please, yours only to love.

Yes, yours only, and vice versa. You two are bijective functions. Each of your factors matches each of hers, without duplicity.

Again, you're hers only. Minnie is here, taking your cock to define you, to differentiate you, to integrate you. Minnie is moaning underneath your body, to be your proof, to be your solution, to be your answer.

"Are you close, babe?"

You were just about to say that you were, but you are so predictable a problem—Minnie knows the exact formulae to use when dealing with you.

"I am,"

Go ahead, she eyes you. You crook your neck to nibble on her ear and whisper,

"I fucking love you, Minnie,"

The moment you cum inside her, you are sent out of this world. The real world means nothing to you. Imaginary sensations feel more real than the real ones. Like you're feeling the i itself—the imaginary number.

1×1=1. You and Minnie just can't get separated. When it comes to you two, it's not the concept of addition or subtraction. You two love each other so it's 1×1=1. When you two are away from each other, you're still one, because 1÷1=1. Being 2 means you and Minnie are two 1s, which makes you two different individuals—right now, and of course always, you'd disprove that proudly.

You already came about half a minute ago, but you don't pull out. You see Minnie's heaving back and it's perfectly symmetrical—her erector muscles being the perfect axis for it.

It's an even function —f(x)=f(-x)— that you get visual, psychological satisfaction from. That way you could split her perfectly into two halves.

Your hands softly rub on her back. She still is in the middle of recovery, as you can feel her bumping heartbeats on your palm.

"Y-you feel so warm, baby,"

Minnie slightly lifts her limp head from the bed and murmurs with such a low tone. You then pull out and lie down next to her. Turning your body to her side you brush her hair behind her ears. She looks at you and gives you a satisfied, satisfying smile that could literally melt anything, everything.

Minnie's hand comes up to touch your face and–

It hits your still hard dick and she looks at it.

"Wanna go for another round?"

Minnie asks you, sitting up and getting on your thighs. It's a question with only one answer, it's another function that defines you two—a constant function, no matter what she says your answer is yes, undeniably.

Minnie grabs it, strokes it softly, and you feel your lust being recharged. Seems like she doesn't even look for your answer, anyway.

"Holy–"

You're inside her. Right after insertion Minnie starts to ride you out fast. You can only gasp at the feeling of her already-fucked inside, which is what you do every fucking time. Minnie kneeling, each of her knees are next to your both sides as she waves her hip and waist on you.

"Ah, fuck… You can last longer, right?"

She again asks you a question, locking fingers with you. Just like before, your answer is undoubtedly yes. Minnie then brings your hands to her lower waist and leans down forward, completely relying her weight on you. She lets your head into her embrace and whispers into your ear.

"Warm me up again, baby."

Your hands go down to grab on her hips and you begin fucking her upward with pace. As soon as your thrusts start to fuck your previous cum deeper into her womanhood, she moans beautifully into your ear.

Her arms tighten around your head, but the only tightness you can recognize is that of what's around your cock. She nibbles on your ear. Hot breaths tickle your ear, and her teeth on it motivates you to go even faster, rougher, and harsher. That's what loving sex is to her, and of course, to you too.

Your right hand detaches from her ass and gets on the back of her head. Minnie then lifts her head slightly up to fall into a dirty kiss. This time, you review the whole session beforehand.

You are drawing sin x, cos x, and infinity randomly with your tongue. Her back begins to arc exponentially, so you strengthen your hand on her head to keep her body tangent to yours. Your fingers on her ass rubs on her another hole, to make her pleasure diverge to infinity.

And everything you're doing earns her uncontrolled scream of ecstasy, as if she's trying to make an auditory definition of orgasm.

She again closes in to lock lips with you, and soon she cums. On your still-moving cock she cums hard, and in your mouth she lets the orgasmic sound out. It travels through your body fast, reverberates in your skull, sending you over the edge in no time.

You push deep into her for the last time before violently cumming inside her again.

It's explosive, you would term it. Minnie's entire body reacts to every spurt you shoot inside her.

"Holy fuck…"

You let out a sigh of words.

"Baby, that was…"

On top of her lungs she says, only to pause for breathing.

That was awesome, yeah. You know that, because you feel just like her too.

You look into her eyes, and find the excellence itself, the perfect, absolute beauty—e^(iπ)+1=0 .

[proof-by-intimidation]

CGWH spaces

As I was reading May's book and was about to enter the realms of homotopy theory, I was stopped by a chapter on point-set topology. I didn't entirely understand the point of this detour but the author insisted that one would truly appreciate it later. But, my curiosity didn't let me be, so I read around it and here's what I found.

1. What is happening?

So, apparently, for various reasons, the category of topological spaces (Top) is not nice enough to work with in homotopy theory (I will try to give a couple of reasons). For a lot of results in homotopy theory, there are these weird topological spaces which don't satisfy them meaning we would have to impose additional conditions every time in the hypotheses.

Instead of doing that, topologists proposed considering a full subcategory of Top which is not too restrictive but also gets rid of these pathological spaces. So, what do we want this subcategory to satisfy?

2. Some Pathologies that arise

2.1 Product of CW complexes

CW complexes are the most important kind of spaces studied in homotopy theory. Essentially, any space can be reduced to a CW complex. I won't go into the construction of CW complexes here, but the issue with the product of CW complexes is that the natural CW decomposition (arising from the CW complex structure of the components) doesn't necessarily coincide with the categorical product of the CW complexes in Top.

So, one motivation could be to find a subcategory (which, of course, contains all CW complexes) where the CW product of CW complexes coincides with the product in that category.

2.2 Mapping spaces and adjoints

Another important class of spaces studied in homotopy theory is the set of continuous functions between two spaces X and Y (denoted as U(X,Y)). There is a conventional topology on this set (I don't know why it's the chosen one) called the compact-open topology. With this, we have a couple of important maps

which are not continuous for all spaces.

Moreover, we have the following adjoint map

which is a bijection for sure, but not necessarily a homeomorphism with our chosen topology on the mapping spaces.

With these motivations in mind, let's list down the axioms for the subcategory.

3. The Category of Spaces

After a long search and coordinated efforts, topologists came up with the category of compactly generated weak Hausdorff spaces which satisfies all the axioms and provides a suitable framework to do homotopy theory.

4. The category of Compactly generated spaces

When I read the first part of this chapter in May's book, which was a bunch of (seemingly arbitrary) topological definitions, I thought I was screwed. Am I supposed to go back to these definitions every time I work with 'spaces' now? Well, not really.

At the categorical level, it's quite elegant and makes your life so much easier. Sure, you might want to think about the topology once in a while (like showing every cofibration is a closed embedding) but a lot of meaningful work can be done using the categorical properties of these spaces (I hope so).

I am not going to intimidate you by listing down the definitions (although it is amazing to me how people came up with this shit) but I do think it is worthwhile to go through the technical constructions at least once. I found the blog post by Dongryul Kim (which I have linked in the end) served this purpose.

Notes : In May's book however, he considers the category of compactly generated spaces as his category of interest. This category is not closed under colimits (which is why one had to add the weak Hausdorff part) but he claims that all the colimits that he will be considering in his book are in CG spaces.

I also think in the axioms, we just want it to be closed under small limits and colimits. Not sure about this.

References

Peter May's A concise course in Algebraic Topology

Jeffery Strom's Modern Classical Homotopy Theory

Dongryul Kim's blog post : link