Definition df-termc 49442

Description: Definition of the proper class (termcnex 49545) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49455). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49484). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49489). See also dftermc3 49500 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49500, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49460.

Followed directly from the definition, terminal categories are thin (termcthin 49446). The opposite category of a terminal category is "almost" itself (oppctermco 49474). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49509).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49470) and terminal objects (dftermo4 49471). The universal properties provide an alternate proof of initoeu1 17979, termoeu1 17986, initoeu2 17984, and termoeu2 49209. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49485).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17655, 0thincg 49427, func0g 49066, 0funcg 49062, and initc 49068.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49441	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1539	. . . . . 6 class 𝑐
4		cbs 17185	. . . . . 6 class Base
5	3, 4	cfv 6513	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1539	. . . . . 6 class 𝑥
8	7	csn 4591	. . . . 5 class {𝑥}
9	5, 8	wceq 1540	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1779	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49386	. . 3 class ThinCat
12	10, 2, 11	crab 3408	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1540	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49443